Based on ENEE631 Spring’04
Section 11
Wavelet Based Image Coding
Spring ’04 Instructor: Min Wu
ECE Department, Univ. of Maryland, College Park
 www.ajconline.umd.edu (select ENEE631 S’04)
 minwu@eng.umd.edu
ENEE631 Digital Image Processing (Spring'04)
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
Wavelet Transform for Image Compression

ENEE631 emphasis
– Focus on conceptual aspects related to image compression
– Wavelet is also useful for denoising, enhancement, and image
analysis
– Build upon filterbank and subband coding from ENEE630
(For more in-depth info. on wavelet: wavelet course offered in Math Dept.)

K-level 1-D wavelet/subband decomposition
– Successive lowpass/highpass filtering and downsampling


on different level: capture transitions of different frequency bands
on the same level: capture transitions at different locations
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [3]
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
Successive Wavelet/Subband Decomposition
Successive lowpass/highpass filtering and downsampling


on different level: capture transitions of different frequency bands
on the same level: capture transitions at different locations
Figure from Matlab Wavelet Toolbox Documentation
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [4]
UMCP ENEE631 Slides (created by M.Wu © 2001)
Examples of 1-D Wavelet Transform
From Matlab
Wavelet Toolbox
Documentation
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [5]
UMCP ENEE631 Slides (created by M.Wu © 2004)
2-D Example
From Usevitch (IEEE
Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [7]
Subband Coding Techniques
UMCP ENEE631 Slides (created by M.Wu © 2001)

General coding approach
– Allocate different bits for coeff. in different frequency bands
– Encode different bands separately
– Example: DCT-based JPEG and early wavelet coding

Some difference between subband coding and early
wavelet coding ~ Choices of filters
– Subband filters aims at (approx.) non-overlapping freq. response
– Wavelet filters has interpretations in terms of basis and typically designed
for certain smoothness constraints
(=> will discuss more )

Shortcomings of subband coding
– Difficult to determine optimal bit allocation for low bit rate applications
– Not easy to accommodate different bit rates with a single code stream
– Difficult to encode at an exact target rate
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [8]
UMCP ENEE631 Slides (created by M.Wu © 2004)
Review: Filterbank & Multiresolution Analysis
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [9]
UMCP ENEE631 Slides (created by M.Wu © 2004)
Smoothness Conditions on Wavelet Filter
– Ensure the low band coefficients obtained by recursive filtering
can provide a smooth approximation of the original signal
From M. Vetterli’s wavelet/filter-bank paper
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [10]
Embedded Zero-Tree Wavelet Coding (EZW)
UMCP ENEE631 Slides (created by M.Wu © 2001)

“Modern” lossy wavelet coding exploits multi-resolution
and self-similar nature of wavelet decomposition
– Energy is compacted into a small number of coeff.
– Significant coeff. tend to cluster at the same spatial location in
each frequency subband

Two set of info. to code:
– Where are the significant
coefficients?
– What values are the significant
coefficients?
From Usevitch (IEEE
Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [11]
Key Concepts in EZW
UMCP ENEE631 Slides (created by M.Wu © 2001)

Parent-children relation among coeff.
– Each parent coeff at level k spatially correlates
with 4 coeff at level (k-1) of same orientation
– A coeff at lowest band correlates with 3 coeff.

Coding significance map via zero-tree
– Encode only high energy coefficients

Need to send location info.
 large overhead
– Encode “insignificance map” w/ zero-trees

Successive approximation
quantization
– Send most-significant-bits first and
gradually refine coeff. value
– “Embedded” nature of coded bit-stream

get higher fidelity image by adding extra
refining bits
ENEE631 Digital Image Processing (Spring'04)
From Usevitch (IEEE
Sig.Proc. Mag. 9/01)
Lec14 – Wavelet Coding [12]
EZW Algorithm and Example
UMCP ENEE631 Slides (created by M.Wu © 2001)



Initial threshold ~ 2 ^ floor(log2 xmax)
– Put all coeff. in dominant list
Dominant Pass (“zig-zag” across bands)
– Assign symbol to each coeff. and
entropy encode symbols




– Significant coeff.


move to subordinate list
put zero in dominant list
Subordinate Pass
– Output one bit for subordinate list


ps – positive significance
ns – negative significance
iz – isolated zero
ztr – zero-tree root
According to position in up/low half of quantization interval
From Usevitch (IEEE
Sig.Proc. Mag. 9/01)
Repeat with half threshold
– Until bit budget achieved
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [13]
Pass
From Usevitch (IEEE
Sig.Proc. Mag. 9/01)
UMCP ENEE631 Slides (created by M.Wu © 2001)
After
1st
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [14]
UMCP ENEE631 Slides (created by M.Wu © 2001)
After 2nd Pass
From Usevitch (IEEE
Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [15]
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
Beyond EZW

Cons of EZW
– Poor error resilience
– Difficult for selective spatial decoding

SPIHT (Set Partitioning in Hierarchal Trees)
– Further improvement over EZW to remove redundancy

EBCOT (Embedded Block Coding with Optimal Truncation)
– Used in JPEG 2000
– Address the shortcomings of EZW (random access, error resilience, …)
– Embedded wavelet coding in each block + bit-allocations among blocks
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [16]
UMCP ENEE631 Slides (created by M.Wu © 2004)
JPEG 2000 Image Compression Standard
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [17]
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Christopoulos
(IEEE Trans. on CE
11/00)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [18]
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
JPEG 2000: A Wavelet-Based New Standard

Targets and features
– Excellent low bit rate performance without sacrifice performance at
higher bit rate
– Progressive decoding to allow from lossy to lossless
– Region-of-interest (ROI) coding
– Error resilience

For details
– David Taubman: “High Performance Scalable Image Compression
with EBCOT”, IEEE Trans. On Image Proc, vol.9(7), 7/2000.
– JPEG2000 Tutorial by Skrodras @ IEEE Sig. Proc Magazine 9/2001
– Taubman’s book on JPEG 2000 (on library reserve)
– Links and tutorials @ http://www.jpeg.org/JPEG2000.htm
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [19]
Examples
UMCP ENEE631 Slides (created by M.Wu © 2001)
JPEG2K
vs.
JPEG
From Christopoulos
(IEEE Trans. on CE
11/00)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [20]
DCT vs. Wavelet: Which is Better?
UMCP ENEE631 Slides (created by M.Wu © 2001)

3dB improvement?
– Wavelet compression was claimed to have 3dB improvement over
DCT-based compression
– Comparison is done on JPEG Baseline

Improvement not all due to transforms
– Main contribution from better rate allocation, advanced entropy
coding, & smarter redundancy reduction via zero-tree
– DCT coder can be improved to decrease the gap
[Ref] "A comparative study of DCT- and wavelet-based image coding",
Z. Xiong, K. Ramchandran, M. Orchard, Y-Q. Zhang,
IEEE Trans. on Circuits and Systems for Video Tech.,
v.9, no.5, 8/99, pp692-695.
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [21]
UMCP ENEE631 Slides (created by M.Wu © 2004)
A Close Look at Wavelet Transform
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [22]
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
Construction of Haar functions

Unique decomposition of integer k  (p, q)
– k = 0, …, N-1 with N =
0 <= p <= n-1
– q = 0, 1 (for p=0); 1 <= q <= 2p (for p>0)
2 n,
e.g., k=0
(0,0)

k=1
(0,1)
k=2
(1,1)
k=3
(1,2)
k=4 …
(2,1) …
power of 2
k = 2p + q – 1
“reminder”
hk(x) = h p,q(x) for x  [0,1]
1
h0 ( x)  h0,0 ( x) 
for x  [0,1]
N
q- 12
q-1
 1 p/2
for p  x  p
 N 2
2
2

q- 12
q
 1 p/2
2
for p  x  p
hk ( x)  h p ,q ( x)  
2
2
N



for other x  [0,1]
0
ENEE631 Digital Image Processing (Spring'04)
x
1
Lec14 – Wavelet Coding [23]
UMCP ENEE631 Slides (created by M.Wu © 2001)
Haar Transform

Haar transform H
– Sample hk(x) at {m/N}

m = 0, …, N-1
– Real and orthogonal
– Transition at each scale p is
localized according to q






1 
8





1
1
1
1
1
1
1
1
 2
 2
0
2 2
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
0
2
1 
1 1 1
 1 
0
0
0
0 

2
2  2  2
0 0 0 0


0 0 0 0

2 2 0 0


0 0 2 2

1
1
1
Basis images of 2-D
(separable) Haar transform
– Outer product of two basis vectors
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [24]
Compare Basis Images of DCT and Haar
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)
See also: Jain’s Fig.5.2 pp136
Lec14 – Wavelet Coding [25]
Summary on Haar Transform
UMCP ENEE631 Slides (created by M.Wu © 2001)

Two major sub-operations
– Scaling captures info. at different frequencies
– Translation captures info. at different locations

Can be represented by filtering and downsampling

Relatively poor energy compaction
x
1
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [26]
UMCP ENEE631 Slides (created by M.Wu © 2001)
Orthonormal Filters

Equiv. to projecting input signal to orthonormal basis

Energy preservation property
– Convenient for quantizer design


MSE by transform domain quantizer is same as reconstruction MSE
Shortcomings: “coefficient expansion”
– Linear filtering with N-element input & M-element filter
 (N+M-1)-element output  (N+M)/2 after downsample
– Length of output per stage grows ~ undesirable for compression

Solutions to coefficient expansion
– Symmetrically extended input (circular convolution) &
Symmetric filter
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [27]
Solutions to Coefficient Expansion
UMCP ENEE631 Slides (created by M.Wu © 2001)

Circular convolution in place of linear convolution
– Periodic extension of input signal
– Problem: artifacts by large discontinuity at borders

Symmetric extension of input
– Reduce border artifacts (note the signal length doubled with symmetry)
From Usevitch (IEEE
– Problem: output at each stage may not be symmetric
Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [28]
UMCP ENEE631 Slides (created by M.Wu © 2001)
Solutions to Coefficient Expansion (cont’d)

Symmetric extension + symmetric filters
– No coefficient expansion and little artifacts
– Symmetric filter (or asymmetric filter) => “linear phase filters”
(no phase distortion except by delays)

Problem
– Only one set of linear phase filters for real FIR orthogonal wavelets
 Haar filters: (1, 1) & (1,-1)
do not give good energy compaction
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [29]
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
Biorthogonal Wavelets
UMCP ENEE631 Slides (created by M.Wu © 2001)

“Biorthogonal”
– Basis in forward and inverse transf.
are not the same but give overall
perfect reconstruction (PR)

recall EE624 PR filterbank
– No strict orthogonality for transf.
filters so energy is not preserved


But could be close to orthogonal
filters’ performance
Advantage
– Covers a much broader class of filters


including symmetric filters that eliminate coefficient expansion
Commonly used filters for compression
– 9/7 biorthogonal symmetric filter
– Efficient implementation: Lifting (see Swelden’s tuotial)
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [30]
Summary of Compression Tools

Lossless compression tools
– Entropy coding


Huffman, Arithmetic, Lempel-Ziv
run-length
– Predictive coding

reduce the dynamic range to code
– Transform


enhance energy compaction
Lossy compression tools
– Discarding and thresholding
– Quantization


( previous lectures ) Scalar quantizer
Quantize more than one component => vector quantization
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [35]
UMCP ENEE408G Slides (created by M.Wu & R.Liu © 2002)
Question for Today: Bring in Motion  Video

Capturing video
– Frame by frame => image sequence
– Image sequence: A 3-D signal



2 spatial dimensions & time dimension
continuous I( x, y, t ) => discrete I( m, n, tk )
Encode digital video
– Simplest way ~ compress each frame image individually


e.g., “motion-JPEG”
only spatial redundancy is explored and reduced
– How about temporal redundancy?

Will differential coding work well?
ENEE631 Digital Image Processing (Spring'04)
Lec14 – Wavelet Coding [40]