Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 476181 11 pageshttpdxdoiorg1011552013476181

Research ArticleReversible Data Hiding Scheme with High Embedding CapacityUsing Semi-Indicator-Free Strategy

Jiann-Der Lee1 Yaw-Hwang Chiou1 and Jing-Ming Guo2

1 Department of Electrical Engineering Chang Gung University Tao-Yuan 333 Taiwan2Department of Electrical Engineering National Taiwan University of Science and Technology Taipei 106 Taiwan

Correspondence should be addressed to Jing-Ming Guo jmguoseednettw

Received 12 June 2013 Accepted 5 September 2013

Academic Editor Marco Perez-Cisneros

Copyright copy 2013 Jiann-Der Lee et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel reversible data-hiding scheme is proposed to embed secret data into a side-matched-vector-quantization- (SMVQ-)compressed image and achieve lossless reconstruction of a vector-quantization- (VQ-) compressed image The rather randomdistributed histogramof aVQ-compressed image can be relocated to locations close to zero by SMVQpredictionWith this strategyfewer bits can be utilized to encode SMVQ indices with very small valuesMoreover no indicator is required to encode these indiceswhich yields extrahiding space to hide secret data Hence high embedding capacity and low bit rate scenarios are deposited Morespecifically in terms of the embedding rate the bit rate and the embedding capacity experimental results show that the performanceof the proposed scheme is superior to those of the former data hiding schemes for VQ-based VQSMVQ-based and search-order-coding- (SOC-) based compressed images

1 Introduction

With the explosive growth of communication through theInternet information processing and management at anytime have become a standard service for most people sub-sequently inducing security problems such as interceptionmodification and montage Thus one critical issue is howto find effective ways to protect information transmissionover the Internet with safety and security To cope with thisdata hiding techniques have attracted attentions for beingable to embed secret data into a cover image with minimalperceptual degradation

In general data hiding techniques can be classifiedinto two categories namely reversible data hiding schemesand irreversible data hiding schemes For irreversible datahiding schemes only secret data can be extracted whilerestoration of cover images is unavailable The irreversibledata hiding scheme is not suited for certain applications thatdemand the original cover image to be unaltered by the datahiding process Conversely reversible data hiding schemescan extract the secret data and recover the original coverimages in the decoder Moreover data hiding schemes canbe performed in three possible formats that is in spatial

domain [1ndash5] in frequency domain [6ndash8] and in compresseddomain [9ndash15] In spatial domain Tian [1] proposed ascheme for reversible data hiding called difference expansion(DE) Tianrsquos work expands the value differences between twoneighboring pixels to embed a bit This scheme is easy torealize yet the embedding capacity heavily depends on thesmoothness of an image Ni et al [2] utilized the zero of theminimum points of the histogram in an image and slightlymodified the pixel values to embed secret data into the peakpoints of the histogram of an image The scheme is simplefor implementation and which supports high visual qualityHowever the embedding capacity is restricted by the peakpoints of the histogram of an image Moreover a cover imagecan be transformed into the frequency domain with variouspossible transform kernels for example DCT DFT or DWTNormally frequency domain techniques embed message bymodulating the transformed coefficients of subbands withjust noticeable difference (JND) according to the sensitivityof the human visual system (HVS)

Data hiding techniques in compressed domain can relishthe both advantages of data hiding and compression for amultimedia distribution Jo and Kim [9] firstly proposed anirreversible VQ-based scheme which is easy to realize yet

2 Mathematical Problems in Engineering

the hiding capacity is rather small Later many VQSMVQ-based schemes with indicators are proposed For exampleChang et al [10] developed a VQ-based data hiding withrecovery capability The scheme has low embedding capacityand high bit rate (BR) because of an indicator added in frontof most encoded indices and only two clusters are used tohide secret data In [11 12] a secret data hiding was proposedbased on the search-order coding compression method ofVQ indices to increase embedding capacityThe search-ordercoding selects the neighboring indices of the encoding indexto form a state codebook and then search-order code (SOC)and original index value (OIV) are employed to hide a secretbit 0 or 1 A 1-bit indicator is always employed to distinguishbetween SOC and OIV For this scheme the original coverimage can be restored completely Shie et al [13] developedan adaptive data hiding based on VQ-compressed imagein which image blocks are classified into embeddable andunembeddable blocks based on the variances and side-matchdistortions (SMDs) The embeddable blocks are employed tohide secret data while those unembeddable blocks remainunchanged It is necessary that indicator be used for the blockjudgment Although this scheme can yield a high embeddingcapacity the quality of the reconstructed image reduces as thequantity of the secret data increases In addition this schemeis a case of irreversible information hiding In [14] both VQand SMVQ are used to hide secret data in which a 1-bitindictor is always required for each index and only 1-bit secretdata can be embedded in each index when the VQ techniqueis applied Conversely more than one bit secret data canbe embedded in each index when the SMVQ technique isapplied This scheme can provide a higher hiding capacityyet it is an irreversible data-hiding scheme In [15] Yang andLin extended the scheme proposed by Chang et al [10] bydividing the VQ codebook into 2BS clusters and half of whichare used to embed secret data where BS denotes the size ofthe secret data embedded into each VQ index In the schemeboth of the VQ and SMVQ are applied to hide secret data anda 1-bit indictor is required for the index identification More-over it is a reversible data-hiding scheme In these schemesindicators are always used which causes a higher bit rate

By exploiting the special indices distribution of a SMVQ-compressed image that frequently used indices are encodedby short codes while rarely-used indices are encoded by longcodes this work proposed a novel reversible data hiding inthe SMVQ-compressed domain The scheme can increasethe hiding capacity and completely restore the cover imagefrom an embedded image after the hiding data is extractedMoreover the indices of a VQ-compressed image can berelocated close to zero by using SMVQ predications SMVQindices located around zero can be encoded using fewer bitsCompared with the previous schemes [10 12 15] in whichthe bit rates are increased because indicators are requiredin most of the indices the proposed scheme employs theSMVQ predication and thus most of the indices are encodedwithout indicator As a result high embedding capacity andlow bit rate scenarios can be achieved As documented inthe experimental results the performance of the proposedscheme is significantly superior to that of the previous works

[10 12 15] in terms of the embedding rate bit rate andembedding capacity

The rest of this paper is organized as follows Section 2presents the related works to briefly introduce the conceptsof VQ and SMVQ Section 3 describes the proposed schemein detail Section 4 gives the experimental results and per-formances comparisons of the proposed scheme with formerapproaches Finally Section 5 draws conclusions

2 Backgrounds

21 Brief Concept of the VQ VQ initially involves codebookconstruction from a set of training images the trainingimages are partitioned into nonoverlapping blocks and themost representative blocks are selected to form codebook inwhich the elements in the codebook are called codewordsIn general the LBG algorithm [16] is employed to producethe desired codebook With the generated codebook eachblock in an image is encoded with the index of the nearestcodeword such that the total storage space for an imageis minimized To measure the similarity between a block119861 = (119887

1 1198872 119887

119896) and a codeword 119862

119895= (1198881198951 1198881198952 119888

119895119896)

where 119862119895is the 119895th codeword in the codebook the squared

Euclidean distance is generally used as below

119863(119861 119862119895) =

119896

sum

119894=1

(119887119894minus 119888119895119894)2

(1)

When the encoding process is completed block 119861 issimply represented by the index min 119895 where 119862min 119895 is thenearest codeword of the block 119861 For decoding process thetable lookup is performed with the same codebook as thatused by the encoder

22 SMVQ Kim initially proposed SMVQ for image coding[17] SMVQ exploits the high correlation among neighboringblocks and side-match prediction to create a state codebookwith a size smaller than that of the original VQ codebookThe side-match prediction assumes that the values of theadjacent pixels between the neighboring blocks are equal InSMVQ blocks located in the first row and first column ofan image are encoded by traditional VQ and the remainingblocks are predicted using the corresponding state codebookFigure 1 shows an example of the relationships among anencoding block119883 its upper neighboring block 119880 and its leftneighboring block 119871 for blocks of size 4times4This work denotesthe border vector and the side vector of block 119883 as 119861(119883) =

(11988311198832119883311988341198835119883911988313) = (1198871 1198872 1198873 1198874 1198875 1198876 1198877) and

119878(119883119880 119871) = ((11988013 + 1198714)2 11988014 11988015 11988016 1198718 11987112 11987116) =

(1199041 11990421199043 1199044 1199045 1199046 1199047) The SMD between a block 119883predicted

by blocks 119880 and 119871 and a codeword cw in the codebookis measured by the squared Euclidean distance denoted asfollows

SMD (119883 cw) = 119863 (S (XU L) B (cw)) =7

sum

119894=1

(119904119894minus 119887119894)2

(2)

For each block SMVQ sorts the codebook CB accordingto SMDs and thus the first 119865

119888codewords in the sorted

Mathematical Problems in Engineering 3

U13 U14 U15 U16L4 X1 X2 X3 X4L8 X5L12 X9L16 X13

Figure 1 An example of SMVQ

codebook are picked to form the state codebook SC in whichSMDs are the smallest for block 119883 Subsequently the SMVQsearches the closest codeword from the state codebook forblock 119883 and then the index of the closest codeword is usedto encode block 119883 Because the size of the state codebook issmaller than that of the VQ codebook the size of the SMVQindex is reduced to improve the coding gain However thereis a distortion between the block decoded by the SMVQindex and the corresponding block decoded by the VQindex

3 The Proposed Scheme

A grayscale cover image 119862 of size 119873 lowast 119872 is partitionedinto nonoverlapped blocks BLKi119894=1119871 of 119908 lowast ℎpixels inwhich 119871 = (119873119908) lowast (119872ℎ) Normally the variables 119908 =

ℎ = 4 Each block of the image can be encoded using VQand thus each block is represented with an index of thenearest codeword in the codebook AVQ-compressed imagedenoted by 119868 consists of indices The size of an index isproportional to the number of codewords in the codebookA SMVQ-compressed image is generated by applying SMVQpredictions to a VQ-compressed image To reconstruct a VQ-compressed image completely the size of the state codebookand that of the VQ codebook are first set as identical Nextthe codebook is sorted according to the SMDs which denotethe differences between an encoding block and the codewordsin the VQ codebook Finally the state codebook is generatedusing all of the codewords in the sorted codebook For a VQindex the corresponding SMVQ index can be obtained usingthe state codebook and no distortion is presented between thetwo codewords decoded by the two indices

Figures 2 and 3 show histograms of the VQ-compressedPeppers image and the corresponding SMVQ-compressedimage respectively with a codebook of size 256 Basedon the SMVQ predication most indices of the SMVQ-compressed image are distributed close to zero For a SMVQ-compressed image a very small value 119873

119911binary digits (119873

119911

bits) representing unsigned integers from 0 to 2119873119911

minus 1 cancover most of the indices In the proposed scheme the 2119873119911numbers are employed as indicators orand encoding codesTable 1 shows the functions of each number of the119873

119911binary

digits (119873119911bits) A binary number among 0 and 2

119873119911minus 3

represents an encoding code and indicator Binary numbers

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250 300Index value

Num

ber o

f ind

ices

Figure 2 Histogram of the VQ-compressed Peppers image

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300Index value

Num

ber o

f ind

ices

9000

Figure 3 Histogram of the SMVQ-compressed Peppers image

Table 1 The functions of each number of the 119873119911binary digits

(119873119911bits)

Number Function0sim2119873119911 minus 3 Indicator and encoding code2119873119911 minus 2 Indicator (indicator 0)

2119873119911 minus 1 Indicator (indicator 1)

2119873119911

minus2 and 2119873119911 minus1 represent indicators denoted by indicator0 and indicator 1 respectively

31 The Encoding Process Suppose the size of a secret datasd is 119904 bits During the encoding process a SMVQ indexwith a value less than 2

119873119911minus 2 which is defined as a tiny-

value index is denoted by119864119894 and the corresponding encoded

index is denoted by 1198641015840

119894 Then 1198641015840

119894is expressed in (3) where

and and denote exponential and concatenation operatorsrespectively

1198641015840

119894= 119864119894 sd (3)

4 Mathematical Problems in Engineering

Table 2 The ranges and symbols of tiny-value middle-value andlarge-value indices

Indices Range SymbolTiny-value indices 0sim2119873119911 minus 3 119864

119894

Middle-value indices 0 + 2119873119911minus 2sim2119873119888 minus 1 + 2

119873119911 minus 2 119866

119894

Large-value indices 2119873119888 + 2119873119911minus 2simcsminus 1 119874

119894

Most of the SMVQ indices can be encoded using (3)including no indicator and the rest are encoded with the helpof indicator 0 or indicator 1 However the adoption of theindicator increases the size of the encoded result A SMVQindex with a value in between (2

119873119911minus 2) + 0 and (2119873119911 minus 2) +

2119873119888minus1 is defined as amiddle-value index Indices withmiddle

values are encoded using indicator 0 concatenated with anencoding space with119873

119888bits Amiddle-value index is denoted

by119866119894 and the corresponding encoded index is denoted by1198661015840

119894

which can be expressed in the following

1198661015840

119894= 119894119899119889119894119888119886119905119900119903 0 119866

119894minus 2119873119911

+ 2 (4)

The119873119888-bit encoding space of (4) is not to hide data but to

collect more indices Indicator 1 is employed to assist encod-ing a SMVQ index which cannot be encoded using (3) or (4)A SMVQ index with a value more than (2119873119911 minus 2) + 2

119873119888minus 1

is defined as a large-value index denoted by 119874119894 and the

corresponding encoded index is denoted by 1198741015840119894which can be

expressed in the following

1198741015840

119894= 119894119899119889119894119888119886119905119900119903 1 119874

119894 (5)

Equation (5) not only hides no data but also increases anadditional indicator with 119873

119911bits For a SMVQ-compressed

image most indices with tiny values are encoded using (3)which hides 119904 bits and generates an encoded result with119873

119911+119904

bits for each index a few indices are encoded using (4)which generates an encoded result with119873

119911+119873119888bits for each

index few indices are encoded using (5) which generates anencoded result with119873

119911+log2(cs) bits for each index where cs

indicates the size of a codebook Table 2 shows the ranges andsymbols of tiny-value middle-value and large-value indicesThe size of an embedded image can be measured with (6)where 119909 denotes the size of 119909 and119868

119911 119868119888 and 119868

119900denote the

number of indices with tiny values middle values or largevalues respectively

1003817100381710038171003817embedded image1003817100381710038171003817 = (119873119911+ 119904) lowast 119868

119911

+ (119873119911+ 119873119888) lowast 119868119888

+ (119873119911+ log2(cs)) lowast 119868

119900

(6)

In (6) (119873119911+ 119904) and (119873

119911+ 119873119888) can be controlled in less

or more than log2(cs) bits and only the third item always

requires a space with more than log2(cs) bits It is remarkable

that a larger 119904 leads to a larger embedding capacity and viceversa Moreover a larger 119873

119888increases 119868

119888 while decreasing

119868119900 and vice versa Based on the special index distribution

property of the SMVQmost indices can be encodedusing (3)

An encoding index with value Q

Is Q in between(2Nz

minus 2) + 0 and (2Nzminus 2) + 2

Ncminus 1

A middle-valueindex encodedusing (4)

A large-valueindex encodedusing (5)

A tiny-valueindex encodedusing (3)

No

No

Yes

YesQ le 2

N119911 minus 3

Figure 4 An example of the index classification and encodingschemes

in which no indicator is required and 119904-bit data are hiddenin each index and thus a high embedding capacity and alow embedded image size can be achieved in the encodingprocess The algorithm of the embedding process is summa-rized as follows and an example of the index classification andencoding schemes is shown in Figure 4

Input A grayscale cover image 119862 of size119873 lowast119872 a codebookCB of size cs a secret data stream and parameters119873

119911and119873

119888

OutputThe code stream in binary form cs 119887 and parameters119873119911and119873

119888

Step 1 Compress the cover image 119862 using VQ to obtain theVQ-compressed image 119868 of size (119873119908) lowast (119872ℎ)

Step 2 Perform SMVQ predictions with the VQ-compressedimage 119868 to create the SMVQ-compressed image 119879 of size(119873119908) lowast (119872ℎ)

Step 3 Read the next SMVQ index denoted as 119876 from theSMVQ-compressed image 119879 with the raster scan order

Step 4 If 119876 le 2119873119911 minus 3 (a tiny-value index) then we have the

following

Step 41 Read 119904-bit data sd from the secret data stream

Step 42 119876 is encoded using (3) cs 119887 = cs 119887 D2B(119876) sdwhere denotes a concatenation operation and D2B denotesa decimal to binary operation

Step 5 If119876 is in between (2119873119911 minus 2) + 0 and (2119873119911 minus 2) + 2

119873119888 minus 1

(a middle-value index) then 119876 is encoded using (4) cs119887=

cs 119887 119894119899119889119894119888119886119905119900119903 0 D2B(119876 minus 2119873119911 + 2)

Mathematical Problems in Engineering 5

(a)

00000 01100 00110 01010 00010 01111 001100 12 6 10 2 15 6

(b) (c)

0 1 2 3

4 5 1 7

8 0 1 10

127 2 1 0 1 || 15

1 || 2

1 || 6

2 || 0

3 || 127

2 || 6

2 || 2

0 || 0

2 || 0

0 || 10

2 || 3

1 || 12

0 || 6

2 || 8

2 || 5

2 || 1

Figure 5 An example of the encoding process for hiding 5-bit secretdata (a) Secret data (b) SMVQ-compressed image and (c) encodedimage

Step 6

If 119876 ge (2119873119911 minus 2) + 2

119873119888 (a large-value index) then

119876 is encoded using (5)cs 119887 = cs 119887 119894119899119889119894119888119886119905119900119903 1 D2B(119876)

Step 7 Repeat Steps 3ndash7 until all of the indices in the SMVQ-compressed image are processed

Step 8 Output the code stream cs 119887 and parameters 119873119911and

119873119888

Figure 5 shows an example of the encoding process forhiding 5-bit secret data Let 119868

0be indicator 0 and 119868

1indicator

1 where 119873119911= 2 119873

119888= 5 119904 = 5 and 119888119904 = 128 and the size

of the state codebook is identical to that of the VQ codebookFor each image (Figure 5) let 119877

1to 1198774be the indices located

in the first row and 1198775to 1198778the indices located in the second

row and so on For the SMVQ-compressed image 1198771has a

value 0 and it is less than 2119873119911 minus2 The index is encoded using(3) Secret data are (00000)

2 According to (3) the index with

119873119911bits followed by secret data (00000)

2is presented as the

encoded result for 1198771 1198774has a value of 3 and it is in between

(2119873119911

minus 2) + 0 and (2119873119911

minus 2) + 2119873119888

minus 1 An indicator 1198680 2

followed by the offset index 1 is the encoded result for119877411987713

has a value of 127 and it is more than (2119873119911

minus 2) + 2119873119888

minus 1 Anindicator 119868

1 3 followed by the index 127 is the encoded result

for 11987713 All of the encoded results are in binary form and the

length of the encoded result of each index is (119873119911+119904) (119873

119911+119873119888)

or (119873119911+ log2(cs)) bits using (3) (4) or (5) respectively

32TheExtraction andReversion Process Thegoal of the dataextraction and reversion process is to extract the embeddedsecret data from an embedded image and then recover theoriginal VQ-compressed imageThe extraction and reversionprocess is introduced as below For an encoded index 119873

119911

bits are firstly read from code stream cs 119887 and the 119873119911-bit

binary value is converted to the decimal value denoted by119889 119873119911 In the first case if 119889 119873

119911is less than 2

119873119911minus 2 the

encoded index is an 119864119894index Then the original index is

directly restored by 119889 119873119911 Next read 119904 bits from the code

stream 119888 119887119904 to reconstruct the secret data In the second case

if 119889 119873119911is 2119873119911 minus 1 the encoded index is an 119874

119894index Then

read the next log2(cs) bits and convert them into decimal

value denoted by 119889 119874119894 the original index is recovered by

119889 119874119894 In the third case if 119889 119873

119911is 2119873119911 minus 2 the encoded index

is a 119866119894index Then read the next 119873

119888bits and convert them

into decimal value 119889 119866119894 the original index is recovered by

119889119866119894

+ (2119873119885 minus 2) When the original SMVQ-compressed image

has been reconstructed SMVQ predictions can be employedto restore the original VQ-compressed image In summarythe extraction and reversion process of the proposed schemeis organized as below

Input The code stream in binary form cs 119887 and parameters119873119911and119873

119888

Output The reconstructed VQ-compressed image 1198681015840 of size119873 lowast119872 and the retrieved secret data stream

Step 1 Set ptr = 1 and the restored sd = Null

Step 2 Read the next119873119911bits from the code stream 119888 119887119904 and

convert the119873119911bits to decimal value denoted by119889 119873

119911119889 119873119911=

B2D(cs 119887(ptr ptr + 119873119911minus 1) ptr = ptr + 119873

119911

Step 3 If 119889 119873119911lt 2119873119911

minus 2 (a tiny-value index) then

restored index = 119889 119873119911

read the next 119904 bits denoted by sd1015840 from code stream119888 119887119904restored sd = restored sd sd1015840ptr = ptr + 119904

Step 4 If 119889 119873119911= 2119873119911

minus 1 (indicator 1) then

read the next log2(cs) bits from the code stream c bs

119889 119874119894= B2D(cs 119887(ptr ptr + log

2(cs) minus 1) ptr = ptr +

log2(cs)

restored index = 119889 119874119894

Step 5 If 119889 119873119911= 2119873119911

minus 2 (indicator 0) then

read the next119873119888bits from the code stream 119888 119887119904

119889 119866119894= B2D(cs 119887(ptr ptr + 119873

119888minus 1) ptr = ptr + 119873

119888

restored index = 119889 119866119894+ (2119873119911

minus 2)

Step 6 Repeat Steps 2ndash6 until all of the bits in the code streamare processed

Step 7 Apply the SMVQ prediction to the reconstructedSMVQ-compressed image to obtain the original VQ-compressed image

Step 8 Output the reconstructed VQ-compressed image andthe restored secret data stream

4 Experimental Results

In this section we present the experimental results forevaluating the performance of the proposed scheme in terms

6 Mathematical Problems in Engineering

(a) Airplane (b) Lena

(c) Peppers (d) Boat

(e) Sailboat (f) Tiffany

(g) House (h) Elaine

(i) Zelda (j) Gold hill

Figure 6 Test images

of the embedding rate bit rate and embedding capacityVarious gray-level images as shown in Figure 6 are employedas the training images or cover images each of which is ofsize 512 lowast 512 The codebooks of sizes 128 256 512 and1024 with codewords of 16 dimensions were trained by theLBG algorithmwith ldquoAirplanerdquo ldquoBoatrdquo ldquoLenardquo ldquoPeppersrdquo andldquoSailboatrdquo as the training images

The secret data in the experiment is in binary format 0and 1 and which are generated by a pseudorandom numbergenerator If a high-level security is required secret data canbe encrypted prior to the embedding using the well-knowncryptographicmethods such as DES or RSA In this work theembedding rate (ER) bit rate (BR) and embedding capacity(EC) as defined in (7)ndash(9) respectively are employed toevaluate the performance of various schemes In general adata-hidingmethod with a small value in BR and large valuesin EC and ER endorses a good performance and vice versa

BR =

1003817100381710038171003817embedded image1003817100381710038171003817119873 lowast119872

(7)

EC = 119904 lowast 119868119911 (8)

ER =EC

1003817100381710038171003817embedded image1003817100381710038171003817 (9)

41 Experiments on Selecting the Appropriate Parameter N119911

To obtain the best performance of the proposed schemethe parameter 119873

119911is needed to be properly determined As

mentioned when 119873119911and 119904 are set to a larger value 119868

119911

EC and BR are increased The encoding length of a VQindex is log

2(cs) bits A VQ index encoded as an 119864

119894or 119866119894

index yields an additional space with log2(cs) minus 119873

119911bits

the space can be used to hide secret data or collect moreindices Experiments are performed to choose an appropriate119873119911value which can achieve the highest embedding rate for

different codebooks and different images First of all 119904 and119873119888are set to log

2(cs) minus 119873

119911 Figure 7 shows the performance

of the proposed algorithmusing different119873119911 codebook sizes

and test images For codebooks of sizes 128 and 256 SMVQprovides accurate prediction and the most of the VQ indicesare predicted by the codewords with smaller indices in theSMVQ state codebook In these cases the highest ERs can beobtained when119873

119911is set to a small value 3 For codebooks of

sizes 512 and 1024 119873119911must be set at greater values 4 or 5

to obtain a high ER For each image larger ERs appear as119873119911

with values of 3 3 4 and 5 for codebooks of sizes 128 256512 and 1024 respectively

The experiments are performed based on the selectedvarious119873

119911with the four codebooks of sizes 128 256 512 and

1024 Experimental results for the test images are shown inTable 3 in which different columns of the simulation resultsare obtained with (cs 119873

119911 119873119888 119904) = (128 3 4 4) (256 3

5 5) (512 4 5 5) and (1024 5 5 5) respectively FromTable 3 it can be seen that the BRs of the proposed schemeare only somewhat larger than that of the original VQ BRsand the embedding capacities of the proposed scheme arevery high Moreover for most images the high embeddingcapacities are still obtained when the sizes of codebooksare increased The smooth image such as ldquoTiffanyrdquo has ahigh embedding capacity than that of the complicated imagesuch as ldquoBoatrdquo since the SMVQ prediction normally presentsgood performance for a smooth image thus many indices areencoded with 119864

119894indices With the proposed method when

the codebooks are of sizes 128 256 512 and 1024 the averageECs are 3246 3637 3848 and 3872 bitsindex respectively

Mathematical Problems in Engineering 7

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the hiding capacity is rather small Later many VQSMVQ-based schemes with indicators are proposed For exampleChang et al [10] developed a VQ-based data hiding withrecovery capability The scheme has low embedding capacityand high bit rate (BR) because of an indicator added in frontof most encoded indices and only two clusters are used tohide secret data In [11 12] a secret data hiding was proposedbased on the search-order coding compression method ofVQ indices to increase embedding capacityThe search-ordercoding selects the neighboring indices of the encoding indexto form a state codebook and then search-order code (SOC)and original index value (OIV) are employed to hide a secretbit 0 or 1 A 1-bit indicator is always employed to distinguishbetween SOC and OIV For this scheme the original coverimage can be restored completely Shie et al [13] developedan adaptive data hiding based on VQ-compressed imagein which image blocks are classified into embeddable andunembeddable blocks based on the variances and side-matchdistortions (SMDs) The embeddable blocks are employed tohide secret data while those unembeddable blocks remainunchanged It is necessary that indicator be used for the blockjudgment Although this scheme can yield a high embeddingcapacity the quality of the reconstructed image reduces as thequantity of the secret data increases In addition this schemeis a case of irreversible information hiding In [14] both VQand SMVQ are used to hide secret data in which a 1-bitindictor is always required for each index and only 1-bit secretdata can be embedded in each index when the VQ techniqueis applied Conversely more than one bit secret data canbe embedded in each index when the SMVQ technique isapplied This scheme can provide a higher hiding capacityyet it is an irreversible data-hiding scheme In [15] Yang andLin extended the scheme proposed by Chang et al [10] bydividing the VQ codebook into 2BS clusters and half of whichare used to embed secret data where BS denotes the size ofthe secret data embedded into each VQ index In the schemeboth of the VQ and SMVQ are applied to hide secret data anda 1-bit indictor is required for the index identification More-over it is a reversible data-hiding scheme In these schemesindicators are always used which causes a higher bit rate

By exploiting the special indices distribution of a SMVQ-compressed image that frequently used indices are encodedby short codes while rarely-used indices are encoded by longcodes this work proposed a novel reversible data hiding inthe SMVQ-compressed domain The scheme can increasethe hiding capacity and completely restore the cover imagefrom an embedded image after the hiding data is extractedMoreover the indices of a VQ-compressed image can berelocated close to zero by using SMVQ predications SMVQindices located around zero can be encoded using fewer bitsCompared with the previous schemes [10 12 15] in whichthe bit rates are increased because indicators are requiredin most of the indices the proposed scheme employs theSMVQ predication and thus most of the indices are encodedwithout indicator As a result high embedding capacity andlow bit rate scenarios can be achieved As documented inthe experimental results the performance of the proposedscheme is significantly superior to that of the previous works

[10 12 15] in terms of the embedding rate bit rate andembedding capacity

The rest of this paper is organized as follows Section 2presents the related works to briefly introduce the conceptsof VQ and SMVQ Section 3 describes the proposed schemein detail Section 4 gives the experimental results and per-formances comparisons of the proposed scheme with formerapproaches Finally Section 5 draws conclusions

2 Backgrounds

21 Brief Concept of the VQ VQ initially involves codebookconstruction from a set of training images the trainingimages are partitioned into nonoverlapping blocks and themost representative blocks are selected to form codebook inwhich the elements in the codebook are called codewordsIn general the LBG algorithm [16] is employed to producethe desired codebook With the generated codebook eachblock in an image is encoded with the index of the nearestcodeword such that the total storage space for an imageis minimized To measure the similarity between a block119861 = (119887

1 1198872 119887

119896) and a codeword 119862

119895= (1198881198951 1198881198952 119888

119895119896)

where 119862119895is the 119895th codeword in the codebook the squared

Euclidean distance is generally used as below

119863(119861 119862119895) =

119896

sum

119894=1

(119887119894minus 119888119895119894)2

(1)

When the encoding process is completed block 119861 issimply represented by the index min 119895 where 119862min 119895 is thenearest codeword of the block 119861 For decoding process thetable lookup is performed with the same codebook as thatused by the encoder

22 SMVQ Kim initially proposed SMVQ for image coding[17] SMVQ exploits the high correlation among neighboringblocks and side-match prediction to create a state codebookwith a size smaller than that of the original VQ codebookThe side-match prediction assumes that the values of theadjacent pixels between the neighboring blocks are equal InSMVQ blocks located in the first row and first column ofan image are encoded by traditional VQ and the remainingblocks are predicted using the corresponding state codebookFigure 1 shows an example of the relationships among anencoding block119883 its upper neighboring block 119880 and its leftneighboring block 119871 for blocks of size 4times4This work denotesthe border vector and the side vector of block 119883 as 119861(119883) =

(11988311198832119883311988341198835119883911988313) = (1198871 1198872 1198873 1198874 1198875 1198876 1198877) and

119878(119883119880 119871) = ((11988013 + 1198714)2 11988014 11988015 11988016 1198718 11987112 11987116) =

(1199041 11990421199043 1199044 1199045 1199046 1199047) The SMD between a block 119883predicted

by blocks 119880 and 119871 and a codeword cw in the codebookis measured by the squared Euclidean distance denoted asfollows

SMD (119883 cw) = 119863 (S (XU L) B (cw)) =7

sum

119894=1

(119904119894minus 119887119894)2

(2)

For each block SMVQ sorts the codebook CB accordingto SMDs and thus the first 119865

119888codewords in the sorted

Mathematical Problems in Engineering 3

U13 U14 U15 U16L4 X1 X2 X3 X4L8 X5L12 X9L16 X13

Figure 1 An example of SMVQ

codebook are picked to form the state codebook SC in whichSMDs are the smallest for block 119883 Subsequently the SMVQsearches the closest codeword from the state codebook forblock 119883 and then the index of the closest codeword is usedto encode block 119883 Because the size of the state codebook issmaller than that of the VQ codebook the size of the SMVQindex is reduced to improve the coding gain However thereis a distortion between the block decoded by the SMVQindex and the corresponding block decoded by the VQindex

3 The Proposed Scheme

A grayscale cover image 119862 of size 119873 lowast 119872 is partitionedinto nonoverlapped blocks BLKi119894=1119871 of 119908 lowast ℎpixels inwhich 119871 = (119873119908) lowast (119872ℎ) Normally the variables 119908 =

ℎ = 4 Each block of the image can be encoded using VQand thus each block is represented with an index of thenearest codeword in the codebook AVQ-compressed imagedenoted by 119868 consists of indices The size of an index isproportional to the number of codewords in the codebookA SMVQ-compressed image is generated by applying SMVQpredictions to a VQ-compressed image To reconstruct a VQ-compressed image completely the size of the state codebookand that of the VQ codebook are first set as identical Nextthe codebook is sorted according to the SMDs which denotethe differences between an encoding block and the codewordsin the VQ codebook Finally the state codebook is generatedusing all of the codewords in the sorted codebook For a VQindex the corresponding SMVQ index can be obtained usingthe state codebook and no distortion is presented between thetwo codewords decoded by the two indices

Figures 2 and 3 show histograms of the VQ-compressedPeppers image and the corresponding SMVQ-compressedimage respectively with a codebook of size 256 Basedon the SMVQ predication most indices of the SMVQ-compressed image are distributed close to zero For a SMVQ-compressed image a very small value 119873

119911binary digits (119873

119911

bits) representing unsigned integers from 0 to 2119873119911

minus 1 cancover most of the indices In the proposed scheme the 2119873119911numbers are employed as indicators orand encoding codesTable 1 shows the functions of each number of the119873

119911binary

digits (119873119911bits) A binary number among 0 and 2

119873119911minus 3

represents an encoding code and indicator Binary numbers

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250 300Index value

Num

ber o

f ind

ices

Figure 2 Histogram of the VQ-compressed Peppers image

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300Index value

Num

ber o

f ind

ices

9000

Figure 3 Histogram of the SMVQ-compressed Peppers image

Table 1 The functions of each number of the 119873119911binary digits

(119873119911bits)

Number Function0sim2119873119911 minus 3 Indicator and encoding code2119873119911 minus 2 Indicator (indicator 0)

2119873119911 minus 1 Indicator (indicator 1)

2119873119911

minus2 and 2119873119911 minus1 represent indicators denoted by indicator0 and indicator 1 respectively

31 The Encoding Process Suppose the size of a secret datasd is 119904 bits During the encoding process a SMVQ indexwith a value less than 2

119873119911minus 2 which is defined as a tiny-

value index is denoted by119864119894 and the corresponding encoded

index is denoted by 1198641015840

119894 Then 1198641015840

119894is expressed in (3) where

and and denote exponential and concatenation operatorsrespectively

1198641015840

119894= 119864119894 sd (3)

4 Mathematical Problems in Engineering

Table 2 The ranges and symbols of tiny-value middle-value andlarge-value indices

Indices Range SymbolTiny-value indices 0sim2119873119911 minus 3 119864

119894

Middle-value indices 0 + 2119873119911minus 2sim2119873119888 minus 1 + 2

119873119911 minus 2 119866

119894

Large-value indices 2119873119888 + 2119873119911minus 2simcsminus 1 119874

119894

Most of the SMVQ indices can be encoded using (3)including no indicator and the rest are encoded with the helpof indicator 0 or indicator 1 However the adoption of theindicator increases the size of the encoded result A SMVQindex with a value in between (2

119873119911minus 2) + 0 and (2119873119911 minus 2) +

2119873119888minus1 is defined as amiddle-value index Indices withmiddle

values are encoded using indicator 0 concatenated with anencoding space with119873

119888bits Amiddle-value index is denoted

by119866119894 and the corresponding encoded index is denoted by1198661015840

119894

which can be expressed in the following

1198661015840

119894= 119894119899119889119894119888119886119905119900119903 0 119866

119894minus 2119873119911

+ 2 (4)

The119873119888-bit encoding space of (4) is not to hide data but to

collect more indices Indicator 1 is employed to assist encod-ing a SMVQ index which cannot be encoded using (3) or (4)A SMVQ index with a value more than (2119873119911 minus 2) + 2

119873119888minus 1

is defined as a large-value index denoted by 119874119894 and the

corresponding encoded index is denoted by 1198741015840119894which can be

expressed in the following

1198741015840

119894= 119894119899119889119894119888119886119905119900119903 1 119874

119894 (5)

Equation (5) not only hides no data but also increases anadditional indicator with 119873

119911bits For a SMVQ-compressed

image most indices with tiny values are encoded using (3)which hides 119904 bits and generates an encoded result with119873

119911+119904

bits for each index a few indices are encoded using (4)which generates an encoded result with119873

119911+119873119888bits for each

index few indices are encoded using (5) which generates anencoded result with119873

119911+log2(cs) bits for each index where cs

indicates the size of a codebook Table 2 shows the ranges andsymbols of tiny-value middle-value and large-value indicesThe size of an embedded image can be measured with (6)where 119909 denotes the size of 119909 and119868

119911 119868119888 and 119868

119900denote the

number of indices with tiny values middle values or largevalues respectively

1003817100381710038171003817embedded image1003817100381710038171003817 = (119873119911+ 119904) lowast 119868

119911

+ (119873119911+ 119873119888) lowast 119868119888

+ (119873119911+ log2(cs)) lowast 119868

119900

(6)

In (6) (119873119911+ 119904) and (119873

119911+ 119873119888) can be controlled in less

or more than log2(cs) bits and only the third item always

requires a space with more than log2(cs) bits It is remarkable

that a larger 119904 leads to a larger embedding capacity and viceversa Moreover a larger 119873

119888increases 119868

119888 while decreasing

119868119900 and vice versa Based on the special index distribution

property of the SMVQmost indices can be encodedusing (3)

An encoding index with value Q

Is Q in between(2Nz

minus 2) + 0 and (2Nzminus 2) + 2

Ncminus 1

A middle-valueindex encodedusing (4)

A large-valueindex encodedusing (5)

A tiny-valueindex encodedusing (3)

No

No

Yes

YesQ le 2

N119911 minus 3

Figure 4 An example of the index classification and encodingschemes

in which no indicator is required and 119904-bit data are hiddenin each index and thus a high embedding capacity and alow embedded image size can be achieved in the encodingprocess The algorithm of the embedding process is summa-rized as follows and an example of the index classification andencoding schemes is shown in Figure 4

Input A grayscale cover image 119862 of size119873 lowast119872 a codebookCB of size cs a secret data stream and parameters119873

119911and119873

119888

OutputThe code stream in binary form cs 119887 and parameters119873119911and119873

119888

Step 1 Compress the cover image 119862 using VQ to obtain theVQ-compressed image 119868 of size (119873119908) lowast (119872ℎ)

Step 2 Perform SMVQ predictions with the VQ-compressedimage 119868 to create the SMVQ-compressed image 119879 of size(119873119908) lowast (119872ℎ)

Step 3 Read the next SMVQ index denoted as 119876 from theSMVQ-compressed image 119879 with the raster scan order

Step 4 If 119876 le 2119873119911 minus 3 (a tiny-value index) then we have the

following

Step 41 Read 119904-bit data sd from the secret data stream

Step 42 119876 is encoded using (3) cs 119887 = cs 119887 D2B(119876) sdwhere denotes a concatenation operation and D2B denotesa decimal to binary operation

Step 5 If119876 is in between (2119873119911 minus 2) + 0 and (2119873119911 minus 2) + 2

119873119888 minus 1

(a middle-value index) then 119876 is encoded using (4) cs119887=

cs 119887 119894119899119889119894119888119886119905119900119903 0 D2B(119876 minus 2119873119911 + 2)

Mathematical Problems in Engineering 5

(a)

00000 01100 00110 01010 00010 01111 001100 12 6 10 2 15 6

(b) (c)

0 1 2 3

4 5 1 7

8 0 1 10

127 2 1 0 1 || 15

1 || 2

1 || 6

2 || 0

3 || 127

2 || 6

2 || 2

0 || 0

2 || 0

0 || 10

2 || 3

1 || 12

0 || 6

2 || 8

2 || 5

2 || 1

Figure 5 An example of the encoding process for hiding 5-bit secretdata (a) Secret data (b) SMVQ-compressed image and (c) encodedimage

Step 6

If 119876 ge (2119873119911 minus 2) + 2

119873119888 (a large-value index) then

119876 is encoded using (5)cs 119887 = cs 119887 119894119899119889119894119888119886119905119900119903 1 D2B(119876)

Step 7 Repeat Steps 3ndash7 until all of the indices in the SMVQ-compressed image are processed

Step 8 Output the code stream cs 119887 and parameters 119873119911and

119873119888

Figure 5 shows an example of the encoding process forhiding 5-bit secret data Let 119868

0be indicator 0 and 119868

1indicator

1 where 119873119911= 2 119873

119888= 5 119904 = 5 and 119888119904 = 128 and the size

of the state codebook is identical to that of the VQ codebookFor each image (Figure 5) let 119877

1to 1198774be the indices located

in the first row and 1198775to 1198778the indices located in the second

row and so on For the SMVQ-compressed image 1198771has a

value 0 and it is less than 2119873119911 minus2 The index is encoded using(3) Secret data are (00000)

2 According to (3) the index with

119873119911bits followed by secret data (00000)

2is presented as the

encoded result for 1198771 1198774has a value of 3 and it is in between

(2119873119911

minus 2) + 0 and (2119873119911

minus 2) + 2119873119888

minus 1 An indicator 1198680 2

followed by the offset index 1 is the encoded result for119877411987713

has a value of 127 and it is more than (2119873119911

minus 2) + 2119873119888

minus 1 Anindicator 119868

1 3 followed by the index 127 is the encoded result

for 11987713 All of the encoded results are in binary form and the

length of the encoded result of each index is (119873119911+119904) (119873

119911+119873119888)

or (119873119911+ log2(cs)) bits using (3) (4) or (5) respectively

32TheExtraction andReversion Process Thegoal of the dataextraction and reversion process is to extract the embeddedsecret data from an embedded image and then recover theoriginal VQ-compressed imageThe extraction and reversionprocess is introduced as below For an encoded index 119873

119911

bits are firstly read from code stream cs 119887 and the 119873119911-bit

binary value is converted to the decimal value denoted by119889 119873119911 In the first case if 119889 119873

119911is less than 2

119873119911minus 2 the

encoded index is an 119864119894index Then the original index is

directly restored by 119889 119873119911 Next read 119904 bits from the code

stream 119888 119887119904 to reconstruct the secret data In the second case

if 119889 119873119911is 2119873119911 minus 1 the encoded index is an 119874

119894index Then

read the next log2(cs) bits and convert them into decimal

value denoted by 119889 119874119894 the original index is recovered by

119889 119874119894 In the third case if 119889 119873

119911is 2119873119911 minus 2 the encoded index

is a 119866119894index Then read the next 119873

119888bits and convert them

into decimal value 119889 119866119894 the original index is recovered by

119889119866119894

+ (2119873119885 minus 2) When the original SMVQ-compressed image

has been reconstructed SMVQ predictions can be employedto restore the original VQ-compressed image In summarythe extraction and reversion process of the proposed schemeis organized as below

Input The code stream in binary form cs 119887 and parameters119873119911and119873

119888

Output The reconstructed VQ-compressed image 1198681015840 of size119873 lowast119872 and the retrieved secret data stream

Step 1 Set ptr = 1 and the restored sd = Null

Step 2 Read the next119873119911bits from the code stream 119888 119887119904 and

convert the119873119911bits to decimal value denoted by119889 119873

119911119889 119873119911=

B2D(cs 119887(ptr ptr + 119873119911minus 1) ptr = ptr + 119873

119911

Step 3 If 119889 119873119911lt 2119873119911

minus 2 (a tiny-value index) then

restored index = 119889 119873119911

read the next 119904 bits denoted by sd1015840 from code stream119888 119887119904restored sd = restored sd sd1015840ptr = ptr + 119904

Step 4 If 119889 119873119911= 2119873119911

minus 1 (indicator 1) then

read the next log2(cs) bits from the code stream c bs

119889 119874119894= B2D(cs 119887(ptr ptr + log

2(cs) minus 1) ptr = ptr +

log2(cs)

restored index = 119889 119874119894

Step 5 If 119889 119873119911= 2119873119911

minus 2 (indicator 0) then

read the next119873119888bits from the code stream 119888 119887119904

119889 119866119894= B2D(cs 119887(ptr ptr + 119873

119888minus 1) ptr = ptr + 119873

119888

restored index = 119889 119866119894+ (2119873119911

minus 2)

Step 6 Repeat Steps 2ndash6 until all of the bits in the code streamare processed

Step 7 Apply the SMVQ prediction to the reconstructedSMVQ-compressed image to obtain the original VQ-compressed image

Step 8 Output the reconstructed VQ-compressed image andthe restored secret data stream

4 Experimental Results

In this section we present the experimental results forevaluating the performance of the proposed scheme in terms

6 Mathematical Problems in Engineering

(a) Airplane (b) Lena

(c) Peppers (d) Boat

(e) Sailboat (f) Tiffany

(g) House (h) Elaine

(i) Zelda (j) Gold hill

Figure 6 Test images

of the embedding rate bit rate and embedding capacityVarious gray-level images as shown in Figure 6 are employedas the training images or cover images each of which is ofsize 512 lowast 512 The codebooks of sizes 128 256 512 and1024 with codewords of 16 dimensions were trained by theLBG algorithmwith ldquoAirplanerdquo ldquoBoatrdquo ldquoLenardquo ldquoPeppersrdquo andldquoSailboatrdquo as the training images

The secret data in the experiment is in binary format 0and 1 and which are generated by a pseudorandom numbergenerator If a high-level security is required secret data canbe encrypted prior to the embedding using the well-knowncryptographicmethods such as DES or RSA In this work theembedding rate (ER) bit rate (BR) and embedding capacity(EC) as defined in (7)ndash(9) respectively are employed toevaluate the performance of various schemes In general adata-hidingmethod with a small value in BR and large valuesin EC and ER endorses a good performance and vice versa

BR =

1003817100381710038171003817embedded image1003817100381710038171003817119873 lowast119872

(7)

EC = 119904 lowast 119868119911 (8)

ER =EC

1003817100381710038171003817embedded image1003817100381710038171003817 (9)

41 Experiments on Selecting the Appropriate Parameter N119911

To obtain the best performance of the proposed schemethe parameter 119873

119911is needed to be properly determined As

mentioned when 119873119911and 119904 are set to a larger value 119868

119911

EC and BR are increased The encoding length of a VQindex is log

2(cs) bits A VQ index encoded as an 119864

119894or 119866119894

index yields an additional space with log2(cs) minus 119873

119911bits

the space can be used to hide secret data or collect moreindices Experiments are performed to choose an appropriate119873119911value which can achieve the highest embedding rate for

different codebooks and different images First of all 119904 and119873119888are set to log

2(cs) minus 119873

119911 Figure 7 shows the performance

of the proposed algorithmusing different119873119911 codebook sizes

and test images For codebooks of sizes 128 and 256 SMVQprovides accurate prediction and the most of the VQ indicesare predicted by the codewords with smaller indices in theSMVQ state codebook In these cases the highest ERs can beobtained when119873

119911is set to a small value 3 For codebooks of

sizes 512 and 1024 119873119911must be set at greater values 4 or 5

to obtain a high ER For each image larger ERs appear as119873119911

with values of 3 3 4 and 5 for codebooks of sizes 128 256512 and 1024 respectively

The experiments are performed based on the selectedvarious119873

119911with the four codebooks of sizes 128 256 512 and

1024 Experimental results for the test images are shown inTable 3 in which different columns of the simulation resultsare obtained with (cs 119873

119911 119873119888 119904) = (128 3 4 4) (256 3

5 5) (512 4 5 5) and (1024 5 5 5) respectively FromTable 3 it can be seen that the BRs of the proposed schemeare only somewhat larger than that of the original VQ BRsand the embedding capacities of the proposed scheme arevery high Moreover for most images the high embeddingcapacities are still obtained when the sizes of codebooksare increased The smooth image such as ldquoTiffanyrdquo has ahigh embedding capacity than that of the complicated imagesuch as ldquoBoatrdquo since the SMVQ prediction normally presentsgood performance for a smooth image thus many indices areencoded with 119864

119894indices With the proposed method when

the codebooks are of sizes 128 256 512 and 1024 the averageECs are 3246 3637 3848 and 3872 bitsindex respectively

Mathematical Problems in Engineering 7

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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Mathematical Problems in Engineering

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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

U13 U14 U15 U16L4 X1 X2 X3 X4L8 X5L12 X9L16 X13

Figure 1 An example of SMVQ

codebook are picked to form the state codebook SC in whichSMDs are the smallest for block 119883 Subsequently the SMVQsearches the closest codeword from the state codebook forblock 119883 and then the index of the closest codeword is usedto encode block 119883 Because the size of the state codebook issmaller than that of the VQ codebook the size of the SMVQindex is reduced to improve the coding gain However thereis a distortion between the block decoded by the SMVQindex and the corresponding block decoded by the VQindex

3 The Proposed Scheme

A grayscale cover image 119862 of size 119873 lowast 119872 is partitionedinto nonoverlapped blocks BLKi119894=1119871 of 119908 lowast ℎpixels inwhich 119871 = (119873119908) lowast (119872ℎ) Normally the variables 119908 =

ℎ = 4 Each block of the image can be encoded using VQand thus each block is represented with an index of thenearest codeword in the codebook AVQ-compressed imagedenoted by 119868 consists of indices The size of an index isproportional to the number of codewords in the codebookA SMVQ-compressed image is generated by applying SMVQpredictions to a VQ-compressed image To reconstruct a VQ-compressed image completely the size of the state codebookand that of the VQ codebook are first set as identical Nextthe codebook is sorted according to the SMDs which denotethe differences between an encoding block and the codewordsin the VQ codebook Finally the state codebook is generatedusing all of the codewords in the sorted codebook For a VQindex the corresponding SMVQ index can be obtained usingthe state codebook and no distortion is presented between thetwo codewords decoded by the two indices

Figures 2 and 3 show histograms of the VQ-compressedPeppers image and the corresponding SMVQ-compressedimage respectively with a codebook of size 256 Basedon the SMVQ predication most indices of the SMVQ-compressed image are distributed close to zero For a SMVQ-compressed image a very small value 119873

119911binary digits (119873

119911

bits) representing unsigned integers from 0 to 2119873119911

minus 1 cancover most of the indices In the proposed scheme the 2119873119911numbers are employed as indicators orand encoding codesTable 1 shows the functions of each number of the119873

119911binary

digits (119873119911bits) A binary number among 0 and 2

119873119911minus 3

represents an encoding code and indicator Binary numbers

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250 300Index value

Num

ber o

f ind

ices

Figure 2 Histogram of the VQ-compressed Peppers image

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300Index value

Num

ber o

f ind

ices

9000

Figure 3 Histogram of the SMVQ-compressed Peppers image

Table 1 The functions of each number of the 119873119911binary digits

(119873119911bits)

Number Function0sim2119873119911 minus 3 Indicator and encoding code2119873119911 minus 2 Indicator (indicator 0)

2119873119911 minus 1 Indicator (indicator 1)

2119873119911

minus2 and 2119873119911 minus1 represent indicators denoted by indicator0 and indicator 1 respectively

31 The Encoding Process Suppose the size of a secret datasd is 119904 bits During the encoding process a SMVQ indexwith a value less than 2

119873119911minus 2 which is defined as a tiny-

value index is denoted by119864119894 and the corresponding encoded

index is denoted by 1198641015840

119894 Then 1198641015840

119894is expressed in (3) where

and and denote exponential and concatenation operatorsrespectively

1198641015840

119894= 119864119894 sd (3)

4 Mathematical Problems in Engineering

Table 2 The ranges and symbols of tiny-value middle-value andlarge-value indices

Indices Range SymbolTiny-value indices 0sim2119873119911 minus 3 119864

119894

Middle-value indices 0 + 2119873119911minus 2sim2119873119888 minus 1 + 2

119873119911 minus 2 119866

119894

Large-value indices 2119873119888 + 2119873119911minus 2simcsminus 1 119874

119894

Most of the SMVQ indices can be encoded using (3)including no indicator and the rest are encoded with the helpof indicator 0 or indicator 1 However the adoption of theindicator increases the size of the encoded result A SMVQindex with a value in between (2

119873119911minus 2) + 0 and (2119873119911 minus 2) +

2119873119888minus1 is defined as amiddle-value index Indices withmiddle

values are encoded using indicator 0 concatenated with anencoding space with119873

119888bits Amiddle-value index is denoted

by119866119894 and the corresponding encoded index is denoted by1198661015840

119894

which can be expressed in the following

1198661015840

119894= 119894119899119889119894119888119886119905119900119903 0 119866

119894minus 2119873119911

+ 2 (4)

The119873119888-bit encoding space of (4) is not to hide data but to

collect more indices Indicator 1 is employed to assist encod-ing a SMVQ index which cannot be encoded using (3) or (4)A SMVQ index with a value more than (2119873119911 minus 2) + 2

119873119888minus 1

is defined as a large-value index denoted by 119874119894 and the

corresponding encoded index is denoted by 1198741015840119894which can be

expressed in the following

1198741015840

119894= 119894119899119889119894119888119886119905119900119903 1 119874

119894 (5)

Equation (5) not only hides no data but also increases anadditional indicator with 119873

119911bits For a SMVQ-compressed

image most indices with tiny values are encoded using (3)which hides 119904 bits and generates an encoded result with119873

119911+119904

bits for each index a few indices are encoded using (4)which generates an encoded result with119873

119911+119873119888bits for each

index few indices are encoded using (5) which generates anencoded result with119873

119911+log2(cs) bits for each index where cs

indicates the size of a codebook Table 2 shows the ranges andsymbols of tiny-value middle-value and large-value indicesThe size of an embedded image can be measured with (6)where 119909 denotes the size of 119909 and119868

119911 119868119888 and 119868

119900denote the

number of indices with tiny values middle values or largevalues respectively

1003817100381710038171003817embedded image1003817100381710038171003817 = (119873119911+ 119904) lowast 119868

119911

+ (119873119911+ 119873119888) lowast 119868119888

+ (119873119911+ log2(cs)) lowast 119868

119900

(6)

In (6) (119873119911+ 119904) and (119873

119911+ 119873119888) can be controlled in less

or more than log2(cs) bits and only the third item always

requires a space with more than log2(cs) bits It is remarkable

that a larger 119904 leads to a larger embedding capacity and viceversa Moreover a larger 119873

119888increases 119868

119888 while decreasing

119868119900 and vice versa Based on the special index distribution

property of the SMVQmost indices can be encodedusing (3)

An encoding index with value Q

Is Q in between(2Nz

minus 2) + 0 and (2Nzminus 2) + 2

Ncminus 1

A middle-valueindex encodedusing (4)

A large-valueindex encodedusing (5)

A tiny-valueindex encodedusing (3)

No

No

Yes

YesQ le 2

N119911 minus 3

Figure 4 An example of the index classification and encodingschemes

in which no indicator is required and 119904-bit data are hiddenin each index and thus a high embedding capacity and alow embedded image size can be achieved in the encodingprocess The algorithm of the embedding process is summa-rized as follows and an example of the index classification andencoding schemes is shown in Figure 4

Input A grayscale cover image 119862 of size119873 lowast119872 a codebookCB of size cs a secret data stream and parameters119873

119911and119873

119888

OutputThe code stream in binary form cs 119887 and parameters119873119911and119873

119888

Step 1 Compress the cover image 119862 using VQ to obtain theVQ-compressed image 119868 of size (119873119908) lowast (119872ℎ)

Step 2 Perform SMVQ predictions with the VQ-compressedimage 119868 to create the SMVQ-compressed image 119879 of size(119873119908) lowast (119872ℎ)

Step 3 Read the next SMVQ index denoted as 119876 from theSMVQ-compressed image 119879 with the raster scan order

Step 4 If 119876 le 2119873119911 minus 3 (a tiny-value index) then we have the

following

Step 41 Read 119904-bit data sd from the secret data stream

Step 42 119876 is encoded using (3) cs 119887 = cs 119887 D2B(119876) sdwhere denotes a concatenation operation and D2B denotesa decimal to binary operation

Step 5 If119876 is in between (2119873119911 minus 2) + 0 and (2119873119911 minus 2) + 2

119873119888 minus 1

(a middle-value index) then 119876 is encoded using (4) cs119887=

cs 119887 119894119899119889119894119888119886119905119900119903 0 D2B(119876 minus 2119873119911 + 2)

Mathematical Problems in Engineering 5

(a)

00000 01100 00110 01010 00010 01111 001100 12 6 10 2 15 6

(b) (c)

0 1 2 3

4 5 1 7

8 0 1 10

127 2 1 0 1 || 15

1 || 2

1 || 6

2 || 0

3 || 127

2 || 6

2 || 2

0 || 0

2 || 0

0 || 10

2 || 3

1 || 12

0 || 6

2 || 8

2 || 5

2 || 1

Figure 5 An example of the encoding process for hiding 5-bit secretdata (a) Secret data (b) SMVQ-compressed image and (c) encodedimage

Step 6

If 119876 ge (2119873119911 minus 2) + 2

119873119888 (a large-value index) then

119876 is encoded using (5)cs 119887 = cs 119887 119894119899119889119894119888119886119905119900119903 1 D2B(119876)

Step 7 Repeat Steps 3ndash7 until all of the indices in the SMVQ-compressed image are processed

Step 8 Output the code stream cs 119887 and parameters 119873119911and

119873119888

Figure 5 shows an example of the encoding process forhiding 5-bit secret data Let 119868

0be indicator 0 and 119868

1indicator

1 where 119873119911= 2 119873

119888= 5 119904 = 5 and 119888119904 = 128 and the size

of the state codebook is identical to that of the VQ codebookFor each image (Figure 5) let 119877

1to 1198774be the indices located

in the first row and 1198775to 1198778the indices located in the second

row and so on For the SMVQ-compressed image 1198771has a

value 0 and it is less than 2119873119911 minus2 The index is encoded using(3) Secret data are (00000)

2 According to (3) the index with

119873119911bits followed by secret data (00000)

2is presented as the

encoded result for 1198771 1198774has a value of 3 and it is in between

(2119873119911

minus 2) + 0 and (2119873119911

minus 2) + 2119873119888

minus 1 An indicator 1198680 2

followed by the offset index 1 is the encoded result for119877411987713

has a value of 127 and it is more than (2119873119911

minus 2) + 2119873119888

minus 1 Anindicator 119868

1 3 followed by the index 127 is the encoded result

for 11987713 All of the encoded results are in binary form and the

length of the encoded result of each index is (119873119911+119904) (119873

119911+119873119888)

or (119873119911+ log2(cs)) bits using (3) (4) or (5) respectively

32TheExtraction andReversion Process Thegoal of the dataextraction and reversion process is to extract the embeddedsecret data from an embedded image and then recover theoriginal VQ-compressed imageThe extraction and reversionprocess is introduced as below For an encoded index 119873

119911

bits are firstly read from code stream cs 119887 and the 119873119911-bit

binary value is converted to the decimal value denoted by119889 119873119911 In the first case if 119889 119873

119911is less than 2

119873119911minus 2 the

encoded index is an 119864119894index Then the original index is

directly restored by 119889 119873119911 Next read 119904 bits from the code

stream 119888 119887119904 to reconstruct the secret data In the second case

if 119889 119873119911is 2119873119911 minus 1 the encoded index is an 119874

119894index Then

read the next log2(cs) bits and convert them into decimal

value denoted by 119889 119874119894 the original index is recovered by

119889 119874119894 In the third case if 119889 119873

119911is 2119873119911 minus 2 the encoded index

is a 119866119894index Then read the next 119873

119888bits and convert them

into decimal value 119889 119866119894 the original index is recovered by

119889119866119894

+ (2119873119885 minus 2) When the original SMVQ-compressed image

has been reconstructed SMVQ predictions can be employedto restore the original VQ-compressed image In summarythe extraction and reversion process of the proposed schemeis organized as below

Input The code stream in binary form cs 119887 and parameters119873119911and119873

119888

Output The reconstructed VQ-compressed image 1198681015840 of size119873 lowast119872 and the retrieved secret data stream

Step 1 Set ptr = 1 and the restored sd = Null

Step 2 Read the next119873119911bits from the code stream 119888 119887119904 and

convert the119873119911bits to decimal value denoted by119889 119873

119911119889 119873119911=

B2D(cs 119887(ptr ptr + 119873119911minus 1) ptr = ptr + 119873

119911

Step 3 If 119889 119873119911lt 2119873119911

minus 2 (a tiny-value index) then

restored index = 119889 119873119911

read the next 119904 bits denoted by sd1015840 from code stream119888 119887119904restored sd = restored sd sd1015840ptr = ptr + 119904

Step 4 If 119889 119873119911= 2119873119911

minus 1 (indicator 1) then

read the next log2(cs) bits from the code stream c bs

119889 119874119894= B2D(cs 119887(ptr ptr + log

2(cs) minus 1) ptr = ptr +

log2(cs)

restored index = 119889 119874119894

Step 5 If 119889 119873119911= 2119873119911

minus 2 (indicator 0) then

read the next119873119888bits from the code stream 119888 119887119904

119889 119866119894= B2D(cs 119887(ptr ptr + 119873

119888minus 1) ptr = ptr + 119873

119888

restored index = 119889 119866119894+ (2119873119911

minus 2)

Step 6 Repeat Steps 2ndash6 until all of the bits in the code streamare processed

Step 7 Apply the SMVQ prediction to the reconstructedSMVQ-compressed image to obtain the original VQ-compressed image

Step 8 Output the reconstructed VQ-compressed image andthe restored secret data stream

4 Experimental Results

In this section we present the experimental results forevaluating the performance of the proposed scheme in terms

6 Mathematical Problems in Engineering

(a) Airplane (b) Lena

(c) Peppers (d) Boat

(e) Sailboat (f) Tiffany

(g) House (h) Elaine

(i) Zelda (j) Gold hill

Figure 6 Test images

of the embedding rate bit rate and embedding capacityVarious gray-level images as shown in Figure 6 are employedas the training images or cover images each of which is ofsize 512 lowast 512 The codebooks of sizes 128 256 512 and1024 with codewords of 16 dimensions were trained by theLBG algorithmwith ldquoAirplanerdquo ldquoBoatrdquo ldquoLenardquo ldquoPeppersrdquo andldquoSailboatrdquo as the training images

The secret data in the experiment is in binary format 0and 1 and which are generated by a pseudorandom numbergenerator If a high-level security is required secret data canbe encrypted prior to the embedding using the well-knowncryptographicmethods such as DES or RSA In this work theembedding rate (ER) bit rate (BR) and embedding capacity(EC) as defined in (7)ndash(9) respectively are employed toevaluate the performance of various schemes In general adata-hidingmethod with a small value in BR and large valuesin EC and ER endorses a good performance and vice versa

BR =

1003817100381710038171003817embedded image1003817100381710038171003817119873 lowast119872

(7)

EC = 119904 lowast 119868119911 (8)

ER =EC

1003817100381710038171003817embedded image1003817100381710038171003817 (9)

41 Experiments on Selecting the Appropriate Parameter N119911

To obtain the best performance of the proposed schemethe parameter 119873

119911is needed to be properly determined As

mentioned when 119873119911and 119904 are set to a larger value 119868

119911

EC and BR are increased The encoding length of a VQindex is log

2(cs) bits A VQ index encoded as an 119864

119894or 119866119894

index yields an additional space with log2(cs) minus 119873

119911bits

the space can be used to hide secret data or collect moreindices Experiments are performed to choose an appropriate119873119911value which can achieve the highest embedding rate for

different codebooks and different images First of all 119904 and119873119888are set to log

2(cs) minus 119873

119911 Figure 7 shows the performance

of the proposed algorithmusing different119873119911 codebook sizes

and test images For codebooks of sizes 128 and 256 SMVQprovides accurate prediction and the most of the VQ indicesare predicted by the codewords with smaller indices in theSMVQ state codebook In these cases the highest ERs can beobtained when119873

119911is set to a small value 3 For codebooks of

sizes 512 and 1024 119873119911must be set at greater values 4 or 5

to obtain a high ER For each image larger ERs appear as119873119911

with values of 3 3 4 and 5 for codebooks of sizes 128 256512 and 1024 respectively

The experiments are performed based on the selectedvarious119873

119911with the four codebooks of sizes 128 256 512 and

1024 Experimental results for the test images are shown inTable 3 in which different columns of the simulation resultsare obtained with (cs 119873

119911 119873119888 119904) = (128 3 4 4) (256 3

5 5) (512 4 5 5) and (1024 5 5 5) respectively FromTable 3 it can be seen that the BRs of the proposed schemeare only somewhat larger than that of the original VQ BRsand the embedding capacities of the proposed scheme arevery high Moreover for most images the high embeddingcapacities are still obtained when the sizes of codebooksare increased The smooth image such as ldquoTiffanyrdquo has ahigh embedding capacity than that of the complicated imagesuch as ldquoBoatrdquo since the SMVQ prediction normally presentsgood performance for a smooth image thus many indices areencoded with 119864

119894indices With the proposed method when

the codebooks are of sizes 128 256 512 and 1024 the averageECs are 3246 3637 3848 and 3872 bitsindex respectively

Mathematical Problems in Engineering 7

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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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

Table 2 The ranges and symbols of tiny-value middle-value andlarge-value indices

Indices Range SymbolTiny-value indices 0sim2119873119911 minus 3 119864

119894

Middle-value indices 0 + 2119873119911minus 2sim2119873119888 minus 1 + 2

119873119911 minus 2 119866

119894

Large-value indices 2119873119888 + 2119873119911minus 2simcsminus 1 119874

119894

Most of the SMVQ indices can be encoded using (3)including no indicator and the rest are encoded with the helpof indicator 0 or indicator 1 However the adoption of theindicator increases the size of the encoded result A SMVQindex with a value in between (2

119873119911minus 2) + 0 and (2119873119911 minus 2) +

2119873119888minus1 is defined as amiddle-value index Indices withmiddle

values are encoded using indicator 0 concatenated with anencoding space with119873

119888bits Amiddle-value index is denoted

by119866119894 and the corresponding encoded index is denoted by1198661015840

119894

which can be expressed in the following

1198661015840

119894= 119894119899119889119894119888119886119905119900119903 0 119866

119894minus 2119873119911

+ 2 (4)

The119873119888-bit encoding space of (4) is not to hide data but to

collect more indices Indicator 1 is employed to assist encod-ing a SMVQ index which cannot be encoded using (3) or (4)A SMVQ index with a value more than (2119873119911 minus 2) + 2

119873119888minus 1

is defined as a large-value index denoted by 119874119894 and the

corresponding encoded index is denoted by 1198741015840119894which can be

expressed in the following

1198741015840

119894= 119894119899119889119894119888119886119905119900119903 1 119874

119894 (5)

Equation (5) not only hides no data but also increases anadditional indicator with 119873

119911bits For a SMVQ-compressed

image most indices with tiny values are encoded using (3)which hides 119904 bits and generates an encoded result with119873

119911+119904

bits for each index a few indices are encoded using (4)which generates an encoded result with119873

119911+119873119888bits for each

index few indices are encoded using (5) which generates anencoded result with119873

119911+log2(cs) bits for each index where cs

indicates the size of a codebook Table 2 shows the ranges andsymbols of tiny-value middle-value and large-value indicesThe size of an embedded image can be measured with (6)where 119909 denotes the size of 119909 and119868

119911 119868119888 and 119868

119900denote the

number of indices with tiny values middle values or largevalues respectively

1003817100381710038171003817embedded image1003817100381710038171003817 = (119873119911+ 119904) lowast 119868

119911

+ (119873119911+ 119873119888) lowast 119868119888

+ (119873119911+ log2(cs)) lowast 119868

119900

(6)

In (6) (119873119911+ 119904) and (119873

119911+ 119873119888) can be controlled in less

or more than log2(cs) bits and only the third item always

requires a space with more than log2(cs) bits It is remarkable

that a larger 119904 leads to a larger embedding capacity and viceversa Moreover a larger 119873

119888increases 119868

119888 while decreasing

119868119900 and vice versa Based on the special index distribution

property of the SMVQmost indices can be encodedusing (3)

An encoding index with value Q

Is Q in between(2Nz

minus 2) + 0 and (2Nzminus 2) + 2

Ncminus 1

A middle-valueindex encodedusing (4)

A large-valueindex encodedusing (5)

A tiny-valueindex encodedusing (3)

No

No

Yes

YesQ le 2

N119911 minus 3

Figure 4 An example of the index classification and encodingschemes

in which no indicator is required and 119904-bit data are hiddenin each index and thus a high embedding capacity and alow embedded image size can be achieved in the encodingprocess The algorithm of the embedding process is summa-rized as follows and an example of the index classification andencoding schemes is shown in Figure 4

Input A grayscale cover image 119862 of size119873 lowast119872 a codebookCB of size cs a secret data stream and parameters119873

119911and119873

119888

OutputThe code stream in binary form cs 119887 and parameters119873119911and119873

119888

Step 1 Compress the cover image 119862 using VQ to obtain theVQ-compressed image 119868 of size (119873119908) lowast (119872ℎ)

Step 2 Perform SMVQ predictions with the VQ-compressedimage 119868 to create the SMVQ-compressed image 119879 of size(119873119908) lowast (119872ℎ)

Step 3 Read the next SMVQ index denoted as 119876 from theSMVQ-compressed image 119879 with the raster scan order

Step 4 If 119876 le 2119873119911 minus 3 (a tiny-value index) then we have the

following

Step 41 Read 119904-bit data sd from the secret data stream

Step 42 119876 is encoded using (3) cs 119887 = cs 119887 D2B(119876) sdwhere denotes a concatenation operation and D2B denotesa decimal to binary operation

Step 5 If119876 is in between (2119873119911 minus 2) + 0 and (2119873119911 minus 2) + 2

119873119888 minus 1

(a middle-value index) then 119876 is encoded using (4) cs119887=

cs 119887 119894119899119889119894119888119886119905119900119903 0 D2B(119876 minus 2119873119911 + 2)

Mathematical Problems in Engineering 5

(a)

00000 01100 00110 01010 00010 01111 001100 12 6 10 2 15 6

(b) (c)

0 1 2 3

4 5 1 7

8 0 1 10

127 2 1 0 1 || 15

1 || 2

1 || 6

2 || 0

3 || 127

2 || 6

2 || 2

0 || 0

2 || 0

0 || 10

2 || 3

1 || 12

0 || 6

2 || 8

2 || 5

2 || 1

Figure 5 An example of the encoding process for hiding 5-bit secretdata (a) Secret data (b) SMVQ-compressed image and (c) encodedimage

Step 6

If 119876 ge (2119873119911 minus 2) + 2

119873119888 (a large-value index) then

119876 is encoded using (5)cs 119887 = cs 119887 119894119899119889119894119888119886119905119900119903 1 D2B(119876)

Step 7 Repeat Steps 3ndash7 until all of the indices in the SMVQ-compressed image are processed

Step 8 Output the code stream cs 119887 and parameters 119873119911and

119873119888

Figure 5 shows an example of the encoding process forhiding 5-bit secret data Let 119868

0be indicator 0 and 119868

1indicator

1 where 119873119911= 2 119873

119888= 5 119904 = 5 and 119888119904 = 128 and the size

of the state codebook is identical to that of the VQ codebookFor each image (Figure 5) let 119877

1to 1198774be the indices located

in the first row and 1198775to 1198778the indices located in the second

row and so on For the SMVQ-compressed image 1198771has a

value 0 and it is less than 2119873119911 minus2 The index is encoded using(3) Secret data are (00000)

2 According to (3) the index with

119873119911bits followed by secret data (00000)

2is presented as the

encoded result for 1198771 1198774has a value of 3 and it is in between

(2119873119911

minus 2) + 0 and (2119873119911

minus 2) + 2119873119888

minus 1 An indicator 1198680 2

followed by the offset index 1 is the encoded result for119877411987713

has a value of 127 and it is more than (2119873119911

minus 2) + 2119873119888

minus 1 Anindicator 119868

1 3 followed by the index 127 is the encoded result

for 11987713 All of the encoded results are in binary form and the

length of the encoded result of each index is (119873119911+119904) (119873

119911+119873119888)

or (119873119911+ log2(cs)) bits using (3) (4) or (5) respectively

32TheExtraction andReversion Process Thegoal of the dataextraction and reversion process is to extract the embeddedsecret data from an embedded image and then recover theoriginal VQ-compressed imageThe extraction and reversionprocess is introduced as below For an encoded index 119873

119911

bits are firstly read from code stream cs 119887 and the 119873119911-bit

binary value is converted to the decimal value denoted by119889 119873119911 In the first case if 119889 119873

119911is less than 2

119873119911minus 2 the

encoded index is an 119864119894index Then the original index is

directly restored by 119889 119873119911 Next read 119904 bits from the code

stream 119888 119887119904 to reconstruct the secret data In the second case

if 119889 119873119911is 2119873119911 minus 1 the encoded index is an 119874

119894index Then

read the next log2(cs) bits and convert them into decimal

value denoted by 119889 119874119894 the original index is recovered by

119889 119874119894 In the third case if 119889 119873

119911is 2119873119911 minus 2 the encoded index

is a 119866119894index Then read the next 119873

119888bits and convert them

into decimal value 119889 119866119894 the original index is recovered by

119889119866119894

+ (2119873119885 minus 2) When the original SMVQ-compressed image

has been reconstructed SMVQ predictions can be employedto restore the original VQ-compressed image In summarythe extraction and reversion process of the proposed schemeis organized as below

Input The code stream in binary form cs 119887 and parameters119873119911and119873

119888

Output The reconstructed VQ-compressed image 1198681015840 of size119873 lowast119872 and the retrieved secret data stream

Step 1 Set ptr = 1 and the restored sd = Null

Step 2 Read the next119873119911bits from the code stream 119888 119887119904 and

convert the119873119911bits to decimal value denoted by119889 119873

119911119889 119873119911=

B2D(cs 119887(ptr ptr + 119873119911minus 1) ptr = ptr + 119873

119911

Step 3 If 119889 119873119911lt 2119873119911

minus 2 (a tiny-value index) then

restored index = 119889 119873119911

read the next 119904 bits denoted by sd1015840 from code stream119888 119887119904restored sd = restored sd sd1015840ptr = ptr + 119904

Step 4 If 119889 119873119911= 2119873119911

minus 1 (indicator 1) then

read the next log2(cs) bits from the code stream c bs

119889 119874119894= B2D(cs 119887(ptr ptr + log

2(cs) minus 1) ptr = ptr +

log2(cs)

restored index = 119889 119874119894

Step 5 If 119889 119873119911= 2119873119911

minus 2 (indicator 0) then

read the next119873119888bits from the code stream 119888 119887119904

119889 119866119894= B2D(cs 119887(ptr ptr + 119873

119888minus 1) ptr = ptr + 119873

119888

restored index = 119889 119866119894+ (2119873119911

minus 2)

Step 6 Repeat Steps 2ndash6 until all of the bits in the code streamare processed

Step 7 Apply the SMVQ prediction to the reconstructedSMVQ-compressed image to obtain the original VQ-compressed image

Step 8 Output the reconstructed VQ-compressed image andthe restored secret data stream

4 Experimental Results

In this section we present the experimental results forevaluating the performance of the proposed scheme in terms

6 Mathematical Problems in Engineering

(a) Airplane (b) Lena

(c) Peppers (d) Boat

(e) Sailboat (f) Tiffany

(g) House (h) Elaine

(i) Zelda (j) Gold hill

Figure 6 Test images

of the embedding rate bit rate and embedding capacityVarious gray-level images as shown in Figure 6 are employedas the training images or cover images each of which is ofsize 512 lowast 512 The codebooks of sizes 128 256 512 and1024 with codewords of 16 dimensions were trained by theLBG algorithmwith ldquoAirplanerdquo ldquoBoatrdquo ldquoLenardquo ldquoPeppersrdquo andldquoSailboatrdquo as the training images

The secret data in the experiment is in binary format 0and 1 and which are generated by a pseudorandom numbergenerator If a high-level security is required secret data canbe encrypted prior to the embedding using the well-knowncryptographicmethods such as DES or RSA In this work theembedding rate (ER) bit rate (BR) and embedding capacity(EC) as defined in (7)ndash(9) respectively are employed toevaluate the performance of various schemes In general adata-hidingmethod with a small value in BR and large valuesin EC and ER endorses a good performance and vice versa

BR =

1003817100381710038171003817embedded image1003817100381710038171003817119873 lowast119872

(7)

EC = 119904 lowast 119868119911 (8)

ER =EC

1003817100381710038171003817embedded image1003817100381710038171003817 (9)

41 Experiments on Selecting the Appropriate Parameter N119911

To obtain the best performance of the proposed schemethe parameter 119873

119911is needed to be properly determined As

mentioned when 119873119911and 119904 are set to a larger value 119868

119911

EC and BR are increased The encoding length of a VQindex is log

2(cs) bits A VQ index encoded as an 119864

119894or 119866119894

index yields an additional space with log2(cs) minus 119873

119911bits

the space can be used to hide secret data or collect moreindices Experiments are performed to choose an appropriate119873119911value which can achieve the highest embedding rate for

different codebooks and different images First of all 119904 and119873119888are set to log

2(cs) minus 119873

119911 Figure 7 shows the performance

of the proposed algorithmusing different119873119911 codebook sizes

and test images For codebooks of sizes 128 and 256 SMVQprovides accurate prediction and the most of the VQ indicesare predicted by the codewords with smaller indices in theSMVQ state codebook In these cases the highest ERs can beobtained when119873

119911is set to a small value 3 For codebooks of

sizes 512 and 1024 119873119911must be set at greater values 4 or 5

to obtain a high ER For each image larger ERs appear as119873119911

with values of 3 3 4 and 5 for codebooks of sizes 128 256512 and 1024 respectively

The experiments are performed based on the selectedvarious119873

119911with the four codebooks of sizes 128 256 512 and

1024 Experimental results for the test images are shown inTable 3 in which different columns of the simulation resultsare obtained with (cs 119873

119911 119873119888 119904) = (128 3 4 4) (256 3

5 5) (512 4 5 5) and (1024 5 5 5) respectively FromTable 3 it can be seen that the BRs of the proposed schemeare only somewhat larger than that of the original VQ BRsand the embedding capacities of the proposed scheme arevery high Moreover for most images the high embeddingcapacities are still obtained when the sizes of codebooksare increased The smooth image such as ldquoTiffanyrdquo has ahigh embedding capacity than that of the complicated imagesuch as ldquoBoatrdquo since the SMVQ prediction normally presentsgood performance for a smooth image thus many indices areencoded with 119864

119894indices With the proposed method when

the codebooks are of sizes 128 256 512 and 1024 the averageECs are 3246 3637 3848 and 3872 bitsindex respectively

Mathematical Problems in Engineering 7

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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

(a)

00000 01100 00110 01010 00010 01111 001100 12 6 10 2 15 6

(b) (c)

0 1 2 3

4 5 1 7

8 0 1 10

127 2 1 0 1 || 15

1 || 2

1 || 6

2 || 0

3 || 127

2 || 6

2 || 2

0 || 0

2 || 0

0 || 10

2 || 3

1 || 12

0 || 6

2 || 8

2 || 5

2 || 1

Figure 5 An example of the encoding process for hiding 5-bit secretdata (a) Secret data (b) SMVQ-compressed image and (c) encodedimage

Step 6

If 119876 ge (2119873119911 minus 2) + 2

119873119888 (a large-value index) then

119876 is encoded using (5)cs 119887 = cs 119887 119894119899119889119894119888119886119905119900119903 1 D2B(119876)

Step 7 Repeat Steps 3ndash7 until all of the indices in the SMVQ-compressed image are processed

Step 8 Output the code stream cs 119887 and parameters 119873119911and

119873119888

Figure 5 shows an example of the encoding process forhiding 5-bit secret data Let 119868

0be indicator 0 and 119868

1indicator

1 where 119873119911= 2 119873

119888= 5 119904 = 5 and 119888119904 = 128 and the size

of the state codebook is identical to that of the VQ codebookFor each image (Figure 5) let 119877

1to 1198774be the indices located

in the first row and 1198775to 1198778the indices located in the second

row and so on For the SMVQ-compressed image 1198771has a

value 0 and it is less than 2119873119911 minus2 The index is encoded using(3) Secret data are (00000)

2 According to (3) the index with

119873119911bits followed by secret data (00000)

2is presented as the

encoded result for 1198771 1198774has a value of 3 and it is in between

(2119873119911

minus 2) + 0 and (2119873119911

minus 2) + 2119873119888

minus 1 An indicator 1198680 2

followed by the offset index 1 is the encoded result for119877411987713

has a value of 127 and it is more than (2119873119911

minus 2) + 2119873119888

minus 1 Anindicator 119868

1 3 followed by the index 127 is the encoded result

for 11987713 All of the encoded results are in binary form and the

length of the encoded result of each index is (119873119911+119904) (119873

119911+119873119888)

or (119873119911+ log2(cs)) bits using (3) (4) or (5) respectively

32TheExtraction andReversion Process Thegoal of the dataextraction and reversion process is to extract the embeddedsecret data from an embedded image and then recover theoriginal VQ-compressed imageThe extraction and reversionprocess is introduced as below For an encoded index 119873

119911

bits are firstly read from code stream cs 119887 and the 119873119911-bit

binary value is converted to the decimal value denoted by119889 119873119911 In the first case if 119889 119873

119911is less than 2

119873119911minus 2 the

encoded index is an 119864119894index Then the original index is

directly restored by 119889 119873119911 Next read 119904 bits from the code

stream 119888 119887119904 to reconstruct the secret data In the second case

if 119889 119873119911is 2119873119911 minus 1 the encoded index is an 119874

119894index Then

read the next log2(cs) bits and convert them into decimal

value denoted by 119889 119874119894 the original index is recovered by

119889 119874119894 In the third case if 119889 119873

119911is 2119873119911 minus 2 the encoded index

is a 119866119894index Then read the next 119873

119888bits and convert them

into decimal value 119889 119866119894 the original index is recovered by

119889119866119894

+ (2119873119885 minus 2) When the original SMVQ-compressed image

has been reconstructed SMVQ predictions can be employedto restore the original VQ-compressed image In summarythe extraction and reversion process of the proposed schemeis organized as below

Input The code stream in binary form cs 119887 and parameters119873119911and119873

119888

Output The reconstructed VQ-compressed image 1198681015840 of size119873 lowast119872 and the retrieved secret data stream

Step 1 Set ptr = 1 and the restored sd = Null

Step 2 Read the next119873119911bits from the code stream 119888 119887119904 and

convert the119873119911bits to decimal value denoted by119889 119873

119911119889 119873119911=

B2D(cs 119887(ptr ptr + 119873119911minus 1) ptr = ptr + 119873

119911

Step 3 If 119889 119873119911lt 2119873119911

minus 2 (a tiny-value index) then

restored index = 119889 119873119911

read the next 119904 bits denoted by sd1015840 from code stream119888 119887119904restored sd = restored sd sd1015840ptr = ptr + 119904

Step 4 If 119889 119873119911= 2119873119911

minus 1 (indicator 1) then

read the next log2(cs) bits from the code stream c bs

119889 119874119894= B2D(cs 119887(ptr ptr + log

2(cs) minus 1) ptr = ptr +

log2(cs)

restored index = 119889 119874119894

Step 5 If 119889 119873119911= 2119873119911

minus 2 (indicator 0) then

read the next119873119888bits from the code stream 119888 119887119904

119889 119866119894= B2D(cs 119887(ptr ptr + 119873

119888minus 1) ptr = ptr + 119873

119888

restored index = 119889 119866119894+ (2119873119911

minus 2)

Step 6 Repeat Steps 2ndash6 until all of the bits in the code streamare processed

Step 7 Apply the SMVQ prediction to the reconstructedSMVQ-compressed image to obtain the original VQ-compressed image

Step 8 Output the reconstructed VQ-compressed image andthe restored secret data stream

4 Experimental Results

In this section we present the experimental results forevaluating the performance of the proposed scheme in terms

6 Mathematical Problems in Engineering

(a) Airplane (b) Lena

(c) Peppers (d) Boat

(e) Sailboat (f) Tiffany

(g) House (h) Elaine

(i) Zelda (j) Gold hill

Figure 6 Test images

of the embedding rate bit rate and embedding capacityVarious gray-level images as shown in Figure 6 are employedas the training images or cover images each of which is ofsize 512 lowast 512 The codebooks of sizes 128 256 512 and1024 with codewords of 16 dimensions were trained by theLBG algorithmwith ldquoAirplanerdquo ldquoBoatrdquo ldquoLenardquo ldquoPeppersrdquo andldquoSailboatrdquo as the training images

The secret data in the experiment is in binary format 0and 1 and which are generated by a pseudorandom numbergenerator If a high-level security is required secret data canbe encrypted prior to the embedding using the well-knowncryptographicmethods such as DES or RSA In this work theembedding rate (ER) bit rate (BR) and embedding capacity(EC) as defined in (7)ndash(9) respectively are employed toevaluate the performance of various schemes In general adata-hidingmethod with a small value in BR and large valuesin EC and ER endorses a good performance and vice versa

BR =

1003817100381710038171003817embedded image1003817100381710038171003817119873 lowast119872

(7)

EC = 119904 lowast 119868119911 (8)

ER =EC

1003817100381710038171003817embedded image1003817100381710038171003817 (9)

41 Experiments on Selecting the Appropriate Parameter N119911

To obtain the best performance of the proposed schemethe parameter 119873

119911is needed to be properly determined As

mentioned when 119873119911and 119904 are set to a larger value 119868

119911

EC and BR are increased The encoding length of a VQindex is log

2(cs) bits A VQ index encoded as an 119864

119894or 119866119894

index yields an additional space with log2(cs) minus 119873

119911bits

the space can be used to hide secret data or collect moreindices Experiments are performed to choose an appropriate119873119911value which can achieve the highest embedding rate for

different codebooks and different images First of all 119904 and119873119888are set to log

2(cs) minus 119873

119911 Figure 7 shows the performance

of the proposed algorithmusing different119873119911 codebook sizes

and test images For codebooks of sizes 128 and 256 SMVQprovides accurate prediction and the most of the VQ indicesare predicted by the codewords with smaller indices in theSMVQ state codebook In these cases the highest ERs can beobtained when119873

119911is set to a small value 3 For codebooks of

sizes 512 and 1024 119873119911must be set at greater values 4 or 5

to obtain a high ER For each image larger ERs appear as119873119911

with values of 3 3 4 and 5 for codebooks of sizes 128 256512 and 1024 respectively

The experiments are performed based on the selectedvarious119873

119911with the four codebooks of sizes 128 256 512 and

1024 Experimental results for the test images are shown inTable 3 in which different columns of the simulation resultsare obtained with (cs 119873

119911 119873119888 119904) = (128 3 4 4) (256 3

5 5) (512 4 5 5) and (1024 5 5 5) respectively FromTable 3 it can be seen that the BRs of the proposed schemeare only somewhat larger than that of the original VQ BRsand the embedding capacities of the proposed scheme arevery high Moreover for most images the high embeddingcapacities are still obtained when the sizes of codebooksare increased The smooth image such as ldquoTiffanyrdquo has ahigh embedding capacity than that of the complicated imagesuch as ldquoBoatrdquo since the SMVQ prediction normally presentsgood performance for a smooth image thus many indices areencoded with 119864

119894indices With the proposed method when

the codebooks are of sizes 128 256 512 and 1024 the averageECs are 3246 3637 3848 and 3872 bitsindex respectively

Mathematical Problems in Engineering 7

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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

(a) Airplane (b) Lena

(c) Peppers (d) Boat

(e) Sailboat (f) Tiffany

(g) House (h) Elaine

(i) Zelda (j) Gold hill

Figure 6 Test images

of the embedding rate bit rate and embedding capacityVarious gray-level images as shown in Figure 6 are employedas the training images or cover images each of which is ofsize 512 lowast 512 The codebooks of sizes 128 256 512 and1024 with codewords of 16 dimensions were trained by theLBG algorithmwith ldquoAirplanerdquo ldquoBoatrdquo ldquoLenardquo ldquoPeppersrdquo andldquoSailboatrdquo as the training images

The secret data in the experiment is in binary format 0and 1 and which are generated by a pseudorandom numbergenerator If a high-level security is required secret data canbe encrypted prior to the embedding using the well-knowncryptographicmethods such as DES or RSA In this work theembedding rate (ER) bit rate (BR) and embedding capacity(EC) as defined in (7)ndash(9) respectively are employed toevaluate the performance of various schemes In general adata-hidingmethod with a small value in BR and large valuesin EC and ER endorses a good performance and vice versa

BR =

1003817100381710038171003817embedded image1003817100381710038171003817119873 lowast119872

(7)

EC = 119904 lowast 119868119911 (8)

ER =EC

1003817100381710038171003817embedded image1003817100381710038171003817 (9)

41 Experiments on Selecting the Appropriate Parameter N119911

To obtain the best performance of the proposed schemethe parameter 119873

119911is needed to be properly determined As

mentioned when 119873119911and 119904 are set to a larger value 119868

119911

EC and BR are increased The encoding length of a VQindex is log

2(cs) bits A VQ index encoded as an 119864

119894or 119866119894

index yields an additional space with log2(cs) minus 119873

119911bits

the space can be used to hide secret data or collect moreindices Experiments are performed to choose an appropriate119873119911value which can achieve the highest embedding rate for

different codebooks and different images First of all 119904 and119873119888are set to log

2(cs) minus 119873

119911 Figure 7 shows the performance

of the proposed algorithmusing different119873119911 codebook sizes

and test images For codebooks of sizes 128 and 256 SMVQprovides accurate prediction and the most of the VQ indicesare predicted by the codewords with smaller indices in theSMVQ state codebook In these cases the highest ERs can beobtained when119873

119911is set to a small value 3 For codebooks of

sizes 512 and 1024 119873119911must be set at greater values 4 or 5

to obtain a high ER For each image larger ERs appear as119873119911

with values of 3 3 4 and 5 for codebooks of sizes 128 256512 and 1024 respectively

The experiments are performed based on the selectedvarious119873

119911with the four codebooks of sizes 128 256 512 and

1024 Experimental results for the test images are shown inTable 3 in which different columns of the simulation resultsare obtained with (cs 119873

119911 119873119888 119904) = (128 3 4 4) (256 3

5 5) (512 4 5 5) and (1024 5 5 5) respectively FromTable 3 it can be seen that the BRs of the proposed schemeare only somewhat larger than that of the original VQ BRsand the embedding capacities of the proposed scheme arevery high Moreover for most images the high embeddingcapacities are still obtained when the sizes of codebooksare increased The smooth image such as ldquoTiffanyrdquo has ahigh embedding capacity than that of the complicated imagesuch as ldquoBoatrdquo since the SMVQ prediction normally presentsgood performance for a smooth image thus many indices areencoded with 119864

119894indices With the proposed method when

the codebooks are of sizes 128 256 512 and 1024 the averageECs are 3246 3637 3848 and 3872 bitsindex respectively

Mathematical Problems in Engineering 7

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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

01

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(a)

02

03

04

05

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(b)

015

025

035

045

ER

2 3 4 5 6Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(c)

01

02

03

04

ER

2 3 4 5 6

Nz

Airplane

LenaPepper

BoatSailboat

TiffanyHouse

Elaine

(d)

Figure 7 The performance of the proposed algorithm using different 119873119911 different test images and different codebook sizes (a) codebook

size 128 (b) codebook size 256 (c) codebook size 512 and (d) codebook size 1024

In addition for applications of secret communication 119904 canbe set to a value smaller (larger) than log

2(cs) to obtain a

lower (higher) embedding capacity and consequently thecorresponding bit rate can be changed as well

42 Experiments on Comparing the Proposed Method withOther Methods To demonstrate the superiority of the pro-posed algorithm it is compared with the VQ-based schemeproposed by Chang et al [10] the VQSMVQ-based scheme

proposed by Yang and Lin [15] and SOC-based schemeproposed by Shie and Lin [12] as detailed below To ensureall of them can be fairly compared the proposed scheme isperformed with the 119873

119911 119873119888 and 119904 values are set to obtain an

embedding capacity close to that of the other schemes andthe codebook size for each comparison is set as identical

The results on the left and the middle of Table 4 showthe ERs BRs and ECs of the schemes developed by Changet al [10] and Yang and Lin [15] respectively The results

8 Mathematical Problems in Engineering

Table 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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 3 Simulation results for various test images and codebook sizes

Images

Codebook sizesVQ BR (in bpp)1280438 256050 5120563 10240625

119873119911119904

34 35 45 55ER

BR (in bpp)EC (in bits)

Airplane0454 0466 0438 03980446 0509 0574 064053144 62095 65905 66785

Lena0478 0471 0441 04030442 0505 0570 063755396 62350 65955 67390

Pepper0483 0483 0456 04120442 0504 0570 063655928 63810 68030 68685

Boat0423 0380 0354 03180446 0509 0579 065049432 50775 53720 54155

Sailboat0435 0416 0391 03440446 0509 0576 064550792 55515 59120 58110

Tiffany0492 0521 0473 04220441 0503 0569 063556860 68625 70475 70220

House0394 0370 0345 03140449 0513 0584 065446392 49780 52745 53840

Elaine0498 0484 0460 04120440 0503 0568 063457492 63775 68445 68385

on the right of Table 4 show the ERs BRs and ECs of theproposed scheme In Table 4 Chang et alrsquos scheme is VQ-based Yang and Linrsquos scheme is VQSMVQ-based and theproposed scheme is SMVQ-based First of all119873

119911119873119888 and 119904 of

the proposed scheme are set to obtain an embedding capacityclose to that of the other schemesThus when the codebooksare of sizes 128 256 512 and 1024 (119873

119911 119904 119873119888) are set at (2

3 log2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911) (33 log

2(cs) minus 119873

119911)

and (43 log2(cs) minus 119873

119911) respectively SMVQ can relocate

the histogram of the VQ indices to locations near zero Inthe proposed scheme the indices with values close to zeroare encoded using short codes without any indicator whilethe indices of values away from zero are encoded using theoriginal indexwith an indicator Based on the characteristic ofthe SMVQmost of the indices are with small values and onlya few indices have large values Thus this property leads tolow bit rate and high embedding capacity with the proposed

scheme In Chang et alrsquos [10] scheme when the numberof embeddable indices is increased the embedding capacityof each index is decreased and vice versa The satisfactoryEC cannot yield Moreover most indices are encoded by anindicator concatenated with the encoding code The lengthof an indicator or an encoding code is identical to that ofthe original VQ index The bit rate of an embedded imageis thus increased In Yang and Linrsquos [15] scheme a 1-bitindicator is always used to distinguish a SMVQ encodingfrom a VQ encoding and other indicators are required forVQ encoding leading to a high bit rate result Comparedwiththe other schemes the proposed scheme has low bit rate yethigh embedding capacity and embedding rate In particularwhen the embedding capacities of the proposed scheme arecontrolled to values close to that of the other schemes theproposed scheme presents much lower BRs even smallerthan that of the original VQ BRs for all of the test images

Mathematical Problems in Engineering 9

Table 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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 4 Performance comparisons among Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme and the proposed scheme with various testimages and codebook sizes

Images

Chang et alrsquos [10] scheme Yang and Linrsquos [15] scheme The proposed schemeCodebook sizes Codebook sizesVQ BR

128 256 512 1024 256 5121280438 2560500 5120563 10240625

119873119911119904

23 33 33 43ER

BR (in bpp)EC (in bits)

Airplane0182 0149 0124 0119 0236 0208 0326 0343 0291 02690580 0670 0760 0850 0500 0550 0364 0414 0442 050327684 26170 24608 26432 30894 29954 31077 37257 33747 35448

Lena0179 0136 0115 0094 0234 0208 0332 0348 0275 02610580 0670 0750 0830 0500 0550 0360 0410 0445 050427164 23884 22626 20502 30736 29958 31308 37410 32067 34428

Peppers0187 0150 0116 0104 0244 0216 0338 0359 0287 02640580 0670 0740 0830 0490 0540 0359 0407 0440 050128372 26284 22592 22676 31346 30576 31743 38286 33180 34677

Toys0195 0160 0136 0119 0254 0234 0397 0373 0332 03020590 0680 0770 0860 0480 0510 0348 0405 0426 048730202 28606 27460 26760 31912 31226 36300 39594 37107 38529

Gold hill0142 0109 0084 0074 0203 0158 0306 0321 0248 02320590 0680 0750 0830 0540 0640 0364 0415 0454 051421928 19420 16548 16116 28792 26502 29211 34866 29451 31287

Zelda0179 0135 0111 0088 0250 0220 0338 0375 0308 02980580 0670 0740 0820 0480 0530 0357 0401 0431 048527258 23754 21506 18842 31444 30634 31704 39462 34809 37842

The results on the left of Table 5 show the ERs BRsand ECs of the SOC-based scheme with codebooks ofsizes 256 and 512 respectively developed by Shie and Lin[12] The results on the right of the Table 5 show the ERsBRs and ECs of the proposed schemewith codebooks of sizes256 and 512 respectively In Shie and Linrsquos [12] scheme a 1-bit indicator is always used to distinguish a SOC index froman OIV index in which a SOC is encoded using 1 + 119889 bitswhile an OIV index is encoded using 1 + log

2(cs) bits The

variable 119889 indicates the length of a search path and which isset to 2 to yield a better performance The proposed schemeencodes 119864

119894and 119866

119894indices using 119873

119911and log

2(cs) minus 119873

119911bits

where 119873119911is set to 3 to obtain embedding capacities close to

that of Shie and Linrsquos method The length of a VQ index islog2(cs) in which 119873

119911or 1 + 119889 bits are used to encode an 119864

119894

index or a SOC index and the remaining bits can be used tohide secret data In Table 5 first of all the encoded result foreach119864

119894or SOC index of the two schemes is set to log

2(cs) bits

which yields a hiding space of log2(cs) minus 1 minus 2 bits for a SOC

index and log2(cs)minus119873

119911bits for a119864

119894index For each test image

with a specific codebook size the bold-number columns inTable 5 show that the EC values of the proposed scheme are

significantly greater than that of Shie and Linrsquos scheme whilelower BR values and higher ER values can be achieved withthe proposed scheme

Next the hiding space of Shie and Linrsquos [12] scheme is setto log

2(cs) bits which can achieve a high embedding space

and the length of the encoded result for a SOC index is 1+ 2 + log

2(cs) bits Because the proposed scheme has good

performance in terms of ER BR and EC 119873119911 119873119888 and the

hiding space of the proposed scheme is set to 3 log2(cs) minus119873

119911

and log2(cs) minus 1 bits the set can offer an EC close to that of

Shie and Linrsquos scheme and decrease the length of the encodedresult of an119864

119894index to 3+ log

2(cs)minus1 bits For each codebook

size and test image the unmarked columns in Table 5 showthat the EC values of the proposed scheme are still muchgreater than that of Shie and Linrsquos scheme under the highhiding space while the proposed scheme can preserve lowBRvalues and high ER values Notably the ldquoZeldardquo and ldquoGoldhillrdquo images are not the training members for proper 119873

119911

selection Yet it till can achieve superior performancesAs mentioned the proposed scheme outperforms the

VQ-based scheme proposed by Chang et al [10] theVQSMVQ-based scheme proposed by Yang and Lin [15]

10 Mathematical Problems in Engineering

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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

Table 5 Performance comparisons between Shie and Linrsquos [12] scheme and the proposed scheme with various test images and codebooksizes

Images

Shie and Linrsquos [12] scheme The proposed schemeCodebook sizesVQ BR

2560500 5120563 2560500 5120563 2560500 5120563 2560500 5120563d119904encoded bits 119873

119911119904encoded bits

258 269 2811 2912 358 369 3710 3811ER

BR (in bpp)EC (in bits)

Airplane0383 0342 0499 0438 0466 0451 0550 05220522 0591 0643 0692 0509 0571 0603 065752500 52998 84000 79497 62095 67494 86933 89992

Lena0371 0308 0485 0400 0471 0431 0555 05030524 0594 0640 0686 0505 0567 0600 064950915 48000 81464 72000 62350 64134 87290 85512

Peppers0369 0298 0484 0389 0483 0447 0567 05180524 0595 0640 0684 0504 0567 0601 065150735 46506 81176 69759 63810 66360 89334 88480

Boat0272 0194 0374 0265 0380 0305 0462 03690533 0605 0621 0664 0509 0573 0587 063138015 30744 60824 46116 50775 45756 71085 61008

Sailboat0296 0214 0402 0289 0416 0347 0499 04140531 0604 0625 0668 0509 0571 0594 063841235 33792 65976 50688 55515 51942 77721 69256

Gold hill0338 0290 0450 0380 0440 0397 0524 04670527 0596 0634 0683 0503 0566 0592 064146695 45378 74712 68067 58110 58902 81354 78536

Zelda0361 0304 0475 0395 0500 0471 0584 05430525 0595 0638 0685 0502 0564 0602 065349665 47340 79464 71010 65770 69618 92078 92824

and SOC-based scheme proposed by Shie and Lin [12] inthree aspects that is embedding rate bit rate and embeddingcapacity Thus the proposed method can be considered aseffective coding candidate for reversible data hiding applica-tion

5 Conclusions

In this paper a novel reversible data-hiding scheme forSMVQ-compressed images is proposed In most VQSMVQhiding schemes the bit rate is increased when an indicatoris required to encode an index The proposed scheme appliesthe SMVQ prediction to relocate most of the indices of a VQ-compressed image to locations close to zero Consequentlythese small-value frequently used indices are encoded usingfewer bits and no indicator is required which thus increasesthe embedding capacity and decreases the bit rate Theexperimental results demonstrate that the proposed schemecan completely extract the secret data and recover the orig-inal VQ-compressed image Moreover comparing to otherschemes the proposed scheme can achieve better efficiencyunder each codebook size for various test images

Acknowledgment

Theworkwas supported byNational ScienceCouncil Repub-lic of China Taiwan under Grant NSC100-2221-E-182-047-MY3

References

[1] J Tian ldquoReversible data embedding using a difference expan-sionrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 13 no 8 pp 890ndash896 2003

[2] Z Ni Y-Q Shi N Ansari and W Su ldquoReversible data hidingrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 16 no 3 pp 354ndash361 2006

[3] Y Hu H-K Lee and J Li ldquoDE-based reversible data hidingwith improved overflow location maprdquo IEEE Transactions onCircuits and Systems for Video Technology vol 19 no 2 pp 250ndash260 2009

[4] CH Lin andC Y Yang ldquoMultipurposewatermarking based onblind vector quantization (BVQ)rdquo Journal of InformationHidingand Multimedia Signal Processing vol 2 no 2 pp 239ndash2462011

Mathematical Problems in Engineering 11

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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

[5] S Weng S C Chu N Cai and R Zhan ldquoInvariability of meanvalue based reversible watermarkingrdquo Journal of InformationHiding andMultimedia Signal Processing vol 4 no 2 pp 90ndash982013

[6] Y Wang J F Doherty and R E Van Dyck ldquoA wavelet-basedwatermarking algorithm for ownership verification of digitalimagesrdquo IEEE Transactions on Image Processing vol 11 no 2pp 77ndash88 2002

[7] J-D Lee Y-H Chiou and J-M Guo ldquoReversible data hidingbased on histogram modification of SMVQ indicesrdquo IEEETransactions on Information Forensics and Security vol 5 no4 pp 638ndash648 2010

[8] C Y Yang C H Lin and W C Hu ldquoReversible data hidingfor high-quality images based on integer wavelet transformrdquoJournal of InformationHiding andMultimedia Signal Processingvol 3 no 2 pp 142ndash150 2012

[9] M Jo andH Kim ldquoA digital image watermarking scheme basedon vector quantisationrdquo IEICE Transactions on Information andSystems vol E85-D no 6 pp 1054ndash1056 2002

[10] C-C Chang W-C Wu and Y-C Hu ldquoLossless recovery of aVQ index table with embedded secret datardquo Journal of VisualCommunication and Image Representation vol 18 no 3 pp207ndash216 2007

[11] C-C Chang G-M Chen and M-H Lin ldquoInformation hidingbased on search-order coding for VQ indicesrdquo Pattern Recogni-tion Letters vol 25 no 11 pp 1253ndash1261 2004

[12] S-C Shie and S D Lin ldquoData hiding based on compressed VQindices of imagesrdquo Computer Standards and Interfaces vol 31no 6 pp 1143ndash1149 2009

[13] S-C Shie S D Lin and C-M Fang ldquoAdaptive data hidingbased on SMVQpredictionrdquo IEICETransactions on Informationand Systems vol E89-D no 1 pp 358ndash362 2006

[14] C-C Chen and C-C Chang ldquoHigh capacity SMVQ-basedhiding scheme using adaptive indexrdquo Signal Processing vol 90no 7 pp 2141ndash2149 2010

[15] C-H Yang and Y-C Lin ldquoReversible data hiding of a VQ indextable based on referred countsrdquo Journal of Visual Communica-tion and Image Representation vol 20 no 6 pp 399ndash407 2009

[16] Y Linde A Buzo and R M Gray ldquoAlgorithm for vector quan-tizer designrdquo IEEE Transactions on Communications Systemsvol 28 no 1 pp 84ndash95 1980

[17] T Kim ldquoSide match and overlap match vector quantizers forimagesrdquo IEEE Transactions of Image Processing vol 1 no 2 pp170ndash185 1992

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