Post on 16-Jan-2016
Discrete Wavelet Transform (DWT)
Presented by
Sharon Shen
UMBC
Overview
Introduction to Video/Image Compression DWT Concepts Compression algorithms using DWT DWT vs. DCT DWT Drawbacks Future image compression standard References
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 MB storage. 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
Principles of Compression
Spatial CorrelationRedundancy among neighboring pixels
Spectral CorrelationRedundancy among different color planes
Temporal CorrelationRedundancy between adjacent frames in a
sequence of image
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 be
redundant Signal is therefore changed from input Achieving much higher compression under normal viewing
conditions no visible loss is perceived (visually lossless)
Predictive vs. Transform coding
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)
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
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
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 image coding standard, JPEG-2000, which has been based upon DWT.
Discrete Wavelet Transform
Wavelet transform decomposes a signal into a set of basis functions.
These basis functions are called wavelets Wavelets are obtained from a single prototype wavelet y(t)
called mother wavelet by dilations and shifting:
(1)
where a is the scaling parameter and b is the shifting parameter
)(1
)(, a
bt
atba
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
Discrete Wavelet Transform
Theory of WT (cont.) Wavelet transform decomposes a signal into a set of basis functions. These basis functions are called wavelets Wavelets are obtained from a single prototype wavelet y(t) called
mother wavelet by dilations and shifting:
(1)
where a is the scaling parameter and b is the shifting parameter
)(1
)(, a
bt
atba
Discrete Wavelet Transform
The 1-D wavelet transform is given by :
Discrete Wavelet Transform
The inverse 1-D wavelet transform is given by:
Discrete Wavelet Transform
Discrete wavelet transform (DWT), which transforms a discrete time signal to a discrete wavelet representation.
it converts an input series x0, x1, ..xm, into one high-pass wavelet
coefficient series and one low-pass wavelet coefficient series (of length n/2 each) given by:
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.
Discrete Wavelet Transform
Lifting schema of DWT has been recognized as a faster approach The basic principle is to factorize the polyphase
matrix 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 of banded-matrix multiplications
Discrete Wavelet Transform
Two Lifting schema:
Discrete Wavelet Transform
where si(z) (primary lifting steps) and ti(z) (dual lifting steps) are filters and K is a constant.
As this factorization is not unique, several {si(z)}, {ti(z)} and K are admissible.
Discrete Wavelet Transform
2-D DWT for Image
Discrete Wavelet Transform
Discrete Wavelet Transform
2-D DWT for Image
Discrete Wavelet Transform
Integer DWT A more efficient approach to lossless compression Whose coefficients are exactly represented by finite
precision numbers Allows for truly lossless encoding IWT can be computed starting from any real valued
wavelet 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 simpler filtering units.
Discrete Wavelet Transform
Integer DWT (cont.)
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
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
Discrete Wavelet Transform
DWT vs. DCT
Discrete Wavelet Transform
Disadvantages of DCT Only spatial correlation of the pixels inside the single
2-D block is considered and the correlation from the pixels of the neighboring blocks is neglected
Impossible to completely decorrelate the blocks at their boundaries using DCT
Undesirable blocking artifacts affect the reconstructed images or video frames. (high compression ratios or very low bit rates)
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 spatial frequencies), followed by brief periods of sharp transitions
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 blocking artifacts.
Allows good localization both in time and spatial frequency domain.
Transformation of the whole image introduces inherent scaling
Better identification of which data is relevant to human perception higher compression ratio
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 inherent scaling
Better identification of which data is relevant to human perception higher compression ratio (64:1 vs. 500:1)
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 and
the reconstructed images are virtually indistinguishable by human observers
MSE
2255
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)
PS
NR
dif
f. (
dB
s)
DWT-JPEG
Discrete Wavelet Transform
Discrete Wavelet Transform
images.
Compression ratios used for 8-bit 512x512 Lena image.
8 16 32 64 128
PSNR (dBs) performance of baseline JPEG using on Lena image.
38.00 35.50 31.70 22.00 2.00
PSNR (dBs) performance of Zero-tree coding using arithmetic coding on Lena image.
39.80 37.00 34.50 33.00 29.90
PSNR (dBs) performance of bi-orthogonal filter bank using VLC on Lena image.
35.00 34.00 32.50 28.20 26.90
PSNR (dBs) performance of bi-orthogonal filter bank using FLC on Lena image.
33.90 32.80 31.70 27.70 26.20
PSNR (dBs) performance of W6 filter bank using VLC on Lena image.
33.60 32.00 30.90 27.00 25.90
PSNR (dBs) performance of W6 filter bank using FLC on Lena image.
29.60 29.00 27.50 25.00 23.90
Comparison of image compression results using DCT and DWT
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
Discrete Wavelet Transform
Implementation Complexity The complexity of calculating wavelet transform depends on the
length of the wavelet filters, which is at least one multiplication per coefficient.
EZW, SPHIT use floating-point demands longer data length which increase the cost of computation
Lifting schemea new method compute DWT using integer arithmetic
DWT has been implemented in hardware such as ASIC and FPGA
Discrete Wavelet Transform
Resources of the ASIC used and data processing rates for DCT and DWT encoders
Type of encoders using ASIC
No. of Logic gates of the ASIC used
Amount of on chip RAM used by the encoders
Data processing rates of the encoders using ASIC
DCT 34000 128 byte 210 MSa/sec
DWT 55000 55 kbyte 150 MSa/sec
Discrete Wavelet Transform
Number of logic gates
No. of logic gates used vs. Compression technique
0
20000
40000
60000
DCT DWT
Compression technique
No
. o
f lo
gic
gate
s
used No. of logic gates
used
Discrete Wavelet Transform
Processing Rate
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
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
Discrete Wavelet Transform
References http://www.ii.metu.edu.tr/em2003/EM2003_presentations/DSD/
benderli.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 integer
wavelet 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
Discrete Wavelet Transform
THANK YOU !
Q & A