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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 !
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