Research ArticleFractal Video Coding Using Fast Normalized CovarianceBased Similarity Measure

Ravindra E Chaudhari and Sanjay B Dhok

Center for VLSI and Nanotechnology Department of Electronics Engineering Visvesvaraya National Institute of TechnologyNagpur 440010 India

Correspondence should be addressed to Ravindra E Chaudhari rec77rediffmailcom

Received 18 July 2016 Revised 13 October 2016 Accepted 1 November 2016

Academic Editor Yakov Strelniker

Copyright copy 2016 R E Chaudhari and S B Dhok This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Fast normalized covariance based similarity measure for fractal video compression with quadtree partitioning is proposed in thispaper To increase the speed of fractal encoding a simplified expression of covariance between range and overlapped domainblocks within a search window is implemented in frequency domain All the covariance coefficients are normalized by usingstandard deviation of overlapped domain blocks and these are efficiently calculated in one computation by using two differentapproaches namely FFTbased and sum table based Results of these two approaches are compared and they are almost equal to eachother in all aspects except the memory requirement Based on proposed simplified similarity measure gray level transformationparameters are computationally modified and isometry transformations are performed using rotationreflection properties ofIFFT Quadtree decompositions are used for the partitions of larger size of range block that is 16 times 16 which is based on targetlevel of motion compensated prediction error Experimental result shows that proposed method can increase the encoding speedand compression ratio by 6649 and 958 respectively as compared to NHEXS method with increase in PSNR by 041 dBCompared to H264 proposed method can save 20 of compression time with marginal variation in PSNR and compressionratio

1 Introduction

The emerging multimedia applications such as video confer-encing video over mobile phones video email and wirelesscommunications require an effective video coding standardto achieve a low bit rate with good quality Performance ofvideo coding standard depends on parameters such as qualityof reconstructed video compression ratio and encodingtime Fractal based video compression [1] is an alternativeto accomplish high compression ratio with good qualityreconstructed output video than the existing video standards(MPEG H263 H264) Recently various researchers pro-posed different algorithms to improve the fractal encodingspeed

Jacquin [2] proposed an innovative technique which isbased on the fractal theory of iterated function system forimage compression It reduces an affine redundancy of animage by using its self-similarity properties Video sequences

contain temporal redundancies between consecutive framesthat can be easily removed by using fractal based techniqueFractal coding has received attention of researcher due to itsadvantages of independent resolution high decoding speedand high compression ratio [3 4] But high encoding timeis main drawback of fractal based method due to this it isnot useful for real time applications The motivation of thismethod is that it gives high compression ratio with goodquality output which is useful for storage and transmissionof bulky videos To increase the encoding speed with keepingmotivational parameters a fast fractal video coder system isproposed

Cube-based [5 6] and frame based [7] fractal compres-sion methods are used frequently for video compression Incube-based compression the video is divided into groupsof frames each of which in turn is partitioned into three-dimensional (3D) domain and range blocks however it hashigh computing complexity and low compression ratio In

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1725051 11 pageshttpdxdoiorg10115520161725051

2 Mathematical Problems in Engineering

frame based compression each frame is encoded using theprevious frame as a domain pool which introduces andspreads the error between the frames and it can be usedto obtain a high compression ratio Wang proposed a fixedblock size hybrid compression algorithm [8] and an adaptivepartition instead of fixed-size partition [9] which merges theadvantages of cube-based and frame based fractal compres-sion method Another hybrid coder scheme which combinesneighborhood vector quantization with fractal coding tocompress the video as a 3D volume was proposed by Yao andWilson [10] Fractal approach for 3D searchless [11] predic-tion of error frame for low bit rate video [12] and wavelettransform based video coding approach [13 14] are alsoconsidered for compression of videos

Circular prediction mapping (CPM) and noncontractiveinterframe mapping (NCIM) are proposed by Kim et al [15]to combine the fractal sequence coder with well-knownmotion estimationmotion compensation algorithm thatexploits the high temporal correlations between the framesFractal video coding using a new cross hexagon search(NHEXS) algorithm is proposed [16] for higher motion esti-mation speed for searching stationary and quasi-stationaryblocks The regions can be defined according to [17 18] apreviously computed segmentation map and are encodedindependently using NHEXS based searching technique Anew object-based method [19] is introduced in the transformdomain using shape-adaptive DCT for stereo video compres-sion Zhu et al proposed an automatic region-based videocoder [20] with asymmetrical hexagon searching algorithmand deblocking loop filter to improve decompression videoquality High efficiency fractal multiview codec is presentedin [21] to encode anchor viewpoint video using intrapre-diction modes and fractal coder with motion compensationtechnique Three-step search algorithm is modified in [22]using two cross search and two cross hexagon search patternsto implement fractal video coder

Block based motion estimation and motion compen-sation algorithms exploit the high temporal correlationsbetween the adjacent frames In frame based fractal videocoding range and domain blocks need to be matched withproper selection of geometrical transformation scaling andluminance factorsNormalized covariance that is ZeroMeanNormalizedCross Correlation (ZNCC) is amethod for deter-mining the structural similarity between two blocks from theimage [23]The best matched domain block having high nor-malized cross correlation [24 25] value may have large aver-age gray level difference This difference is reduced to zero orvery small value by selecting a proper fractal encoding para-meters But the direct computation of ZNCC for every rangeblock is computationally very expensive Sum table basedmethod significantly minimizes the computation complexityof ZNCC It is a precalculated running sum discrete structureof the entire image and acts as a look-up table for thecalculation of definite sum according to the size of block

In this paper a fast fractal based video coder is proposedusing the normalized covariance algorithm as a similaritymeasure It uses three levels of quadtree partition for motionestimation which provides good balance degree of variationto picture content and helps to improve the compression

ratio The complexity of covariance between range and alldomain blocks is simplified and implemented in one com-putation using FFT algorithm Computational complexityof scaling factor and brightness factor are also minimizedwith new simple expression based on normalized covarianceconcept instead of traditional mean square error (MSE) Thespeed of fractal encoding process is further increased byincorporating a few steps such as FFT based or sum tablebased method either one is used to perform the normal-ization of covariance component eight isometry transfor-mationsrsquo operation using 2D IFFT properties and the earlysearch termination technique Performance of video com-pression using FFT based and sum table based methods isseparately verified and they are nearly equal to each otherThese techniques can be used to improve the subjectivequality of video and coding efficiency

The rest of the paper is organized as follows The basicfractal block coding for the image is described in Section 2Normalized covariance based motion estimation and quad-tree partition are explained in Section 3 Fast fractal videocoding using FFT is presented in Section 4The experimentalresults and comparative study of the proposed algorithmwithexisting algorithms are presented in Section 5The conclusionis outlined in Section 6

2 Fractal Image Coding Theory

Fractal image coding is based on the theory of the partitionediterated function system (PIFS) [2] It consists of a set ofcontractive transformations when this transformation isapplied iteratively to an arbitrary image it will converge toan approximation of the original image Images are storedas a collection of transformations which will result in imagecompression

The original image of size119872 times119872 is initially partitionedinto nonoverlapping range blocks (119877119894) of each size119873times119873 (119894 =1 2 11987221198732) Similarly the same image is partitionedinto overlapping domain blocks (119863119895) of each size 2119873times 2119873 asa domain pool with one pixel shift in horizontal and verticaldirection (119895 = 1 2 (119872minus 2119873+ 1)2) For each range blocklocate the best matching domain block from the domain pooland then apply contractive mapping which minimizes theMSE between range and contractive domain block A range-domainmapping consists of three operations [3] sequentiallyon each domain block of size 2119873times2119873 (1) spatial contractionof the domain block (119863119895) by downsampling or averaging thefour neighboring pixels of disjoint group forming a block(119863119888119895) of size 119873 times 119873 (2) taking 8 geometrical transforma-tions of each blockwhich includes 4 rotations with 90 degreesand 4mirror reflections (3) for each geometrical transformedblock perform contractive affine transformation on the gray-scale values and select the parameters which give lowestMSEThe error between range (ℎ = 119877119894) and one of the domain(119892 = 119863119888119895) blocks is measured by equation (1) and scalingfactor ldquo119904rdquo and brightness factor ldquo119900rdquo of an affine transforma-tion are calculated by (2) and (3) respectively

Mathematical Problems in Engineering 3

MSE = 11198732119873minus1sum119894=0

119873minus1sum119895=0

(119904 sdot 119892 (119894 119895) + 119900 minus ℎ (119894 119895))2 (1)

119904 = 1198732 (sum119873minus1119894=0 sum119873minus1119895=0 ℎ (119894 119895) sdot 119892 (119894 119895)) minus (sum119873minus1119894=0 sum119873minus1119895=0 ℎ (119894 119895)) sdot (sum119873minus1119895=0 sum119873minus1119895=0 119892 (119894 119895))1198732sum119873minus1119894=0 sum119873minus1119895=0 1198922 (119894 119895) minus (sum119873minus1119894=0 sum119873minus1119895=0 119892 (119894 119895))2 (2)

119900 = 11198732 [[119873minus1sum119894=0

119873minus1sum119895=0

ℎ (119894 119895) minus 119904119873minus1sum119894=0

119873minus1sum119895=0

119892 (119894 119895)]] (3)

where 1198732 is the number of pixels in the block and ℎ(119894 119895)119892(119894 119895) are the pixel values of range block and contractivedomain block at coordinates (119894 119895) For each range block theparameters which need to be stored as a fractal encoded dataare the coordinates of domain block along with 119904 and 119900 andgeometric transformation index Gray level transformationparameters ldquo119904rdquo and ldquo119900rdquo should be in the range of minus12 to12 and minus255 to 255 respectively [3] to make sure that thetransformation is contractive At the decoder these fractalparameters are iteratively applied to an arbitrary initial imageaccording to the encoding block size which will finallyconverge to a reconstruction of the original image aftercertain number of iterations

3 Normalized Covariance BasedMotion Estimation

The ZNCC is a recognized similarity measure criterion andis considered as one of the accurate motion estimators in

video compression In fractal video coding the best matcheddomain block is decided after applying the proper affinetransformation which gives least mean square error TheZNCC method is more robust under uniform illuminationchanges and less sensitive to noise hence it can help toincrease the coding efficiency and improve subjective visualquality of output video For identical regions it may give ahigh NCC value for the best match but with a large averagegray level difference This difference is minimized by select-ing the accurate luminance and geometrical transformationparameters These fractal parameters are interpreted as akind ofmotion compensation technique due to unavailabilityof error frame In discrete domain the range block (ℎ119905)of current frame is shifted pixel by pixel across the searchwindow (119892119905minus1) of the reference frame The estimation ofmotion relies on the detection of maximum ZNCC functionbetween ℎ119905 and 119892119905minus1 The ZNCC is defined as

119862 (119909 119910) = sum119873minus1119894=0 sum119873minus1119895=0 (ℎ119905 (119894 119895) minus ℎ1199050) sdot (119892119905minus1 (119909 + 119894 119910 + 119895) minus 119892119905minus10(119909119910))radicsum119873minus1119894=0 sum119873minus1119895=0 (119892119905minus1 (119909 + 119894 119910 + 119895) minus 119892119905minus10(119909119910))2 sdot radicsum119873minus1119894=0 sum119873minus1119895=0 (ℎ119905 (119894 119895) minus ℎ1199050)2

(0 le 119909 119910 le 2119881) (4)

In (4) 119892119905minus10(119909119910) denotes the mean value of 119892119905minus1 withinthe area of the range block ℎ119905 shifted to (119909 119910) position andsimilarly ℎ1199050 denote the mean value of the range block ℎ119905119862(119909 119910) represents the ZNCC surface matrix between thecurrent macroblock and reference search window If thesearch range is plusmn119881 pixels in both directions from the samelocation of ℎ119905 then the size of the 119862(119909 119910) will be (2119881 +1 2119881 + 1) Fractal encoding is itself complex method anduse of ZNCC function to estimate motion vector can makethe combined technique computationally expensive and timeconsuming To overcome these complexity problems anefficient method of ZNCC calculation has been proposedwhich is based on new optimal 119904 and 119900 transformationparameters

31 Quadtree Based Partition Quadtree decompositionmethod initially partitions the current frame into a set of

larger size (16 times 16 pixels) range blocks ℎ119905 at level-1 (119871 = 1)Motion vector of the matched block is decided after verifyingthe highest three peak locations of ZNCC surface matrixThis verification is required because after quantizing graylevel transformation parameters the error between the blocksmay vary If the lowest quantizing error of larger size block(ℎ119905) is above the prespecified threshold then the block ℎ119905is partitioned in four quadrants ℎ119905119894 119894 = 1 4 Thepartitioning scheme [26] can be recursively continued untilthe error becomes smaller than the threshold or 4 times 4 pixelsblock size (119871 = 3) is reached

Here 3 levels of quadtree partitioning are employed withblock sizes from 16 times 16 to 4 times 4 pixels All subsequentpartitions of 16 times 16 block size are represented by one codeword as shown in Figure 1 The length of code word can beeither 1 bit or 5 bits it depends on only level-1 and level-2nodes If the level-1 block is not partitioned then the code

4 Mathematical Problems in Engineering

Code word 10010

R1

R13R12 R14R11

R133R132 R134R131

Level-1

Level-2

Level-3

1

0 00 18 times 8

4 times 4

16 times 16

Figure 1 Quadtree structure with code word

word is 1 bit otherwise it is assigned 5 bits Similarly if thecorresponding block is partitioned then it is represented bybit 1 otherwise it is represented by bit 0 as shown in Figure 1

4 Fast Fractal Video Coding Algorithm

The most important factor that affects the speed of fractalencoding is the searching ofmatched domain block In fractalvideo compression searching process is limited to the sizeof search window but it is also expensive in comparisonwith standard video coder (H264MPEG4) We proposeda FFT based ZNCC method with new matching criteriawhich reduced the block searching time Current frame (119891119905)is initially partitioned into nonoverlapped range blocks ldquoℎ =ℎ119905rdquo of size 119873 times 119873 pixels The search window ldquo119892 = 119892119905minus1rdquoof size (119873 + 16) times (119873 + 16) is defined on the previouslydecoded frame (119891119905minus1) that is reference frame of size1198721times1198722The flow diagram of the proposed fast fractal video coderis shown in Figure 2(a) The ZNCC matching process mayfind the wrong matched domain block if the range block ishomogeneous If the variance of any range block is below thesmallest predefined threshold then only mean value of thathomogeneous block needs to be encoded otherwise blockbelongs to nonhomogeneous group The root mean square(RMS) measure is used to find the prediction error (119864119898)between range (ℎ) and matched domain (119892119909119910) block Allthe fractal parameters along with corresponding error (119864119898)are the output of fast fractal searching algorithm as shownin Figure 2(b) The given block ldquoℎrdquo is partitioned into fourquadrants when 119864119898 is above the specified threshold (119905ℎ119871)otherwise encode and save the fractal parameters In thispaper depth of quadtree is three levels that is 119871max = 3 sotwo error thresholds are specified that is 119905ℎ1 and 119905ℎ2

Due to the high temporal correlation in video sequencesrange-domain mapping becomes more effective if the sizesof range and domain block are the same [15] The quality ofreference frame plays an important role while mapping twosame size blocks If interframe motion vector prediction is

based on good quality reference frame (previous frame) thenthe prediction error will be low But for subsequent framesthis errormonotonically increases due to cumulative processSo to get a better quality reference frame at the beginningintraframe is compressed using DCT transformation andquantization technique [27 28]

41 Efficient Searching with Simplified Similarity MeasureThe high computational complexity is the main drawback ofZNCC similarity measure method in spatial domain There-fore (4) is simplified to minimize the complexity Numeratorcomponent of (4) is represented as 119899(119909 119910) and is rewritten asfollows

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)

minus 1198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

ℎ1 (119894 119895) (5)

where ℎ1(119894 119895) = (ℎ(119894 119895) minus ℎ0) it has zero mean and becausethe sum of ℎ1(119894 119895) is zero the term 1198920119909119910sum119873minus1119894=0 sum119873minus1119895=0 ℎ1(119894 119895)will also be zero and (5) can be written as

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)= ⟨119892119909119910 ℎ1⟩

(6)

Equation (6) is independent of the mean (1198920119909119910) value ofdomain block All the domain blocks are overlapped and1198920119909119910value of each block is different 119899(119909 119910) is the cross correlationof ℎ1 and 119892119909119910 block where 119909 and119910 are the location of domainblock 119892119909119910 It is equivalent to the inner product operation oftwo bocks represented by ⟨sdot sdot⟩ operatorThis cross correlationis equal to the complex conjugate multiplications of twofrequency domain components The complex conjugate partcan be avoided by taking the inverse FFT (IFFT) of one of theinput instead of FFT because conj(FFT(ℎ1)) = (119873 + 16)2 timesIFFT(ℎ1) where (119873 + 16)2 is a constant term In frequencydomain size of both the input blocks must be equal The sizeof block ℎ1 is increased to the size of 119892 by padding zeros onthe left and down the side of the block The calculation of (6)for all the blocks can be computed in one computation usingFFT as given below

119899 = IFFT [FFT (119892) sdot IFFT (ℎ1)] = IFFT [119866 sdot 1198671] (7)

Similarly the denominator component of (4) is the product ofstandard deviation of 119892119909119910 and ℎ blocks Due to cancellationof (11198732) factor of standard deviation with numerator termit turns into 1198712 norm as (8) and (9) The norm of 1198921119909119910 (8)is also expensive because it is repeated for each overlapped

Mathematical Problems in Engineering 5

Range block ldquohrdquofrom current frame

Homogeneousdetection

Fast fractal searchingalgorithm

Search window ldquogrdquofrom ref frame

Encode and storefractal parameters

All subblockof ldquohrdquo processed

StopYes

Yes

No

No

No

Yes

Start

Partition ldquohrdquo into 4quadrant (L = L + 1)

orEm le thL

L = Lmax

(a)

FFT of searchwindow (g)

tth isometryusing DFTproperties

Computation of ZNCCcoefficients matrix ldquoCrdquo

E

No

Yes

No

FFT or sum table

mean andstd deviation of ldquogrdquo

IFFT of block

padding(h1) with zero

t = 0

Norm (g1)

g0x119898y119898

If t lt 8Next isometryindex t = t + 1

Em = E and tm = t

Fractal parametersxm ym sm om tm Em

Quantised s and o based onZNCC peak locationg(xm ym) block

If (E lt Em) or(E le th)

E le th

E lt Em

(b)

Figure 2 Flow diagrams of (a) fast fractal video coder and (b) efficient searching algorithm

domain block whereas the norm of ℎ1 (9) is unique for alldomain blocks

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(119892 (119909 + 119894 119910 + 119895) minus 1198920119909119910)2]]12

where 1198921119909119910 = 119892119909119910 minus 1198920119909119910

(8)

1003817100381710038171003817ℎ11003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(ℎ (119894 119895) minus ℎ0)2]]12

where ℎ1 = ℎ minus ℎ0

(9)

The norm of 1198921119909119910 expression is simplified and is written as(10) in the following

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 21198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)

+ 119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910]]12

(10)

119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910 = 1198732( 11198732119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2

(11)

Equation (10) can be simplified using (11) to

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 11198732 (119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2]]12

(12)

This can be represented in terms of mean value of domainblock as10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817

= [[(119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)) minus 119873211989220119909119910]]12

(13)

All the overlapped domain blocks which are shifted by onepixel also require similar computations and mostly these willbe repeated from the previous blocks Two different methodsare proposed to prevent unnecessary computations of themean and norm of all overlapped blocks that is FFT based

6 Mathematical Problems in Engineering

FFT

Square FFT

IFFT

IFFT

Square

Squareroot

Inputldquogrdquo

++

minustimes

times

with zeropadding

IFFT of INtimesN

1(N2)

(N2)

1ldquog rdquo

0ldquog rdquo

Figure 3 FFT based mean and norm calculation of all domainblocks

and sum table based The ZNCC (4) can be rewritten using(7) (8) (9) and (13) to (14)

119862 (119909 119910) = 119899 (119909 119910)10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 sdot 1003817100381710038171003817ℎ11003817100381710038171003817 (14)

The flow diagram of fast efficient searching algorithm withfast computation of ZNCC (14) is shown in Figure 2(b) Invideo sequences more number of the blocks can be observedas stationary blocks with zerononzero motions Such blockscan be identified when the highest value on ZNCC surfaceis nearly equal to one (119862(119909119898 119910119898) asymp 1) and correspondingquantized value of 119904 and 119900 is one and zero respectively Afterfinding the motion vector (119909119898 119910119898) of the stationary domainblock searching process can be terminated if the RMS error(119864) is below the threshold (119905ℎ) with isometry index 119905 = 0 Dueto early termination of searching process coding efficiencyis increased with unchanged quality The norm of ℎ1 can beneglected from (14) because it is constant and divisional termfor all domain blocks and it does not change the final ZNCCresult

411 FFT Based Method for Mean and Norm Calculation ofOverlapped Blocks The sum of squared pixels and the sumof pixels according to the size of range block are similar tothe convolution with unit (119868) matrix of same size FFT andIFFT combinations are used for fast computation The flowdiagram of the mean and norm calculation of all the domainblocks in one computation as matrices using FFT is shown inFigure 3 FFT of input 119892 is readily available from (7) and theIFFT of unit (119868)matrix with zero padding is constant only onetime computation So only FFT of squared pixels and last twoIFFT are required along with others blocks The mean (1198920)and norm (1198921) of all the blocks are required during gray leveltransformation and normalized covariance calculations

412 Sum Table Based Method for Mean and Norm Calcu-lation of Overlapped Blocks The sum table based methodalso can be used to reduce the number of computationsrequired to compute themean andnormof all domain blocksThe precomputation of two sum tables st(119896 119897) and st2(119896 119897)over the previous frame 119891119905minus1(119896 119897) and squared pixel 1198912119905minus1(119896 119897)frame respectively acts as look-up tables These tables are

recursively constructed for each frame before the beginningof encoding process defined by

st (119896 119897) = 119891119905minus1 (119896 119897) + st (119896 minus 1 119897) + st (119896 119897 minus 1)minus st (119896 minus 1 119897 minus 1)

st2 (119896 119897) = 1198912119905minus1 (119896 119897) + st2 (119896 minus 1 119897) + st2 (119896 119897 minus 1)minus st2 (119896 minus 1 119897 minus 1)

(15)

where 119896 and 119897 are the pixel coordinates of frame 119891119905minus1 with 119896 =0 1 1198721 minus 1 and 119897 = 0 1 1198722 minus 1 So initial conditionst(119896 119897) = st2(119896 119897) = 0 when either 119896 or 119897 = minus1 These two sumtables are partitioned according to the size of search windowto 1199041 and 11990421 subtables of size (119873 + 17 times 119873 + 17) It consistsof one additional row and column at the initial position incomparison with search window size The sum expressionsin (12) over 119892(119909+119894 119910+119895) and 1198922(119909+119894 119910+119895) can be calculatedefficiently using subtables

119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)= 1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 1199041 (119909 minus 1 119910 minus 1)minus 1199041 (119909 minus 1 119910 + 119873 minus 1) minus 1199041 (119909 + 119873 minus 1 119910 minus 1)

(16)

119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)= 11990421 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 11990421 (119909 minus 1 119910 minus 1)minus 11990421 (119909 minus 1 119910 + 119873 minus 1) minus 11990421 (119909 + 119873 minus 1 119910 minus 1)

(17)

Using (16) and (17) the norm (12) and the mean (18) of alldomain blocks can be calculated as matrices of size 17 times 17119909 isin 0 1 16 and 119910 isin 0 1 16

1198920119909119910 = 11198732 (1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1)+ 1199041 (119909 minus 1 119910 minus 1) minus 1199041 (119909 minus 1 119910 + 119873 minus 1)minus 1199041 (119909 + 119873 minus 1 119910 minus 1))

(18)

According to the three levelsrsquo quadtree partition it mayrequire total six memories to store these tables of each size1198721 times 1198722 But instead of six only two memory spaces aredefined according to the size of level three blocks and basedon that remaining two subtables are calculated

42 New Gray Level and Isometry Transformation The graylevel transformation parameters such as scaling factor ldquo119904rdquo andbrightness factor ldquo119900rdquo for each range block ℎ is based on certaindomain block 119892119909119910 which strongly matches (19) when MSE iszero

ℎ = 119904 sdot 119892119909119910 + 119900 (19)

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

frame based compression each frame is encoded using theprevious frame as a domain pool which introduces andspreads the error between the frames and it can be usedto obtain a high compression ratio Wang proposed a fixedblock size hybrid compression algorithm [8] and an adaptivepartition instead of fixed-size partition [9] which merges theadvantages of cube-based and frame based fractal compres-sion method Another hybrid coder scheme which combinesneighborhood vector quantization with fractal coding tocompress the video as a 3D volume was proposed by Yao andWilson [10] Fractal approach for 3D searchless [11] predic-tion of error frame for low bit rate video [12] and wavelettransform based video coding approach [13 14] are alsoconsidered for compression of videos

Circular prediction mapping (CPM) and noncontractiveinterframe mapping (NCIM) are proposed by Kim et al [15]to combine the fractal sequence coder with well-knownmotion estimationmotion compensation algorithm thatexploits the high temporal correlations between the framesFractal video coding using a new cross hexagon search(NHEXS) algorithm is proposed [16] for higher motion esti-mation speed for searching stationary and quasi-stationaryblocks The regions can be defined according to [17 18] apreviously computed segmentation map and are encodedindependently using NHEXS based searching technique Anew object-based method [19] is introduced in the transformdomain using shape-adaptive DCT for stereo video compres-sion Zhu et al proposed an automatic region-based videocoder [20] with asymmetrical hexagon searching algorithmand deblocking loop filter to improve decompression videoquality High efficiency fractal multiview codec is presentedin [21] to encode anchor viewpoint video using intrapre-diction modes and fractal coder with motion compensationtechnique Three-step search algorithm is modified in [22]using two cross search and two cross hexagon search patternsto implement fractal video coder

Block based motion estimation and motion compen-sation algorithms exploit the high temporal correlationsbetween the adjacent frames In frame based fractal videocoding range and domain blocks need to be matched withproper selection of geometrical transformation scaling andluminance factorsNormalized covariance that is ZeroMeanNormalizedCross Correlation (ZNCC) is amethod for deter-mining the structural similarity between two blocks from theimage [23]The best matched domain block having high nor-malized cross correlation [24 25] value may have large aver-age gray level difference This difference is reduced to zero orvery small value by selecting a proper fractal encoding para-meters But the direct computation of ZNCC for every rangeblock is computationally very expensive Sum table basedmethod significantly minimizes the computation complexityof ZNCC It is a precalculated running sum discrete structureof the entire image and acts as a look-up table for thecalculation of definite sum according to the size of block

In this paper a fast fractal based video coder is proposedusing the normalized covariance algorithm as a similaritymeasure It uses three levels of quadtree partition for motionestimation which provides good balance degree of variationto picture content and helps to improve the compression

ratio The complexity of covariance between range and alldomain blocks is simplified and implemented in one com-putation using FFT algorithm Computational complexityof scaling factor and brightness factor are also minimizedwith new simple expression based on normalized covarianceconcept instead of traditional mean square error (MSE) Thespeed of fractal encoding process is further increased byincorporating a few steps such as FFT based or sum tablebased method either one is used to perform the normal-ization of covariance component eight isometry transfor-mationsrsquo operation using 2D IFFT properties and the earlysearch termination technique Performance of video com-pression using FFT based and sum table based methods isseparately verified and they are nearly equal to each otherThese techniques can be used to improve the subjectivequality of video and coding efficiency

The rest of the paper is organized as follows The basicfractal block coding for the image is described in Section 2Normalized covariance based motion estimation and quad-tree partition are explained in Section 3 Fast fractal videocoding using FFT is presented in Section 4The experimentalresults and comparative study of the proposed algorithmwithexisting algorithms are presented in Section 5The conclusionis outlined in Section 6

2 Fractal Image Coding Theory

Fractal image coding is based on the theory of the partitionediterated function system (PIFS) [2] It consists of a set ofcontractive transformations when this transformation isapplied iteratively to an arbitrary image it will converge toan approximation of the original image Images are storedas a collection of transformations which will result in imagecompression

The original image of size119872 times119872 is initially partitionedinto nonoverlapping range blocks (119877119894) of each size119873times119873 (119894 =1 2 11987221198732) Similarly the same image is partitionedinto overlapping domain blocks (119863119895) of each size 2119873times 2119873 asa domain pool with one pixel shift in horizontal and verticaldirection (119895 = 1 2 (119872minus 2119873+ 1)2) For each range blocklocate the best matching domain block from the domain pooland then apply contractive mapping which minimizes theMSE between range and contractive domain block A range-domainmapping consists of three operations [3] sequentiallyon each domain block of size 2119873times2119873 (1) spatial contractionof the domain block (119863119895) by downsampling or averaging thefour neighboring pixels of disjoint group forming a block(119863119888119895) of size 119873 times 119873 (2) taking 8 geometrical transforma-tions of each blockwhich includes 4 rotations with 90 degreesand 4mirror reflections (3) for each geometrical transformedblock perform contractive affine transformation on the gray-scale values and select the parameters which give lowestMSEThe error between range (ℎ = 119877119894) and one of the domain(119892 = 119863119888119895) blocks is measured by equation (1) and scalingfactor ldquo119904rdquo and brightness factor ldquo119900rdquo of an affine transforma-tion are calculated by (2) and (3) respectively

Mathematical Problems in Engineering 3

MSE = 11198732119873minus1sum119894=0

119873minus1sum119895=0

(119904 sdot 119892 (119894 119895) + 119900 minus ℎ (119894 119895))2 (1)

119904 = 1198732 (sum119873minus1119894=0 sum119873minus1119895=0 ℎ (119894 119895) sdot 119892 (119894 119895)) minus (sum119873minus1119894=0 sum119873minus1119895=0 ℎ (119894 119895)) sdot (sum119873minus1119895=0 sum119873minus1119895=0 119892 (119894 119895))1198732sum119873minus1119894=0 sum119873minus1119895=0 1198922 (119894 119895) minus (sum119873minus1119894=0 sum119873minus1119895=0 119892 (119894 119895))2 (2)

119900 = 11198732 [[119873minus1sum119894=0

119873minus1sum119895=0

ℎ (119894 119895) minus 119904119873minus1sum119894=0

119873minus1sum119895=0

119892 (119894 119895)]] (3)

where 1198732 is the number of pixels in the block and ℎ(119894 119895)119892(119894 119895) are the pixel values of range block and contractivedomain block at coordinates (119894 119895) For each range block theparameters which need to be stored as a fractal encoded dataare the coordinates of domain block along with 119904 and 119900 andgeometric transformation index Gray level transformationparameters ldquo119904rdquo and ldquo119900rdquo should be in the range of minus12 to12 and minus255 to 255 respectively [3] to make sure that thetransformation is contractive At the decoder these fractalparameters are iteratively applied to an arbitrary initial imageaccording to the encoding block size which will finallyconverge to a reconstruction of the original image aftercertain number of iterations

3 Normalized Covariance BasedMotion Estimation

The ZNCC is a recognized similarity measure criterion andis considered as one of the accurate motion estimators in

video compression In fractal video coding the best matcheddomain block is decided after applying the proper affinetransformation which gives least mean square error TheZNCC method is more robust under uniform illuminationchanges and less sensitive to noise hence it can help toincrease the coding efficiency and improve subjective visualquality of output video For identical regions it may give ahigh NCC value for the best match but with a large averagegray level difference This difference is minimized by select-ing the accurate luminance and geometrical transformationparameters These fractal parameters are interpreted as akind ofmotion compensation technique due to unavailabilityof error frame In discrete domain the range block (ℎ119905)of current frame is shifted pixel by pixel across the searchwindow (119892119905minus1) of the reference frame The estimation ofmotion relies on the detection of maximum ZNCC functionbetween ℎ119905 and 119892119905minus1 The ZNCC is defined as

119862 (119909 119910) = sum119873minus1119894=0 sum119873minus1119895=0 (ℎ119905 (119894 119895) minus ℎ1199050) sdot (119892119905minus1 (119909 + 119894 119910 + 119895) minus 119892119905minus10(119909119910))radicsum119873minus1119894=0 sum119873minus1119895=0 (119892119905minus1 (119909 + 119894 119910 + 119895) minus 119892119905minus10(119909119910))2 sdot radicsum119873minus1119894=0 sum119873minus1119895=0 (ℎ119905 (119894 119895) minus ℎ1199050)2

(0 le 119909 119910 le 2119881) (4)

In (4) 119892119905minus10(119909119910) denotes the mean value of 119892119905minus1 withinthe area of the range block ℎ119905 shifted to (119909 119910) position andsimilarly ℎ1199050 denote the mean value of the range block ℎ119905119862(119909 119910) represents the ZNCC surface matrix between thecurrent macroblock and reference search window If thesearch range is plusmn119881 pixels in both directions from the samelocation of ℎ119905 then the size of the 119862(119909 119910) will be (2119881 +1 2119881 + 1) Fractal encoding is itself complex method anduse of ZNCC function to estimate motion vector can makethe combined technique computationally expensive and timeconsuming To overcome these complexity problems anefficient method of ZNCC calculation has been proposedwhich is based on new optimal 119904 and 119900 transformationparameters

31 Quadtree Based Partition Quadtree decompositionmethod initially partitions the current frame into a set of

larger size (16 times 16 pixels) range blocks ℎ119905 at level-1 (119871 = 1)Motion vector of the matched block is decided after verifyingthe highest three peak locations of ZNCC surface matrixThis verification is required because after quantizing graylevel transformation parameters the error between the blocksmay vary If the lowest quantizing error of larger size block(ℎ119905) is above the prespecified threshold then the block ℎ119905is partitioned in four quadrants ℎ119905119894 119894 = 1 4 Thepartitioning scheme [26] can be recursively continued untilthe error becomes smaller than the threshold or 4 times 4 pixelsblock size (119871 = 3) is reached

Here 3 levels of quadtree partitioning are employed withblock sizes from 16 times 16 to 4 times 4 pixels All subsequentpartitions of 16 times 16 block size are represented by one codeword as shown in Figure 1 The length of code word can beeither 1 bit or 5 bits it depends on only level-1 and level-2nodes If the level-1 block is not partitioned then the code

4 Mathematical Problems in Engineering

Code word 10010

R1

R13R12 R14R11

R133R132 R134R131

Level-1

Level-2

Level-3

1

0 00 18 times 8

4 times 4

16 times 16

Figure 1 Quadtree structure with code word

word is 1 bit otherwise it is assigned 5 bits Similarly if thecorresponding block is partitioned then it is represented bybit 1 otherwise it is represented by bit 0 as shown in Figure 1

4 Fast Fractal Video Coding Algorithm

The most important factor that affects the speed of fractalencoding is the searching ofmatched domain block In fractalvideo compression searching process is limited to the sizeof search window but it is also expensive in comparisonwith standard video coder (H264MPEG4) We proposeda FFT based ZNCC method with new matching criteriawhich reduced the block searching time Current frame (119891119905)is initially partitioned into nonoverlapped range blocks ldquoℎ =ℎ119905rdquo of size 119873 times 119873 pixels The search window ldquo119892 = 119892119905minus1rdquoof size (119873 + 16) times (119873 + 16) is defined on the previouslydecoded frame (119891119905minus1) that is reference frame of size1198721times1198722The flow diagram of the proposed fast fractal video coderis shown in Figure 2(a) The ZNCC matching process mayfind the wrong matched domain block if the range block ishomogeneous If the variance of any range block is below thesmallest predefined threshold then only mean value of thathomogeneous block needs to be encoded otherwise blockbelongs to nonhomogeneous group The root mean square(RMS) measure is used to find the prediction error (119864119898)between range (ℎ) and matched domain (119892119909119910) block Allthe fractal parameters along with corresponding error (119864119898)are the output of fast fractal searching algorithm as shownin Figure 2(b) The given block ldquoℎrdquo is partitioned into fourquadrants when 119864119898 is above the specified threshold (119905ℎ119871)otherwise encode and save the fractal parameters In thispaper depth of quadtree is three levels that is 119871max = 3 sotwo error thresholds are specified that is 119905ℎ1 and 119905ℎ2

Due to the high temporal correlation in video sequencesrange-domain mapping becomes more effective if the sizesof range and domain block are the same [15] The quality ofreference frame plays an important role while mapping twosame size blocks If interframe motion vector prediction is

based on good quality reference frame (previous frame) thenthe prediction error will be low But for subsequent framesthis errormonotonically increases due to cumulative processSo to get a better quality reference frame at the beginningintraframe is compressed using DCT transformation andquantization technique [27 28]

41 Efficient Searching with Simplified Similarity MeasureThe high computational complexity is the main drawback ofZNCC similarity measure method in spatial domain There-fore (4) is simplified to minimize the complexity Numeratorcomponent of (4) is represented as 119899(119909 119910) and is rewritten asfollows

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)

minus 1198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

ℎ1 (119894 119895) (5)

where ℎ1(119894 119895) = (ℎ(119894 119895) minus ℎ0) it has zero mean and becausethe sum of ℎ1(119894 119895) is zero the term 1198920119909119910sum119873minus1119894=0 sum119873minus1119895=0 ℎ1(119894 119895)will also be zero and (5) can be written as

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)= ⟨119892119909119910 ℎ1⟩

(6)

Equation (6) is independent of the mean (1198920119909119910) value ofdomain block All the domain blocks are overlapped and1198920119909119910value of each block is different 119899(119909 119910) is the cross correlationof ℎ1 and 119892119909119910 block where 119909 and119910 are the location of domainblock 119892119909119910 It is equivalent to the inner product operation oftwo bocks represented by ⟨sdot sdot⟩ operatorThis cross correlationis equal to the complex conjugate multiplications of twofrequency domain components The complex conjugate partcan be avoided by taking the inverse FFT (IFFT) of one of theinput instead of FFT because conj(FFT(ℎ1)) = (119873 + 16)2 timesIFFT(ℎ1) where (119873 + 16)2 is a constant term In frequencydomain size of both the input blocks must be equal The sizeof block ℎ1 is increased to the size of 119892 by padding zeros onthe left and down the side of the block The calculation of (6)for all the blocks can be computed in one computation usingFFT as given below

119899 = IFFT [FFT (119892) sdot IFFT (ℎ1)] = IFFT [119866 sdot 1198671] (7)

Similarly the denominator component of (4) is the product ofstandard deviation of 119892119909119910 and ℎ blocks Due to cancellationof (11198732) factor of standard deviation with numerator termit turns into 1198712 norm as (8) and (9) The norm of 1198921119909119910 (8)is also expensive because it is repeated for each overlapped

Mathematical Problems in Engineering 5

Range block ldquohrdquofrom current frame

Homogeneousdetection

Fast fractal searchingalgorithm

Search window ldquogrdquofrom ref frame

Encode and storefractal parameters

All subblockof ldquohrdquo processed

StopYes

Yes

No

No

No

Yes

Start

Partition ldquohrdquo into 4quadrant (L = L + 1)

orEm le thL

L = Lmax

(a)

FFT of searchwindow (g)

tth isometryusing DFTproperties

Computation of ZNCCcoefficients matrix ldquoCrdquo

E

No

Yes

No

FFT or sum table

mean andstd deviation of ldquogrdquo

IFFT of block

padding(h1) with zero

t = 0

Norm (g1)

g0x119898y119898

If t lt 8Next isometryindex t = t + 1

Em = E and tm = t

Fractal parametersxm ym sm om tm Em

Quantised s and o based onZNCC peak locationg(xm ym) block

If (E lt Em) or(E le th)

E le th

E lt Em

(b)

Figure 2 Flow diagrams of (a) fast fractal video coder and (b) efficient searching algorithm

domain block whereas the norm of ℎ1 (9) is unique for alldomain blocks

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(119892 (119909 + 119894 119910 + 119895) minus 1198920119909119910)2]]12

where 1198921119909119910 = 119892119909119910 minus 1198920119909119910

(8)

1003817100381710038171003817ℎ11003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(ℎ (119894 119895) minus ℎ0)2]]12

where ℎ1 = ℎ minus ℎ0

(9)

The norm of 1198921119909119910 expression is simplified and is written as(10) in the following

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 21198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)

+ 119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910]]12

(10)

119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910 = 1198732( 11198732119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2

(11)

Equation (10) can be simplified using (11) to

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 11198732 (119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2]]12

(12)

This can be represented in terms of mean value of domainblock as10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817

= [[(119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)) minus 119873211989220119909119910]]12

(13)

All the overlapped domain blocks which are shifted by onepixel also require similar computations and mostly these willbe repeated from the previous blocks Two different methodsare proposed to prevent unnecessary computations of themean and norm of all overlapped blocks that is FFT based

6 Mathematical Problems in Engineering

FFT

Square FFT

IFFT

IFFT

Square

Squareroot

Inputldquogrdquo

++

minustimes

times

with zeropadding

IFFT of INtimesN

1(N2)

(N2)

1ldquog rdquo

0ldquog rdquo

Figure 3 FFT based mean and norm calculation of all domainblocks

and sum table based The ZNCC (4) can be rewritten using(7) (8) (9) and (13) to (14)

119862 (119909 119910) = 119899 (119909 119910)10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 sdot 1003817100381710038171003817ℎ11003817100381710038171003817 (14)

The flow diagram of fast efficient searching algorithm withfast computation of ZNCC (14) is shown in Figure 2(b) Invideo sequences more number of the blocks can be observedas stationary blocks with zerononzero motions Such blockscan be identified when the highest value on ZNCC surfaceis nearly equal to one (119862(119909119898 119910119898) asymp 1) and correspondingquantized value of 119904 and 119900 is one and zero respectively Afterfinding the motion vector (119909119898 119910119898) of the stationary domainblock searching process can be terminated if the RMS error(119864) is below the threshold (119905ℎ) with isometry index 119905 = 0 Dueto early termination of searching process coding efficiencyis increased with unchanged quality The norm of ℎ1 can beneglected from (14) because it is constant and divisional termfor all domain blocks and it does not change the final ZNCCresult

411 FFT Based Method for Mean and Norm Calculation ofOverlapped Blocks The sum of squared pixels and the sumof pixels according to the size of range block are similar tothe convolution with unit (119868) matrix of same size FFT andIFFT combinations are used for fast computation The flowdiagram of the mean and norm calculation of all the domainblocks in one computation as matrices using FFT is shown inFigure 3 FFT of input 119892 is readily available from (7) and theIFFT of unit (119868)matrix with zero padding is constant only onetime computation So only FFT of squared pixels and last twoIFFT are required along with others blocks The mean (1198920)and norm (1198921) of all the blocks are required during gray leveltransformation and normalized covariance calculations

412 Sum Table Based Method for Mean and Norm Calcu-lation of Overlapped Blocks The sum table based methodalso can be used to reduce the number of computationsrequired to compute themean andnormof all domain blocksThe precomputation of two sum tables st(119896 119897) and st2(119896 119897)over the previous frame 119891119905minus1(119896 119897) and squared pixel 1198912119905minus1(119896 119897)frame respectively acts as look-up tables These tables are

recursively constructed for each frame before the beginningof encoding process defined by

st (119896 119897) = 119891119905minus1 (119896 119897) + st (119896 minus 1 119897) + st (119896 119897 minus 1)minus st (119896 minus 1 119897 minus 1)

st2 (119896 119897) = 1198912119905minus1 (119896 119897) + st2 (119896 minus 1 119897) + st2 (119896 119897 minus 1)minus st2 (119896 minus 1 119897 minus 1)

(15)

where 119896 and 119897 are the pixel coordinates of frame 119891119905minus1 with 119896 =0 1 1198721 minus 1 and 119897 = 0 1 1198722 minus 1 So initial conditionst(119896 119897) = st2(119896 119897) = 0 when either 119896 or 119897 = minus1 These two sumtables are partitioned according to the size of search windowto 1199041 and 11990421 subtables of size (119873 + 17 times 119873 + 17) It consistsof one additional row and column at the initial position incomparison with search window size The sum expressionsin (12) over 119892(119909+119894 119910+119895) and 1198922(119909+119894 119910+119895) can be calculatedefficiently using subtables

119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)= 1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 1199041 (119909 minus 1 119910 minus 1)minus 1199041 (119909 minus 1 119910 + 119873 minus 1) minus 1199041 (119909 + 119873 minus 1 119910 minus 1)

(16)

119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)= 11990421 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 11990421 (119909 minus 1 119910 minus 1)minus 11990421 (119909 minus 1 119910 + 119873 minus 1) minus 11990421 (119909 + 119873 minus 1 119910 minus 1)

(17)

Using (16) and (17) the norm (12) and the mean (18) of alldomain blocks can be calculated as matrices of size 17 times 17119909 isin 0 1 16 and 119910 isin 0 1 16

1198920119909119910 = 11198732 (1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1)+ 1199041 (119909 minus 1 119910 minus 1) minus 1199041 (119909 minus 1 119910 + 119873 minus 1)minus 1199041 (119909 + 119873 minus 1 119910 minus 1))

(18)

According to the three levelsrsquo quadtree partition it mayrequire total six memories to store these tables of each size1198721 times 1198722 But instead of six only two memory spaces aredefined according to the size of level three blocks and basedon that remaining two subtables are calculated

42 New Gray Level and Isometry Transformation The graylevel transformation parameters such as scaling factor ldquo119904rdquo andbrightness factor ldquo119900rdquo for each range block ℎ is based on certaindomain block 119892119909119910 which strongly matches (19) when MSE iszero

ℎ = 119904 sdot 119892119909119910 + 119900 (19)

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

MSE = 11198732119873minus1sum119894=0

119873minus1sum119895=0

(119904 sdot 119892 (119894 119895) + 119900 minus ℎ (119894 119895))2 (1)

119904 = 1198732 (sum119873minus1119894=0 sum119873minus1119895=0 ℎ (119894 119895) sdot 119892 (119894 119895)) minus (sum119873minus1119894=0 sum119873minus1119895=0 ℎ (119894 119895)) sdot (sum119873minus1119895=0 sum119873minus1119895=0 119892 (119894 119895))1198732sum119873minus1119894=0 sum119873minus1119895=0 1198922 (119894 119895) minus (sum119873minus1119894=0 sum119873minus1119895=0 119892 (119894 119895))2 (2)

119900 = 11198732 [[119873minus1sum119894=0

119873minus1sum119895=0

ℎ (119894 119895) minus 119904119873minus1sum119894=0

119873minus1sum119895=0

119892 (119894 119895)]] (3)

where 1198732 is the number of pixels in the block and ℎ(119894 119895)119892(119894 119895) are the pixel values of range block and contractivedomain block at coordinates (119894 119895) For each range block theparameters which need to be stored as a fractal encoded dataare the coordinates of domain block along with 119904 and 119900 andgeometric transformation index Gray level transformationparameters ldquo119904rdquo and ldquo119900rdquo should be in the range of minus12 to12 and minus255 to 255 respectively [3] to make sure that thetransformation is contractive At the decoder these fractalparameters are iteratively applied to an arbitrary initial imageaccording to the encoding block size which will finallyconverge to a reconstruction of the original image aftercertain number of iterations

3 Normalized Covariance BasedMotion Estimation

The ZNCC is a recognized similarity measure criterion andis considered as one of the accurate motion estimators in

video compression In fractal video coding the best matcheddomain block is decided after applying the proper affinetransformation which gives least mean square error TheZNCC method is more robust under uniform illuminationchanges and less sensitive to noise hence it can help toincrease the coding efficiency and improve subjective visualquality of output video For identical regions it may give ahigh NCC value for the best match but with a large averagegray level difference This difference is minimized by select-ing the accurate luminance and geometrical transformationparameters These fractal parameters are interpreted as akind ofmotion compensation technique due to unavailabilityof error frame In discrete domain the range block (ℎ119905)of current frame is shifted pixel by pixel across the searchwindow (119892119905minus1) of the reference frame The estimation ofmotion relies on the detection of maximum ZNCC functionbetween ℎ119905 and 119892119905minus1 The ZNCC is defined as

119862 (119909 119910) = sum119873minus1119894=0 sum119873minus1119895=0 (ℎ119905 (119894 119895) minus ℎ1199050) sdot (119892119905minus1 (119909 + 119894 119910 + 119895) minus 119892119905minus10(119909119910))radicsum119873minus1119894=0 sum119873minus1119895=0 (119892119905minus1 (119909 + 119894 119910 + 119895) minus 119892119905minus10(119909119910))2 sdot radicsum119873minus1119894=0 sum119873minus1119895=0 (ℎ119905 (119894 119895) minus ℎ1199050)2

(0 le 119909 119910 le 2119881) (4)

In (4) 119892119905minus10(119909119910) denotes the mean value of 119892119905minus1 withinthe area of the range block ℎ119905 shifted to (119909 119910) position andsimilarly ℎ1199050 denote the mean value of the range block ℎ119905119862(119909 119910) represents the ZNCC surface matrix between thecurrent macroblock and reference search window If thesearch range is plusmn119881 pixels in both directions from the samelocation of ℎ119905 then the size of the 119862(119909 119910) will be (2119881 +1 2119881 + 1) Fractal encoding is itself complex method anduse of ZNCC function to estimate motion vector can makethe combined technique computationally expensive and timeconsuming To overcome these complexity problems anefficient method of ZNCC calculation has been proposedwhich is based on new optimal 119904 and 119900 transformationparameters

31 Quadtree Based Partition Quadtree decompositionmethod initially partitions the current frame into a set of

larger size (16 times 16 pixels) range blocks ℎ119905 at level-1 (119871 = 1)Motion vector of the matched block is decided after verifyingthe highest three peak locations of ZNCC surface matrixThis verification is required because after quantizing graylevel transformation parameters the error between the blocksmay vary If the lowest quantizing error of larger size block(ℎ119905) is above the prespecified threshold then the block ℎ119905is partitioned in four quadrants ℎ119905119894 119894 = 1 4 Thepartitioning scheme [26] can be recursively continued untilthe error becomes smaller than the threshold or 4 times 4 pixelsblock size (119871 = 3) is reached

Here 3 levels of quadtree partitioning are employed withblock sizes from 16 times 16 to 4 times 4 pixels All subsequentpartitions of 16 times 16 block size are represented by one codeword as shown in Figure 1 The length of code word can beeither 1 bit or 5 bits it depends on only level-1 and level-2nodes If the level-1 block is not partitioned then the code

4 Mathematical Problems in Engineering

Code word 10010

R1

R13R12 R14R11

R133R132 R134R131

Level-1

Level-2

Level-3

1

0 00 18 times 8

4 times 4

16 times 16

Figure 1 Quadtree structure with code word

word is 1 bit otherwise it is assigned 5 bits Similarly if thecorresponding block is partitioned then it is represented bybit 1 otherwise it is represented by bit 0 as shown in Figure 1

4 Fast Fractal Video Coding Algorithm

The most important factor that affects the speed of fractalencoding is the searching ofmatched domain block In fractalvideo compression searching process is limited to the sizeof search window but it is also expensive in comparisonwith standard video coder (H264MPEG4) We proposeda FFT based ZNCC method with new matching criteriawhich reduced the block searching time Current frame (119891119905)is initially partitioned into nonoverlapped range blocks ldquoℎ =ℎ119905rdquo of size 119873 times 119873 pixels The search window ldquo119892 = 119892119905minus1rdquoof size (119873 + 16) times (119873 + 16) is defined on the previouslydecoded frame (119891119905minus1) that is reference frame of size1198721times1198722The flow diagram of the proposed fast fractal video coderis shown in Figure 2(a) The ZNCC matching process mayfind the wrong matched domain block if the range block ishomogeneous If the variance of any range block is below thesmallest predefined threshold then only mean value of thathomogeneous block needs to be encoded otherwise blockbelongs to nonhomogeneous group The root mean square(RMS) measure is used to find the prediction error (119864119898)between range (ℎ) and matched domain (119892119909119910) block Allthe fractal parameters along with corresponding error (119864119898)are the output of fast fractal searching algorithm as shownin Figure 2(b) The given block ldquoℎrdquo is partitioned into fourquadrants when 119864119898 is above the specified threshold (119905ℎ119871)otherwise encode and save the fractal parameters In thispaper depth of quadtree is three levels that is 119871max = 3 sotwo error thresholds are specified that is 119905ℎ1 and 119905ℎ2

Due to the high temporal correlation in video sequencesrange-domain mapping becomes more effective if the sizesof range and domain block are the same [15] The quality ofreference frame plays an important role while mapping twosame size blocks If interframe motion vector prediction is

based on good quality reference frame (previous frame) thenthe prediction error will be low But for subsequent framesthis errormonotonically increases due to cumulative processSo to get a better quality reference frame at the beginningintraframe is compressed using DCT transformation andquantization technique [27 28]

41 Efficient Searching with Simplified Similarity MeasureThe high computational complexity is the main drawback ofZNCC similarity measure method in spatial domain There-fore (4) is simplified to minimize the complexity Numeratorcomponent of (4) is represented as 119899(119909 119910) and is rewritten asfollows

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)

minus 1198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

ℎ1 (119894 119895) (5)

where ℎ1(119894 119895) = (ℎ(119894 119895) minus ℎ0) it has zero mean and becausethe sum of ℎ1(119894 119895) is zero the term 1198920119909119910sum119873minus1119894=0 sum119873minus1119895=0 ℎ1(119894 119895)will also be zero and (5) can be written as

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)= ⟨119892119909119910 ℎ1⟩

(6)

Equation (6) is independent of the mean (1198920119909119910) value ofdomain block All the domain blocks are overlapped and1198920119909119910value of each block is different 119899(119909 119910) is the cross correlationof ℎ1 and 119892119909119910 block where 119909 and119910 are the location of domainblock 119892119909119910 It is equivalent to the inner product operation oftwo bocks represented by ⟨sdot sdot⟩ operatorThis cross correlationis equal to the complex conjugate multiplications of twofrequency domain components The complex conjugate partcan be avoided by taking the inverse FFT (IFFT) of one of theinput instead of FFT because conj(FFT(ℎ1)) = (119873 + 16)2 timesIFFT(ℎ1) where (119873 + 16)2 is a constant term In frequencydomain size of both the input blocks must be equal The sizeof block ℎ1 is increased to the size of 119892 by padding zeros onthe left and down the side of the block The calculation of (6)for all the blocks can be computed in one computation usingFFT as given below

119899 = IFFT [FFT (119892) sdot IFFT (ℎ1)] = IFFT [119866 sdot 1198671] (7)

Similarly the denominator component of (4) is the product ofstandard deviation of 119892119909119910 and ℎ blocks Due to cancellationof (11198732) factor of standard deviation with numerator termit turns into 1198712 norm as (8) and (9) The norm of 1198921119909119910 (8)is also expensive because it is repeated for each overlapped

Mathematical Problems in Engineering 5

Range block ldquohrdquofrom current frame

Homogeneousdetection

Fast fractal searchingalgorithm

Search window ldquogrdquofrom ref frame

Encode and storefractal parameters

All subblockof ldquohrdquo processed

StopYes

Yes

No

No

No

Yes

Start

Partition ldquohrdquo into 4quadrant (L = L + 1)

orEm le thL

L = Lmax

(a)

FFT of searchwindow (g)

tth isometryusing DFTproperties

Computation of ZNCCcoefficients matrix ldquoCrdquo

E

No

Yes

No

FFT or sum table

mean andstd deviation of ldquogrdquo

IFFT of block

padding(h1) with zero

t = 0

Norm (g1)

g0x119898y119898

If t lt 8Next isometryindex t = t + 1

Em = E and tm = t

Fractal parametersxm ym sm om tm Em

Quantised s and o based onZNCC peak locationg(xm ym) block

If (E lt Em) or(E le th)

E le th

E lt Em

(b)

Figure 2 Flow diagrams of (a) fast fractal video coder and (b) efficient searching algorithm

domain block whereas the norm of ℎ1 (9) is unique for alldomain blocks

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(119892 (119909 + 119894 119910 + 119895) minus 1198920119909119910)2]]12

where 1198921119909119910 = 119892119909119910 minus 1198920119909119910

(8)

1003817100381710038171003817ℎ11003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(ℎ (119894 119895) minus ℎ0)2]]12

where ℎ1 = ℎ minus ℎ0

(9)

The norm of 1198921119909119910 expression is simplified and is written as(10) in the following

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 21198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)

+ 119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910]]12

(10)

119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910 = 1198732( 11198732119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2

(11)

Equation (10) can be simplified using (11) to

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 11198732 (119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2]]12

(12)

This can be represented in terms of mean value of domainblock as10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817

= [[(119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)) minus 119873211989220119909119910]]12

(13)

All the overlapped domain blocks which are shifted by onepixel also require similar computations and mostly these willbe repeated from the previous blocks Two different methodsare proposed to prevent unnecessary computations of themean and norm of all overlapped blocks that is FFT based

6 Mathematical Problems in Engineering

FFT

Square FFT

IFFT

IFFT

Square

Squareroot

Inputldquogrdquo

++

minustimes

times

with zeropadding

IFFT of INtimesN

1(N2)

(N2)

1ldquog rdquo

0ldquog rdquo

Figure 3 FFT based mean and norm calculation of all domainblocks

and sum table based The ZNCC (4) can be rewritten using(7) (8) (9) and (13) to (14)

119862 (119909 119910) = 119899 (119909 119910)10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 sdot 1003817100381710038171003817ℎ11003817100381710038171003817 (14)

The flow diagram of fast efficient searching algorithm withfast computation of ZNCC (14) is shown in Figure 2(b) Invideo sequences more number of the blocks can be observedas stationary blocks with zerononzero motions Such blockscan be identified when the highest value on ZNCC surfaceis nearly equal to one (119862(119909119898 119910119898) asymp 1) and correspondingquantized value of 119904 and 119900 is one and zero respectively Afterfinding the motion vector (119909119898 119910119898) of the stationary domainblock searching process can be terminated if the RMS error(119864) is below the threshold (119905ℎ) with isometry index 119905 = 0 Dueto early termination of searching process coding efficiencyis increased with unchanged quality The norm of ℎ1 can beneglected from (14) because it is constant and divisional termfor all domain blocks and it does not change the final ZNCCresult

411 FFT Based Method for Mean and Norm Calculation ofOverlapped Blocks The sum of squared pixels and the sumof pixels according to the size of range block are similar tothe convolution with unit (119868) matrix of same size FFT andIFFT combinations are used for fast computation The flowdiagram of the mean and norm calculation of all the domainblocks in one computation as matrices using FFT is shown inFigure 3 FFT of input 119892 is readily available from (7) and theIFFT of unit (119868)matrix with zero padding is constant only onetime computation So only FFT of squared pixels and last twoIFFT are required along with others blocks The mean (1198920)and norm (1198921) of all the blocks are required during gray leveltransformation and normalized covariance calculations

412 Sum Table Based Method for Mean and Norm Calcu-lation of Overlapped Blocks The sum table based methodalso can be used to reduce the number of computationsrequired to compute themean andnormof all domain blocksThe precomputation of two sum tables st(119896 119897) and st2(119896 119897)over the previous frame 119891119905minus1(119896 119897) and squared pixel 1198912119905minus1(119896 119897)frame respectively acts as look-up tables These tables are

recursively constructed for each frame before the beginningof encoding process defined by

st (119896 119897) = 119891119905minus1 (119896 119897) + st (119896 minus 1 119897) + st (119896 119897 minus 1)minus st (119896 minus 1 119897 minus 1)

st2 (119896 119897) = 1198912119905minus1 (119896 119897) + st2 (119896 minus 1 119897) + st2 (119896 119897 minus 1)minus st2 (119896 minus 1 119897 minus 1)

(15)

where 119896 and 119897 are the pixel coordinates of frame 119891119905minus1 with 119896 =0 1 1198721 minus 1 and 119897 = 0 1 1198722 minus 1 So initial conditionst(119896 119897) = st2(119896 119897) = 0 when either 119896 or 119897 = minus1 These two sumtables are partitioned according to the size of search windowto 1199041 and 11990421 subtables of size (119873 + 17 times 119873 + 17) It consistsof one additional row and column at the initial position incomparison with search window size The sum expressionsin (12) over 119892(119909+119894 119910+119895) and 1198922(119909+119894 119910+119895) can be calculatedefficiently using subtables

119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)= 1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 1199041 (119909 minus 1 119910 minus 1)minus 1199041 (119909 minus 1 119910 + 119873 minus 1) minus 1199041 (119909 + 119873 minus 1 119910 minus 1)

(16)

119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)= 11990421 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 11990421 (119909 minus 1 119910 minus 1)minus 11990421 (119909 minus 1 119910 + 119873 minus 1) minus 11990421 (119909 + 119873 minus 1 119910 minus 1)

(17)

Using (16) and (17) the norm (12) and the mean (18) of alldomain blocks can be calculated as matrices of size 17 times 17119909 isin 0 1 16 and 119910 isin 0 1 16

1198920119909119910 = 11198732 (1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1)+ 1199041 (119909 minus 1 119910 minus 1) minus 1199041 (119909 minus 1 119910 + 119873 minus 1)minus 1199041 (119909 + 119873 minus 1 119910 minus 1))

(18)

According to the three levelsrsquo quadtree partition it mayrequire total six memories to store these tables of each size1198721 times 1198722 But instead of six only two memory spaces aredefined according to the size of level three blocks and basedon that remaining two subtables are calculated

42 New Gray Level and Isometry Transformation The graylevel transformation parameters such as scaling factor ldquo119904rdquo andbrightness factor ldquo119900rdquo for each range block ℎ is based on certaindomain block 119892119909119910 which strongly matches (19) when MSE iszero

ℎ = 119904 sdot 119892119909119910 + 119900 (19)

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Code word 10010

R1

R13R12 R14R11

R133R132 R134R131

Level-1

Level-2

Level-3

1

0 00 18 times 8

4 times 4

16 times 16

Figure 1 Quadtree structure with code word

word is 1 bit otherwise it is assigned 5 bits Similarly if thecorresponding block is partitioned then it is represented bybit 1 otherwise it is represented by bit 0 as shown in Figure 1

4 Fast Fractal Video Coding Algorithm

The most important factor that affects the speed of fractalencoding is the searching ofmatched domain block In fractalvideo compression searching process is limited to the sizeof search window but it is also expensive in comparisonwith standard video coder (H264MPEG4) We proposeda FFT based ZNCC method with new matching criteriawhich reduced the block searching time Current frame (119891119905)is initially partitioned into nonoverlapped range blocks ldquoℎ =ℎ119905rdquo of size 119873 times 119873 pixels The search window ldquo119892 = 119892119905minus1rdquoof size (119873 + 16) times (119873 + 16) is defined on the previouslydecoded frame (119891119905minus1) that is reference frame of size1198721times1198722The flow diagram of the proposed fast fractal video coderis shown in Figure 2(a) The ZNCC matching process mayfind the wrong matched domain block if the range block ishomogeneous If the variance of any range block is below thesmallest predefined threshold then only mean value of thathomogeneous block needs to be encoded otherwise blockbelongs to nonhomogeneous group The root mean square(RMS) measure is used to find the prediction error (119864119898)between range (ℎ) and matched domain (119892119909119910) block Allthe fractal parameters along with corresponding error (119864119898)are the output of fast fractal searching algorithm as shownin Figure 2(b) The given block ldquoℎrdquo is partitioned into fourquadrants when 119864119898 is above the specified threshold (119905ℎ119871)otherwise encode and save the fractal parameters In thispaper depth of quadtree is three levels that is 119871max = 3 sotwo error thresholds are specified that is 119905ℎ1 and 119905ℎ2

Due to the high temporal correlation in video sequencesrange-domain mapping becomes more effective if the sizesof range and domain block are the same [15] The quality ofreference frame plays an important role while mapping twosame size blocks If interframe motion vector prediction is

based on good quality reference frame (previous frame) thenthe prediction error will be low But for subsequent framesthis errormonotonically increases due to cumulative processSo to get a better quality reference frame at the beginningintraframe is compressed using DCT transformation andquantization technique [27 28]

41 Efficient Searching with Simplified Similarity MeasureThe high computational complexity is the main drawback ofZNCC similarity measure method in spatial domain There-fore (4) is simplified to minimize the complexity Numeratorcomponent of (4) is represented as 119899(119909 119910) and is rewritten asfollows

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)

minus 1198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

ℎ1 (119894 119895) (5)

where ℎ1(119894 119895) = (ℎ(119894 119895) minus ℎ0) it has zero mean and becausethe sum of ℎ1(119894 119895) is zero the term 1198920119909119910sum119873minus1119894=0 sum119873minus1119895=0 ℎ1(119894 119895)will also be zero and (5) can be written as

119899 (119909 119910) = 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895) sdot ℎ1 (119894 119895)= ⟨119892119909119910 ℎ1⟩

(6)

Equation (6) is independent of the mean (1198920119909119910) value ofdomain block All the domain blocks are overlapped and1198920119909119910value of each block is different 119899(119909 119910) is the cross correlationof ℎ1 and 119892119909119910 block where 119909 and119910 are the location of domainblock 119892119909119910 It is equivalent to the inner product operation oftwo bocks represented by ⟨sdot sdot⟩ operatorThis cross correlationis equal to the complex conjugate multiplications of twofrequency domain components The complex conjugate partcan be avoided by taking the inverse FFT (IFFT) of one of theinput instead of FFT because conj(FFT(ℎ1)) = (119873 + 16)2 timesIFFT(ℎ1) where (119873 + 16)2 is a constant term In frequencydomain size of both the input blocks must be equal The sizeof block ℎ1 is increased to the size of 119892 by padding zeros onthe left and down the side of the block The calculation of (6)for all the blocks can be computed in one computation usingFFT as given below

119899 = IFFT [FFT (119892) sdot IFFT (ℎ1)] = IFFT [119866 sdot 1198671] (7)

Similarly the denominator component of (4) is the product ofstandard deviation of 119892119909119910 and ℎ blocks Due to cancellationof (11198732) factor of standard deviation with numerator termit turns into 1198712 norm as (8) and (9) The norm of 1198921119909119910 (8)is also expensive because it is repeated for each overlapped

Mathematical Problems in Engineering 5

Range block ldquohrdquofrom current frame

Homogeneousdetection

Fast fractal searchingalgorithm

Search window ldquogrdquofrom ref frame

Encode and storefractal parameters

All subblockof ldquohrdquo processed

StopYes

Yes

No

No

No

Yes

Start

Partition ldquohrdquo into 4quadrant (L = L + 1)

orEm le thL

L = Lmax

(a)

FFT of searchwindow (g)

tth isometryusing DFTproperties

Computation of ZNCCcoefficients matrix ldquoCrdquo

E

No

Yes

No

FFT or sum table

mean andstd deviation of ldquogrdquo

IFFT of block

padding(h1) with zero

t = 0

Norm (g1)

g0x119898y119898

If t lt 8Next isometryindex t = t + 1

Em = E and tm = t

Fractal parametersxm ym sm om tm Em

Quantised s and o based onZNCC peak locationg(xm ym) block

If (E lt Em) or(E le th)

E le th

E lt Em

(b)

Figure 2 Flow diagrams of (a) fast fractal video coder and (b) efficient searching algorithm

domain block whereas the norm of ℎ1 (9) is unique for alldomain blocks

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(119892 (119909 + 119894 119910 + 119895) minus 1198920119909119910)2]]12

where 1198921119909119910 = 119892119909119910 minus 1198920119909119910

(8)

1003817100381710038171003817ℎ11003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(ℎ (119894 119895) minus ℎ0)2]]12

where ℎ1 = ℎ minus ℎ0

(9)

The norm of 1198921119909119910 expression is simplified and is written as(10) in the following

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 21198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)

+ 119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910]]12

(10)

119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910 = 1198732( 11198732119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2

(11)

Equation (10) can be simplified using (11) to

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 11198732 (119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2]]12

(12)

This can be represented in terms of mean value of domainblock as10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817

= [[(119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)) minus 119873211989220119909119910]]12

(13)

All the overlapped domain blocks which are shifted by onepixel also require similar computations and mostly these willbe repeated from the previous blocks Two different methodsare proposed to prevent unnecessary computations of themean and norm of all overlapped blocks that is FFT based

6 Mathematical Problems in Engineering

FFT

Square FFT

IFFT

IFFT

Square

Squareroot

Inputldquogrdquo

++

minustimes

times

with zeropadding

IFFT of INtimesN

1(N2)

(N2)

1ldquog rdquo

0ldquog rdquo

Figure 3 FFT based mean and norm calculation of all domainblocks

and sum table based The ZNCC (4) can be rewritten using(7) (8) (9) and (13) to (14)

119862 (119909 119910) = 119899 (119909 119910)10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 sdot 1003817100381710038171003817ℎ11003817100381710038171003817 (14)

The flow diagram of fast efficient searching algorithm withfast computation of ZNCC (14) is shown in Figure 2(b) Invideo sequences more number of the blocks can be observedas stationary blocks with zerononzero motions Such blockscan be identified when the highest value on ZNCC surfaceis nearly equal to one (119862(119909119898 119910119898) asymp 1) and correspondingquantized value of 119904 and 119900 is one and zero respectively Afterfinding the motion vector (119909119898 119910119898) of the stationary domainblock searching process can be terminated if the RMS error(119864) is below the threshold (119905ℎ) with isometry index 119905 = 0 Dueto early termination of searching process coding efficiencyis increased with unchanged quality The norm of ℎ1 can beneglected from (14) because it is constant and divisional termfor all domain blocks and it does not change the final ZNCCresult

411 FFT Based Method for Mean and Norm Calculation ofOverlapped Blocks The sum of squared pixels and the sumof pixels according to the size of range block are similar tothe convolution with unit (119868) matrix of same size FFT andIFFT combinations are used for fast computation The flowdiagram of the mean and norm calculation of all the domainblocks in one computation as matrices using FFT is shown inFigure 3 FFT of input 119892 is readily available from (7) and theIFFT of unit (119868)matrix with zero padding is constant only onetime computation So only FFT of squared pixels and last twoIFFT are required along with others blocks The mean (1198920)and norm (1198921) of all the blocks are required during gray leveltransformation and normalized covariance calculations

412 Sum Table Based Method for Mean and Norm Calcu-lation of Overlapped Blocks The sum table based methodalso can be used to reduce the number of computationsrequired to compute themean andnormof all domain blocksThe precomputation of two sum tables st(119896 119897) and st2(119896 119897)over the previous frame 119891119905minus1(119896 119897) and squared pixel 1198912119905minus1(119896 119897)frame respectively acts as look-up tables These tables are

recursively constructed for each frame before the beginningof encoding process defined by

st (119896 119897) = 119891119905minus1 (119896 119897) + st (119896 minus 1 119897) + st (119896 119897 minus 1)minus st (119896 minus 1 119897 minus 1)

st2 (119896 119897) = 1198912119905minus1 (119896 119897) + st2 (119896 minus 1 119897) + st2 (119896 119897 minus 1)minus st2 (119896 minus 1 119897 minus 1)

(15)

where 119896 and 119897 are the pixel coordinates of frame 119891119905minus1 with 119896 =0 1 1198721 minus 1 and 119897 = 0 1 1198722 minus 1 So initial conditionst(119896 119897) = st2(119896 119897) = 0 when either 119896 or 119897 = minus1 These two sumtables are partitioned according to the size of search windowto 1199041 and 11990421 subtables of size (119873 + 17 times 119873 + 17) It consistsof one additional row and column at the initial position incomparison with search window size The sum expressionsin (12) over 119892(119909+119894 119910+119895) and 1198922(119909+119894 119910+119895) can be calculatedefficiently using subtables

119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)= 1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 1199041 (119909 minus 1 119910 minus 1)minus 1199041 (119909 minus 1 119910 + 119873 minus 1) minus 1199041 (119909 + 119873 minus 1 119910 minus 1)

(16)

119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)= 11990421 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 11990421 (119909 minus 1 119910 minus 1)minus 11990421 (119909 minus 1 119910 + 119873 minus 1) minus 11990421 (119909 + 119873 minus 1 119910 minus 1)

(17)

Using (16) and (17) the norm (12) and the mean (18) of alldomain blocks can be calculated as matrices of size 17 times 17119909 isin 0 1 16 and 119910 isin 0 1 16

1198920119909119910 = 11198732 (1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1)+ 1199041 (119909 minus 1 119910 minus 1) minus 1199041 (119909 minus 1 119910 + 119873 minus 1)minus 1199041 (119909 + 119873 minus 1 119910 minus 1))

(18)

According to the three levelsrsquo quadtree partition it mayrequire total six memories to store these tables of each size1198721 times 1198722 But instead of six only two memory spaces aredefined according to the size of level three blocks and basedon that remaining two subtables are calculated

42 New Gray Level and Isometry Transformation The graylevel transformation parameters such as scaling factor ldquo119904rdquo andbrightness factor ldquo119900rdquo for each range block ℎ is based on certaindomain block 119892119909119910 which strongly matches (19) when MSE iszero

ℎ = 119904 sdot 119892119909119910 + 119900 (19)

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Range block ldquohrdquofrom current frame

Homogeneousdetection

Fast fractal searchingalgorithm

Search window ldquogrdquofrom ref frame

Encode and storefractal parameters

All subblockof ldquohrdquo processed

StopYes

Yes

No

No

No

Yes

Start

Partition ldquohrdquo into 4quadrant (L = L + 1)

orEm le thL

L = Lmax

(a)

FFT of searchwindow (g)

tth isometryusing DFTproperties

Computation of ZNCCcoefficients matrix ldquoCrdquo

E

No

Yes

No

FFT or sum table

mean andstd deviation of ldquogrdquo

IFFT of block

padding(h1) with zero

t = 0

Norm (g1)

g0x119898y119898

If t lt 8Next isometryindex t = t + 1

Em = E and tm = t

Fractal parametersxm ym sm om tm Em

Quantised s and o based onZNCC peak locationg(xm ym) block

If (E lt Em) or(E le th)

E le th

E lt Em

(b)

Figure 2 Flow diagrams of (a) fast fractal video coder and (b) efficient searching algorithm

domain block whereas the norm of ℎ1 (9) is unique for alldomain blocks

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(119892 (119909 + 119894 119910 + 119895) minus 1198920119909119910)2]]12

where 1198921119909119910 = 119892119909119910 minus 1198920119909119910

(8)

1003817100381710038171003817ℎ11003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

(ℎ (119894 119895) minus ℎ0)2]]12

where ℎ1 = ℎ minus ℎ0

(9)

The norm of 1198921119909119910 expression is simplified and is written as(10) in the following

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 21198920119909119910 119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)

+ 119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910]]12

(10)

119873minus1sum119894=0

119873minus1sum119895=0

11989220119909119910 = 1198732( 11198732119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2

(11)

Equation (10) can be simplified using (11) to

10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 = [[119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)

minus 11198732 (119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895))2]]12

(12)

This can be represented in terms of mean value of domainblock as10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817

= [[(119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)) minus 119873211989220119909119910]]12

(13)

All the overlapped domain blocks which are shifted by onepixel also require similar computations and mostly these willbe repeated from the previous blocks Two different methodsare proposed to prevent unnecessary computations of themean and norm of all overlapped blocks that is FFT based

6 Mathematical Problems in Engineering

FFT

Square FFT

IFFT

IFFT

Square

Squareroot

Inputldquogrdquo

++

minustimes

times

with zeropadding

IFFT of INtimesN

1(N2)

(N2)

1ldquog rdquo

0ldquog rdquo

Figure 3 FFT based mean and norm calculation of all domainblocks

and sum table based The ZNCC (4) can be rewritten using(7) (8) (9) and (13) to (14)

119862 (119909 119910) = 119899 (119909 119910)10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 sdot 1003817100381710038171003817ℎ11003817100381710038171003817 (14)

The flow diagram of fast efficient searching algorithm withfast computation of ZNCC (14) is shown in Figure 2(b) Invideo sequences more number of the blocks can be observedas stationary blocks with zerononzero motions Such blockscan be identified when the highest value on ZNCC surfaceis nearly equal to one (119862(119909119898 119910119898) asymp 1) and correspondingquantized value of 119904 and 119900 is one and zero respectively Afterfinding the motion vector (119909119898 119910119898) of the stationary domainblock searching process can be terminated if the RMS error(119864) is below the threshold (119905ℎ) with isometry index 119905 = 0 Dueto early termination of searching process coding efficiencyis increased with unchanged quality The norm of ℎ1 can beneglected from (14) because it is constant and divisional termfor all domain blocks and it does not change the final ZNCCresult

411 FFT Based Method for Mean and Norm Calculation ofOverlapped Blocks The sum of squared pixels and the sumof pixels according to the size of range block are similar tothe convolution with unit (119868) matrix of same size FFT andIFFT combinations are used for fast computation The flowdiagram of the mean and norm calculation of all the domainblocks in one computation as matrices using FFT is shown inFigure 3 FFT of input 119892 is readily available from (7) and theIFFT of unit (119868)matrix with zero padding is constant only onetime computation So only FFT of squared pixels and last twoIFFT are required along with others blocks The mean (1198920)and norm (1198921) of all the blocks are required during gray leveltransformation and normalized covariance calculations

412 Sum Table Based Method for Mean and Norm Calcu-lation of Overlapped Blocks The sum table based methodalso can be used to reduce the number of computationsrequired to compute themean andnormof all domain blocksThe precomputation of two sum tables st(119896 119897) and st2(119896 119897)over the previous frame 119891119905minus1(119896 119897) and squared pixel 1198912119905minus1(119896 119897)frame respectively acts as look-up tables These tables are

recursively constructed for each frame before the beginningof encoding process defined by

st (119896 119897) = 119891119905minus1 (119896 119897) + st (119896 minus 1 119897) + st (119896 119897 minus 1)minus st (119896 minus 1 119897 minus 1)

st2 (119896 119897) = 1198912119905minus1 (119896 119897) + st2 (119896 minus 1 119897) + st2 (119896 119897 minus 1)minus st2 (119896 minus 1 119897 minus 1)

(15)

where 119896 and 119897 are the pixel coordinates of frame 119891119905minus1 with 119896 =0 1 1198721 minus 1 and 119897 = 0 1 1198722 minus 1 So initial conditionst(119896 119897) = st2(119896 119897) = 0 when either 119896 or 119897 = minus1 These two sumtables are partitioned according to the size of search windowto 1199041 and 11990421 subtables of size (119873 + 17 times 119873 + 17) It consistsof one additional row and column at the initial position incomparison with search window size The sum expressionsin (12) over 119892(119909+119894 119910+119895) and 1198922(119909+119894 119910+119895) can be calculatedefficiently using subtables

119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)= 1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 1199041 (119909 minus 1 119910 minus 1)minus 1199041 (119909 minus 1 119910 + 119873 minus 1) minus 1199041 (119909 + 119873 minus 1 119910 minus 1)

(16)

119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)= 11990421 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 11990421 (119909 minus 1 119910 minus 1)minus 11990421 (119909 minus 1 119910 + 119873 minus 1) minus 11990421 (119909 + 119873 minus 1 119910 minus 1)

(17)

Using (16) and (17) the norm (12) and the mean (18) of alldomain blocks can be calculated as matrices of size 17 times 17119909 isin 0 1 16 and 119910 isin 0 1 16

1198920119909119910 = 11198732 (1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1)+ 1199041 (119909 minus 1 119910 minus 1) minus 1199041 (119909 minus 1 119910 + 119873 minus 1)minus 1199041 (119909 + 119873 minus 1 119910 minus 1))

(18)

According to the three levelsrsquo quadtree partition it mayrequire total six memories to store these tables of each size1198721 times 1198722 But instead of six only two memory spaces aredefined according to the size of level three blocks and basedon that remaining two subtables are calculated

42 New Gray Level and Isometry Transformation The graylevel transformation parameters such as scaling factor ldquo119904rdquo andbrightness factor ldquo119900rdquo for each range block ℎ is based on certaindomain block 119892119909119910 which strongly matches (19) when MSE iszero

ℎ = 119904 sdot 119892119909119910 + 119900 (19)

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

FFT

Square FFT

IFFT

IFFT

Square

Squareroot

Inputldquogrdquo

++

minustimes

times

with zeropadding

IFFT of INtimesN

1(N2)

(N2)

1ldquog rdquo

0ldquog rdquo

Figure 3 FFT based mean and norm calculation of all domainblocks

and sum table based The ZNCC (4) can be rewritten using(7) (8) (9) and (13) to (14)

119862 (119909 119910) = 119899 (119909 119910)10038171003817100381710038171003817119892111990911991010038171003817100381710038171003817 sdot 1003817100381710038171003817ℎ11003817100381710038171003817 (14)

The flow diagram of fast efficient searching algorithm withfast computation of ZNCC (14) is shown in Figure 2(b) Invideo sequences more number of the blocks can be observedas stationary blocks with zerononzero motions Such blockscan be identified when the highest value on ZNCC surfaceis nearly equal to one (119862(119909119898 119910119898) asymp 1) and correspondingquantized value of 119904 and 119900 is one and zero respectively Afterfinding the motion vector (119909119898 119910119898) of the stationary domainblock searching process can be terminated if the RMS error(119864) is below the threshold (119905ℎ) with isometry index 119905 = 0 Dueto early termination of searching process coding efficiencyis increased with unchanged quality The norm of ℎ1 can beneglected from (14) because it is constant and divisional termfor all domain blocks and it does not change the final ZNCCresult

411 FFT Based Method for Mean and Norm Calculation ofOverlapped Blocks The sum of squared pixels and the sumof pixels according to the size of range block are similar tothe convolution with unit (119868) matrix of same size FFT andIFFT combinations are used for fast computation The flowdiagram of the mean and norm calculation of all the domainblocks in one computation as matrices using FFT is shown inFigure 3 FFT of input 119892 is readily available from (7) and theIFFT of unit (119868)matrix with zero padding is constant only onetime computation So only FFT of squared pixels and last twoIFFT are required along with others blocks The mean (1198920)and norm (1198921) of all the blocks are required during gray leveltransformation and normalized covariance calculations

412 Sum Table Based Method for Mean and Norm Calcu-lation of Overlapped Blocks The sum table based methodalso can be used to reduce the number of computationsrequired to compute themean andnormof all domain blocksThe precomputation of two sum tables st(119896 119897) and st2(119896 119897)over the previous frame 119891119905minus1(119896 119897) and squared pixel 1198912119905minus1(119896 119897)frame respectively acts as look-up tables These tables are

recursively constructed for each frame before the beginningof encoding process defined by

st (119896 119897) = 119891119905minus1 (119896 119897) + st (119896 minus 1 119897) + st (119896 119897 minus 1)minus st (119896 minus 1 119897 minus 1)

st2 (119896 119897) = 1198912119905minus1 (119896 119897) + st2 (119896 minus 1 119897) + st2 (119896 119897 minus 1)minus st2 (119896 minus 1 119897 minus 1)

(15)

where 119896 and 119897 are the pixel coordinates of frame 119891119905minus1 with 119896 =0 1 1198721 minus 1 and 119897 = 0 1 1198722 minus 1 So initial conditionst(119896 119897) = st2(119896 119897) = 0 when either 119896 or 119897 = minus1 These two sumtables are partitioned according to the size of search windowto 1199041 and 11990421 subtables of size (119873 + 17 times 119873 + 17) It consistsof one additional row and column at the initial position incomparison with search window size The sum expressionsin (12) over 119892(119909+119894 119910+119895) and 1198922(119909+119894 119910+119895) can be calculatedefficiently using subtables

119873minus1sum119894=0

119873minus1sum119895=0

119892 (119909 + 119894 119910 + 119895)= 1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 1199041 (119909 minus 1 119910 minus 1)minus 1199041 (119909 minus 1 119910 + 119873 minus 1) minus 1199041 (119909 + 119873 minus 1 119910 minus 1)

(16)

119873minus1sum119894=0

119873minus1sum119895=0

1198922 (119909 + 119894 119910 + 119895)= 11990421 (119909 + 119873 minus 1 119910 + 119873 minus 1) + 11990421 (119909 minus 1 119910 minus 1)minus 11990421 (119909 minus 1 119910 + 119873 minus 1) minus 11990421 (119909 + 119873 minus 1 119910 minus 1)

(17)

Using (16) and (17) the norm (12) and the mean (18) of alldomain blocks can be calculated as matrices of size 17 times 17119909 isin 0 1 16 and 119910 isin 0 1 16

1198920119909119910 = 11198732 (1199041 (119909 + 119873 minus 1 119910 + 119873 minus 1)+ 1199041 (119909 minus 1 119910 minus 1) minus 1199041 (119909 minus 1 119910 + 119873 minus 1)minus 1199041 (119909 + 119873 minus 1 119910 minus 1))

(18)

According to the three levelsrsquo quadtree partition it mayrequire total six memories to store these tables of each size1198721 times 1198722 But instead of six only two memory spaces aredefined according to the size of level three blocks and basedon that remaining two subtables are calculated

42 New Gray Level and Isometry Transformation The graylevel transformation parameters such as scaling factor ldquo119904rdquo andbrightness factor ldquo119900rdquo for each range block ℎ is based on certaindomain block 119892119909119910 which strongly matches (19) when MSE iszero

ℎ = 119904 sdot 119892119909119910 + 119900 (19)

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Using (8) and (9) (19) can be represented as

ℎ1 + ℎ0 = 119904 (1198921119909119910 + 1198920119909119910) + 119900 (20)

so ℎ1 = 119904 sdot 1198921119909119910 and ℎ0 = 119904 sdot 1198920119909119910 + 119900When the highest peak of ZNCC is close to one that is119862(119909 119910) asymp 1 as per (14) then 119909 and 119910 denote the coordinates

of matched domain blockThe gray level parameters (119904 and 119900)are estimated by substituting 1198921119909119910 = ℎ1119904 in (14) which arecomputationally more efficient form as given below

119904 = 1003817100381710038171003817ℎ110038171003817100381710038172119899 (119909 119910) (21)

119900 = ℎ0 minus 119904 sdot 1198920119909119910 (22)

As per (21) both numerator and denominator terms areavailable in (14) except square operation and in (22) ℎ0 and1198920119909119910 are also available while finding ℎ1 and 1198921119909119910 So thecomputational complexity of these both the equations areminimal

The searching algorithm has the difficulty of applying 8different isometry transformations to the individual domainblock since it operates on the entire search window using2D FFT Hence the isometry transformations are performedon the range block instead of domain block The initialIFFT of zero padded range block acts as a IFFT of zerodegree transformation block The remaining seven isometrytransformations along with their IFFTs are calculated basedon the previous IFFT of range block by applying rotationand reflection properties of 2D IFFT [26] This helps toavoid the repeated IFFT calculations due to this it increasesthe searching speed of the algorithm In fractal based fastmotion estimation using simplified similarity measure withquadtree partition operating in frequency domain is one ofthe features of this paper

5 Experimental Results

The performance of the proposed fast normalized covariancebased fractal video coder with simplified similarity measureis evaluated The denominator of (14) is implemented byusing two different methods one FFT based method isrepresented as proposed-FFT and other sum table basedmethod is represented as proposed-ST The popular videosequences (352 times 288 pixels of each sequence) [29] foremancarphone Tennis news and coastguard are used to evaluatethe performance of proposed methods Range blocks areformed according to three-level quadtree partition criterionwith block size 16 times 16 pixels at level 1 8 times 8 pixels at level 2and 4 times 4 pixels at level 3 The smallest predefined thresholdto detect a homogeneous block at each level is (025 times119873) andthe remaining two thresholds are 119905ℎ1 = 64 plusmn 05 and 119905ℎ2 =84 plusmn 06 to obtain good quality output video A search areaon the reference frame is plusmn8 pixels in both vertical andhorizontal directions from the same position as of rangeblock on the target frame Along with proposed methodsCPMNCIM and NHEXS algorithms are also implementedIn CPMNCIM method first 3 frames are set for CPM and

the remaining frames are using NCIM The video sequencesare also compressed using H264 JM 186 reference software[30] to compare the performances The parameters of H264coder are defined as high profile quantization parameterOPP between 28 and 36 selected to ensure good qualitysearch range 16 macroblock partitioned 4 times 4 8 times 8 and16 times 16 group of pictures (GOP) 12 or 15 and entropy basedcodingmethod universal variable length coding (UVLC) Allthe methods including proposed methods are implementedin MATLAB 714 and simulated on a PC (Intel Core i5-2400CPU 310GHz 316GB RAM)

Fractal parameters of each range block are quantizedseparately gray scale factors s and o are quantized by assign-ing 5 bits and 7 bits respectively coordinates of matcheddomain block (119909 119910) are encodedwith 4-bit length codewordsand 3 bits for the indexing of isometry transformations Incomparison with all the presented methods only sum tablebased method requires two additional memories of each sizeof (1198721 times1198722)2 bytes

To evaluate the performance the results of proposedalgorithms are compared with traditional CPMNCIM andNHEXS algorithms All these algorithms are also comparedwith H264 BY keeping the fixed value of PSNR for eachvideo sequence encoding time and compression ratio (CR)are calculated using all the methods Table 1 shows thecomparison of average coding results of five video sequencesIn each sequence PSNR and CR of proposed methods arethe references for othermethods analysisThe average codingtime of the proposed method is decreased by 9817 and6649 of the CPMNCIM and NHEXS methods respec-tively In comparison with proposed method the averagecompression ratio of CPMNCIM and NHEXS is decreasedby 5575 and 958 with 412 dB and 041 dB reduction inPSNR The proposed methods present an average of 20reduction in coding time and 2 decrease in compressionratio with marginal degradation of PSNR by 015 dB ifcompared to the H264

Table 2 shows the comparison of performances of theproposed methods with existing fractal video coder methods[17 18 20] In each existing method the different videosequences with different GOPs are used for analysis Thesequence highway (15 frames 352times 288 pixels) is used and thecompression ratio is 355 timeswith PSNR 172 dBhigher thanaverage of [17 18]The silent andmother-daughter (20 frameseach 352 times 288 pixels) sequences are used and their averagePSNR increased by 411 dB and bit rate decreased by 33 incomparison with algorithm in [20] For low bit rate videosthese proposed methods give much better performance thanH264 In Table 2 the proposed method can save the com-pression time by 55 with marginal reduction of PSNR incomparison with H264 The performance of proposed-STand proposed-FFT based methods are almost equal in termsof PSNR encoding time and compression ratio as shown inTables 1 and 2 For low bit rate videos the proposed methodsgive high compression ratio and very less time in comparisonwith H264 News highway mother-daughter and silent arethe low bit rate videos and others are high bit rate videos

A statistical measures were used to compute the scoredistribution range of result parameters Confidence interval

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 Comparison of average video coding results using different methods

Videos Methods PSNR (dB) Time (sec) CR

Foreman

CPMNCIM 3063 4281 4841NHEXS 3419 246 8850H264 3433 120 9700

Proposed-ST 3425 087 9511Proposed-FFT 3425 088 9443

Carphone

CPMNCIM 3132 3590 5242NHEXS 3497 195 10831H264 3512 104 12080

Proposed-ST 3500 068 11924Proposed-FFT 3500 069 11893

Tennis

CPMNCIM 2922 6658 2828NHEXS 3127 491 5381H264 3172 149 6366

Proposed-ST 3150 139 6169Proposed-FFT 3153 142 6158

News

CPMNCIM 2931 4799 5570NHEXS 3707 154 12221H264 3812 090 10584

Proposed-ST 3841 066 12631Proposed-FFT 3840 066 12682

Coastguard

CPMNCIM 2821 9036 1880NHEXS 2985 651 4529H264 3030 164 5244

Proposed-ST 3013 158 5271Proposed-FFT 3012 160 5351

Table 2 Results comparison between proposed methods with other methods(a)

Video Methods GOP PSNR (dB) Compression ratio

Highway

Object-based [18]

15

3553 8118Region-based [17] 3538 8276Proposed-ST 3730 28848Proposed-FFT 3727 28729

H264 3705 20214(b)

Videos Methods GOP PSNR (dB) Bit rate (KBPS)

Silent

Region-based [20]

20

323 39833Proposed-ST 3706 33062Proposed-FFT 3705 33176

H264 3701 39048

Mother-daughter

Region-based [20]

20

3705 314Proposed-ST 4050 16172Proposed-FFT 4048 16180

H264 4004 18787

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 3 Comparison of 90 confidence interval width for mean of result test parameters

Videos PSNR (dB) Time (Sec) Compression ratioH264 Proposed H264 Proposed H264 Proposed

Foreman 3398ndash3468 3384ndash3425 113ndash127 083ndash091 9049ndash10351 8642ndash1038Carphone 3475ndash3549 3461ndash3539 097ndash111 064ndash072 11271ndash12889 1087ndash12978Tennis 3124ndash3220 3097ndash3209 138ndash160 129ndash149 5805ndash6935 5450ndash6888News 3771ndash3853 3812ndash3870 083ndash097 062ndash070 9561ndash11607 11217ndash14148Mother-daughter 3838ndash3930 3874ndash3942 080ndash096 059ndash063 21286ndash23044 21382ndash2381Highway 3590ndash3666 3566ndash3652 117ndash129 041ndash047 17644ndash19608 19518ndash21512Coastguard 2979ndash3081 2950ndash3076 151ndash178 145ndash171 4176ndash6312 4045ndash6497Silent 3624ndash3756 3634ndash3778 062ndash070 048ndash052 17542ndash19858 17861ndash20339

30

32

34

36

38

40

42

44

1 21 41 61 81 101 121 141 161 181 201

PSN

R (d

B)

Frame number

ProposedH264

(a)

0

40

80

120

160

200

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

ratio

Frame number

ProposedH264

(b)

02040608

112141618

1 21 41 61 81 101 121 141 161 181 201

Com

pres

sion

time (

sec)

Frame number

ProposedH264

(c)

Figure 4 Frame to frame performance comparison using proposed method and H264

(CI) generates an upper and lower limit for the mean it givesan indication of how much uncertainty there is with truemeanThe 119905-distributions for each of the test parameters withsample size 119877 sample mean 120583 sample standard deviationsd and desired significance level 120572 are used to define theconfidence limits as follows

CI = 120583 plusmn 119905(1minus1205722119877minus1) sdot sdradic119877 (23)

The 100 times (1 minus 1205722) confidence interval can be computedfor different value of 120572 and (119877 minus 1) degrees of freedom In

the result analysis the 90 confidence interval is calculatedfor each test parameter with the value of 119905 in (23) being 1667Table 3 shows a 90 CI width for a mean of PSNR encodingtime and compression ratio for various video sequences Dueto fixed value of PSNR the average half-width CI of PSNR isalmost equal for both the methods The 90 interval widthof encoding time and compression ratio is narrow it meansproposed method also gives higher accuracy as compared tostandard video coder with less encoding time

Figure 4 shows a frame-wise performance comparison for204 frames of foreman video sequence between the proposed

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

(a) Decoded frame with proposed method (b) Original frame

(c) Decoded frame with H264

Figure 5 Original and decoded 62nd frame of ldquoTennisrdquo sequence

method and H264 Due to cumulative error the PSNRof decoded frames slightly decreases as the frame numberincreases from every intraframe in proposed method Thiserror is minimized by using proper selection of gray leveltransformation as shown in Figure 4(a)The results show thatthe compression ratio and PSNR of the proposed method foreach frame are marginal changes proportional to the H264results The compression time of the proposed method asshown in Figure 4(c) for each frame is on average 06 sec(27) lesser than H264 High encoding time drawback hasbeen overcome by using proposed fast fractal video codermethod In addition to this it gives good quality outputand high compression ratio approximately equal to standard(JM v186) video coder as shown in Table 1 Human visualsystem (HVS) does not perceive the smallest change in PSNR(le08 dB) between the H264 and proposed method Figure 5shows the 62nd original and decoded frame of ldquoTennisrdquosequence using H264 with 3226 dB and proposed methodwith 3212 dB

6 Conclusion

In this paper a quadtree partition based fast normalizedcovariance for fractal video compression is presented A sim-plified normalized covariance for similarity measure eight

isometry transformations using IFFT properties and mod-ified new gray level transformation parameters are proposedand estimated using FFT to improve the encoding speed andoutput quality Meanwhile this method can use FFT basedor sum table based approaches to normalize the covariancematrix which further increases the encoding speed signifi-cantlyThey are used for the calculation ofmean and standarddeviation of all overlapped blocks in one computation Theresults of using these approaches are almost equal in allperspectiveThemain drawback of sum table basedmethod isthat it required largememory space to store the tables as com-pared to the FFT based method Quadtree partition helps toachieve high compression ratios with good quality outputThe proposed methods can save the encoding time by 9817and 6649 compression ratio is increased by 129 and958 and the output quality increased by 412 dB and 041 dBin comparison with CPMNCIM and NHEXS methodsrespectively In comparison to H264 this method saves 20of compression time with marginal degradation in framequality and compression ratio

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

References

[1] M S Lazar and L T Bruton ldquoFractal block coding of digitalvideordquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 4 no 3 pp 297ndash308 1994

[2] A E Jacquin ldquoImage coding based on a fractal theory of iteratedcontractive image transformationsrdquo IEEE Transactions of ImageProcessing vol 1 no 1 pp 18ndash30 1992

[3] Y Fisher Fractal Image Compression Theory and ApplicationSpringer New York NY USA 1995

[4] Y Zheng G Liu and X Niu ldquoAn improved fractal image com-pression approach by using iterated function system and geneticalgorithmrdquoComputers ampMathematics with Applications vol 51no 11 pp 1727ndash1740 2006

[5] K U Barthel and T Voye ldquoThree-dimensional fractal videocodingrdquo in Proceedings of the IEEE International Conference onImage Processing vol 3 pp 260ndash263 IEEE Washington DCUSA 1995

[6] C-C Wang and C-H Hsieh ldquoEfficient fractal video codingalgorithm using intercube correlation searchrdquoOptical Engineer-ing vol 39 no 8 pp 2058ndash2064 2000

[7] M Wang and C-H Lai ldquoA hybrid fractal video compressionmethodrdquoComputers andMathematics withApplications vol 50no 3-4 pp 611ndash621 2005

[8] M Wang R Liu and C-H Lai ldquoAdaptive partition and hybridmethod in fractal video compressionrdquoComputers ampMathemat-ics with Applications vol 51 no 11 pp 1715ndash1726 2006

[9] M Wang and C-H Lai ldquoGrey video compression methodsusing fractalsrdquo International Journal of Computer Mathematicsvol 84 no 11 pp 1567ndash1590 2007

[10] Z Yao and R Wilson ldquoHybrid 3D fractal coding with neigh-bourhood vector quantisationrdquo EURASIP Journal on AppliedSignal Processing vol 16 pp 2571ndash2579 2004

[11] D V Lima W R Schwartz and H Pedrini ldquo3D searchlessfractal video encoding at low bit ratesrdquo Journal of MathematicalImaging and Vision vol 45 no 3 pp 239ndash250 2013

[12] Y Brijmohan and S H Mneney ldquoLow bit-rate video codingusing fractal compression of wavelet subtreesrdquo in Proceedings ofthe 7th IEEE AFRICON Conference in Africa Technology Inno-vation pp 39ndash44 September 2004

[13] Y Zhang L M Po and Y L Yu ldquoWavelet transform basedvariable tree size fractal video codingrdquo in Proceedings of theIEEE International Conference on Image Processing pp 294ndash297IEEE Santa Barbara Calif USA 1997

[14] R Yu J Zhou S Yu and D Chi ldquoFractal-based wavelet trans-form coding for low-bit-rate videordquo in Electronic Imaging andMultimedia Systems vol 2898 of Proceedings of SPIE pp 226ndash237 Beijing China November 1996

[15] C-S Kim R-C Kim and S-U Lee ldquoFractal coding of videosequence using circular prediction mapping and noncontrac-tive interframe mappingrdquo IEEE Transactions on Image Process-ing vol 7 no 4 pp 601ndash605 1998

[16] K Belloulata S Zhu and Z Wang ldquoA fast fractal videocoding algorithm using cross-hexagon search for block motionestimationrdquo ISRN Signal Processing vol 2011 Article ID 38612810 pages 2011

[17] S Zhu Y Hou Z Wang and K Belloulata ldquoFractal video seq-uences coding with region-based functionalityrdquoAppliedMathe-matical Modelling Simulation and Computation for Engineeringand Environmental Systems vol 36 no 11 pp 5633ndash5641 2012

[18] S Zhu L Li and Z Wang ldquoA novel fractal monocular andstereo video codec with object-based functionalityrdquo EurasipJournal on Advances in Signal Processing vol 2012 article 2272012

[19] K Belloulata A Belalia and S Zhu ldquoObject-based stereo videocompression using fractals and shape-adaptive DCTrdquo AEUmdashInternational Journal of Electronics and Communications vol68 no 7 pp 687ndash697 2014

[20] S Zhu L Li J Chen and K Belloulata ldquoAn automatic region-based video sequence codec based on fractal compressionrdquoInternational Journal of Electronics and Communications vol68 no 8 pp 795ndash805 2014

[21] S Zhu D Zhao and L Zhang ldquoA novel high efficiency fractalmultiview video codecrdquoMathematical Problems in Engineeringvol 2015 Article ID 613714 12 pages 2015

[22] S D Kamble N VThakur L G Malik and P R Bajaj ldquoFractalvideo coding using modified three step search algorithm forblock matching motion estimationrdquo Advances in IntelligentSystems and Computing vol 332 pp 151ndash162 2015

[23] A J H Hii C E Hann J G Chase and E E W Van HoutenldquoFast normalized cross correlation for motion tracking usingbasis functionsrdquo Computer Methods and Programs in Biomedi-cine vol 82 no 2 pp 144ndash156 2006

[24] S B Dhok R B Deshmukh and A G Keskar ldquoEfficient fractalimage coding using fast fourier transformrdquo International Jour-nal on Computing vol 1 no 2 2011

[25] R E Chaudhari and S B Dhok ldquoAcceleration of fractal videocompression using FFTrdquo in Proceedings of the 15th InternationalConference on Advanced Computing Technologies (ICACT rsquo13)pp 1ndash4 September 2013

[26] G J Sullivan and R L Baker ldquoEfficient quadtree coding ofimages and videordquo IEEE Transactions on Image Processing vol3 no 3 pp 327ndash331 1994

[27] A K Jain Fundamentals of Digital Image Processing PHI Pub-lications 1989

[28] Y-M Zhou C Zhang and Z-K Zhang ldquoAn efficient fractalimage coding algorithm using unified feature and DCTrdquo ChaosSolitons amp Fractals vol 39 no 4 pp 1823ndash1830 2009

[29] CIPR Video Sequences httpwwwciprrpieduresourceseq-uences

[30] H264AVC Software Coordination httpiphomehhidesue-hringtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of