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    Discrete Wavelet Transform (DWT)

    Presented bySharon Shen

    UMBC

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    Overview

    Introduction to Video/Image Compression

    DWT Concepts

    Compression algorithms using DWT

    DWT vs. DCT

    DWT Drawbacks

    Future image compression standard

    References

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    Need for Compression

    Transmission and storage of uncompressed

    video would be extremely costly and

    impractical. Frame with 352x288 contains 202,752 bytes of information

    Recoding of uncompressed version of this video at 15 frames

    per second would require 3 MB. One minute180 MBstorage. One 24-hour day262 GB

    Using compression, 15 frames/second for 24 hour1.4 GB,187 days of video could be stored using the same disk space

    that uncompressed video would use in one day

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    Principles of Compression

    Spatial Correlation

    Redundancy among neighboring pixels

    Spectral Correlation

    Redundancy among different color planes

    Temporal Correlation

    Redundancy between adjacent frames in a

    sequence of image

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    Classification of Compression

    Lossless vs. Lossy Compression Lossless

    Digitally identical to the original image

    Only achieve a modest amount of compression

    Lossy

    Discards components of the signal that are known to beredundant

    Signal is therefore changed from input

    Achieving much higher compression under normal viewingconditions no visible loss is perceived (visually lossless)

    Predictive vs. Transform coding

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    Classification of Compression

    Predictive coding Information already received (in transmission) is used

    to predict future values

    Difference between predicted and actual is stored

    Easily implemented in spatial (image) domain

    Example: Differential Pulse Code Modulation(DPCM)

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    Classification of Compression

    Transform Coding

    Transform signal from spatial domain to other space

    using a well-known transform

    Encode signal in new domain (by string coefficients)

    Higher compression, in general than predictive, but

    requires more computation (apply quantization)

    Subband Coding

    Split the frequency band of a signal in various

    subbands

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    Classification of Compression

    Subband Coding (cont.)

    The filters used in subband coding are known as

    quadrature mirror filter(QMF)

    Use octave tree decomposition of an image data into

    various frequency subbands.

    The output of each decimated subbands quantized

    and encoded separately

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    Discrete Wavelet Transform

    The wavelet transform (WT) has gained widespread

    acceptance in signal processing and image compression.

    Because of their inherent multi-resolution nature,wavelet-coding schemes are especially suitable for

    applications where scalability and tolerable degradation

    are important

    Recently the JPEG committee has released its new imagecoding standard, JPEG-2000, which has been based

    upon DWT.

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    Discrete Wavelet Transform

    Wavelet transform decomposes a signal into a set ofbasis functions.

    These basis functions are called wavelets

    Wavelets are obtained from a single prototype wavelety(t) called mother wavelet bydilations andshifting:

    (1)

    wherea is the scaling parameter andb is the shiftingparameter

    )(

    1

    )(, a

    bt

    atba

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    Discrete Wavelet Transform

    Theory of WT

    The wavelet transform is computed separately for different

    segments of the time-domain signal at different frequencies.

    Multi-resolution analysis: analyzes the signal at different

    frequencies giving different resolutions

    MRA is designed to give good time resolution and poor

    frequency resolution at high frequencies and good frequency

    resolution and poor time resolution at low frequencies Good for signal having high frequency components for short

    durations and low frequency components for long duration.e.g.

    images and video frames

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    Discrete Wavelet Transform

    Theory of WT (cont.)

    Wavelet transform decomposes a signal into a set of basisfunctions.

    These basis functions are called wavelets

    Wavelets are obtained from a single prototype wavelety(t) calledmother wavelet bydilations andshifting:

    (1)

    wherea is the scaling parameter andb is the shifting

    parameter

    )(

    1

    )(, a

    bt

    atba

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    Discrete Wavelet Transform

    The 1-D wavelet transform is given by :

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    Discrete Wavelet Transform

    The inverse 1-D wavelet transform is givenby:

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    Discrete Wavelet Transform

    Discrete wavelet transform (DWT), which transforms a discrete

    time signal to a discrete wavelet representation.

    it converts an input series x0, x

    1, ..x

    m, into one high-pass wavelet

    coefficient series and one low-pass wavelet coefficient series (of

    length n/2 each) given by:

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    Discrete Wavelet Transform

    where sm

    (Z) and tm

    (Z) are called wavelet filters, K is the length of the

    filter, and i=0, ..., [n/2]-1.

    In practice, such transformation will be applied recursively on the

    low-pass series until the desired number of iterations is reached.

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    Discrete Wavelet Transform

    Lifting schema of DWT has been recognized as a faster

    approach

    The basic principle is to factorize the polyphasematrix of a wavelet filter into a sequence of

    alternating upper and lower triangular matrices and

    a diagonal matrix .

    This leads to the wavelet implementation by means ofbanded-matrix multiplications

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    Discrete Wavelet Transform

    Two Lifting schema:

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    Discrete Wavelet Transform

    wheresi(z) (primary lifting steps) andti(z) (dual lifting steps) are filters

    andKis a constant.

    As this factorization is not unique, several {si(z)}, {ti(z)} andKare

    admissible.

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    Discrete Wavelet Transform

    2-D DWT for Image

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    Discrete Wavelet Transform

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    Discrete Wavelet Transform

    2-D DWT for Image

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    Discrete Wavelet Transform

    Integer DWT

    A more efficient approach to lossless compression

    Whose coefficients are exactly represented by finiteprecision numbers

    Allows for truly lossless encoding

    IWT can be computed starting from any real valuedwavelet filter by means of a straightforward modification

    of the lifting schema Be able to reduce the number of bits for the sample

    storage (memories, registers and etc.) and to use simplerfiltering units.

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    Discrete Wavelet Transform

    Integer DWT (cont.)

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    Discrete Wavelet Transform

    Compression algorithms using DWT

    Embedded zero-tree (EZW)

    Use DWT for the decomposition of an image at each level Scans wavelet coefficients subband by subband in a zigzag

    manner

    Set partitioning in hierarchical trees (SPHIT)

    Highly refined version of EZW

    Perform better at higher compression ratio for a wide

    variety of images than EZW

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    Discrete Wavelet Transform

    Compression algorithms using DWT (cont.)

    Zero-tree entropy (ZTE)

    Quantized wavelet coefficients into wavelet trees to reduce

    the number of bits required to represent those trees

    Quantization is explicit instead of implicit, make it possible to

    adjust the quantization according to where the transform

    coefficient lies and what it represents in the frame

    Coefficient scanning, tree growing, and coding are done in

    one pass

    Coefficient scanning is a depth first traversal of each tree

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    Discrete Wavelet Transform

    DWT vs. DCT

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    Discrete Wavelet Transform

    Disadvantages of DCT

    Only spatial correlation of the pixels inside the single

    2-D block is considered and the correlation from thepixels of the neighboring blocks is neglected

    Impossible to completely decorrelate the blocks at

    their boundaries using DCT

    Undesirable blocking artifacts affect thereconstructed images or video frames. (high

    compression ratios or very low bit rates)

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    Discrete Wavelet Transform

    Disadvantages of DCT(cont.)

    Scaling as add-onadditional effort DCT function is fixedcan not be adapted to source

    data

    Does not perform efficiently for binary images (fax or

    pictures of fingerprints) characterized by large

    periods of constant amplitude (low spatialfrequencies), followed by brief periods of sharp

    transitions

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    Discrete Wavelet Transform

    Advantages of DWT over DCT

    No need to divide the input coding into non-overlapping 2-D

    blocks, it has higher compression ratios avoid blockingartifacts.

    Allows good localization both in time and spatial frequency

    domain.

    Transformation of the whole image introduces inherentscaling

    Better identification of which data is relevant to human

    perception higher compression ratio

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    Discrete Wavelet Transform

    Advantages of DWT over DCT (cont.) Higher flexibility: Wavelet function can be freely chosen

    No need to divide the input coding into non-overlapping 2-D blocks, it has higher compression ratios avoid blocking

    artifacts.

    Transformation of the whole image introduces inherentscaling

    Better identification of which data is relevant to humanperception higher compression ratio (64:1 vs. 500:1)

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    Discrete Wavelet Transform

    Performance

    Peak Signal to Noise ratio used to be a measure of image

    quality The PSNR between two images having 8 bits per pixel or sample

    in terms of decibels (dBs) is given by:

    PSNR = 10 log10

    mean square error (MSE)

    Generally when PSNR is 40 dB or greater, then the original andthe reconstructed images are virtually indistinguishable by

    human observers

    MSE

    2255

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    Discrete Wavelet Transform

    Improvement in PSNR using DWT-JEPG over DCT-JEPG at S = 4

    PSNR Difference vs. Bit rate

    0

    0.5

    1

    1.5

    2

    2.5

    0.2 0.3 0.4 0.5 0.6

    Bit rate (bps)

    PSNR

    diff.

    (dBs)

    DWT-JPEG

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    Discrete Wavelet Transform

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    Discrete Wavelet Transform

    images.

    Compression ratios used for 8-bit 512x512Lena image.

    8 16 32 64 128

    PSNR (dBs) performance of baseline JPEGusing on Lena image.

    38.00 35.50 31.70 22.00 2.00

    PSNR (dBs) performance of Zero-treecoding using arithmetic coding on Lenaimage.

    39.80 37.00 34.50 33.00 29.90

    PSNR (dBs) performance of bi-orthogonalfilter bank using VLC on Lena image.

    35.00 34.00 32.50 28.20 26.90

    PSNR (dBs) performance of bi-orthogonalfilter bank using FLC on Lena image. 33.90 32.80 31.70 27.70 26.20

    PSNR (dBs) performance of W6 filter bankusing VLC on Lena image.

    33.60 32.00 30.90 27.00 25.90

    PSNR (dBs) performance of W6 filter bankusing FLC on Lena image.

    29.60 29.00 27.50 25.00 23.90

    Comparison of image compression results using DCT and DWT

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    Discrete Wavelet Transform

    Visual Comparison

    (a) (b) (c)

    (a) Original Image256x256Pixels, 24-BitRGB (b) JPEG (DCT)Compressed with compression ratio 43:1(c) JPEG2000 (DWT)

    Compressed with compression ratio 43:1

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    Discrete Wavelet Transform

    Implementation Complexity

    The complexity of calculating wavelet transform depends on the

    length of the wavelet filters, which is at least one multiplicationper coefficient.

    EZW, SPHIT use floating-point demands longer data length

    which increase the cost of computation

    Lifting schemea new method compute DWT using integerarithmetic

    DWT has been implemented in hardware such as ASIC and

    FPGA

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    Discrete Wavelet Transform

    Resources of the ASIC used and data processing rates for DCTand DWT encoders

    Type ofencodersusingASIC

    No. ofLogicgates ofthe ASICused

    Amount of onchip RAMused by theencoders

    Dataprocessingrates of theencodersusing ASIC

    DCT 34000 128 byte 210 MSa/sec

    DWT 55000 55 kbyte 150 MSa/sec

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    Discrete Wavelet Transform

    Number of logic gates

    No. of logic gates used vs. Compression

    technique

    0

    20000

    40000

    60000

    DCT DWT

    Compression technique

    No.

    oflogicgates

    used No. of logic gates

    used

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    Discrete Wavelet Transform

    Processing Rate

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    Discrete Wavelet Transform

    Disadvantages of DWT

    The cost of computing DWT as compared to DCT

    may be higher.

    The use of larger DWT basis functions or wavelet

    filters produces blurring and ringing noise near edge

    regions in images or video frames

    Longer compression time

    Lower quality than JPEG at low compression rates

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    Discrete Wavelet Transform

    Future video/image compression Improved low bit-rate compression performance

    Improved lossless and lossy compression Improved continuous-tone and bi-level compression

    Be able to compress large images

    Use single decompression architecture

    Transmission in noisy environments

    Robustness to bit-errors

    Progressive transmission by pixel accuracy and resolution

    Protective image security

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    Discrete Wavelet Transform

    References http://www.ii.metu.edu.tr/em2003/EM2003_presentations/DSD/b

    enderli.pdf

    http://www.etro.vub.ac.be/Members/munteanu.adrian/_private/Conferences/WaveletLosslessCompression_IWSSIP1998.pdf

    http://www.vlsi.ee.upatras.gr/~sklavos/Papers02/DSP02_JPEG200.pdf

    http://www.vlsilab.polito.it/Articles/mwscas00.pdf

    M. Martina, G. Masera , A novel VLSI architecture for integerwavelet transform via lifting scheme, Internal report, VLSI Lab.,Politecnico diTor i no, Jan. 2000, unpublished.

    http://www.ee.vt.edu/~ha/research/publications/islped01.pdf

    http://www.ii.metu.edu.tr/em2003/EM2003_presentations/DSD/benderli.pdfhttp://www.ii.metu.edu.tr/em2003/EM2003_presentations/DSD/benderli.pdfhttp://www.etro.vub.ac.be/Members/munteanu.adrian/_private/Conferences/WaveletLosslessCompression_IWSSIP1998.pdfhttp://www.etro.vub.ac.be/Members/munteanu.adrian/_private/Conferences/WaveletLosslessCompression_IWSSIP1998.pdfhttp://www.vlsi.ee.upatras.gr/~sklavos/Papers02/DSP02_JPEG200.pdfhttp://www.vlsi.ee.upatras.gr/~sklavos/Papers02/DSP02_JPEG200.pdfhttp://www.vlsilab.polito.it/Articles/mwscas00.pdfhttp://www.ee.vt.edu/~ha/research/publications/islped01.pdfhttp://www.ee.vt.edu/~ha/research/publications/islped01.pdfhttp://www.vlsilab.polito.it/Articles/mwscas00.pdfhttp://www.vlsi.ee.upatras.gr/~sklavos/Papers02/DSP02_JPEG200.pdfhttp://www.vlsi.ee.upatras.gr/~sklavos/Papers02/DSP02_JPEG200.pdfhttp://www.etro.vub.ac.be/Members/munteanu.adrian/_private/Conferences/WaveletLosslessCompression_IWSSIP1998.pdfhttp://www.etro.vub.ac.be/Members/munteanu.adrian/_private/Conferences/WaveletLosslessCompression_IWSSIP1998.pdfhttp://www.ii.metu.edu.tr/em2003/EM2003_presentations/DSD/benderli.pdfhttp://www.ii.metu.edu.tr/em2003/EM2003_presentations/DSD/benderli.pdf
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    Discrete Wavelet Transform

    THANK YOU !

    Q & A