1. Wavelet Based Image Coding
12/9/2018
Based on ENEE631 Spring’04 Section 11
Wavelet Based Image Coding
1st Lecture:
Course logistics, etc. – 7 slides, about 15min
Why img. proc. and why digital – 3 slides, about 8min
Examples – 20min
What’s an img? math. modeling of img, sampling and quantization – 5 slides, 15min
10min – fill out questionnaire
Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland, College Park
(select ENEE631 S’04)

2. 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
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
ENEE631 Digital Image Processing (Spring'04)

3. Successive Wavelet/Subband Decomposition
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
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)

4. Examples of 1-D Wavelet Transform
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Matlab Wavelet Toolbox Documentation
ENEE631 Digital Image Processing (Spring'04)

5. UMCP ENEE631 Slides (created by M.Wu © 2004)
2-D Example
UMCP ENEE631 Slides (created by M.Wu © 2004)
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)

6. Subband Coding Techniques
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
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

7. Review: Filterbank & Multiresolution Analysis
UMCP ENEE631 Slides (created by M.Wu © 2004)
ENEE631 Digital Image Processing (Spring'04)

8. Smoothness Conditions on Wavelet Filter
Ensure the low band coefficients obtained by recursive filtering can provide a smooth approximation of the original signal
UMCP ENEE631 Slides (created by M.Wu © 2004)
From M. Vetterli’s wavelet/filter-bank paper
ENEE631 Digital Image Processing (Spring'04)

9. Embedded Zero-Tree Wavelet Coding (EZW)
“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?
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)

10. UMCP ENEE631 Slides (created by M.Wu © 2001)
Key Concepts in EZW
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
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)

11. EZW Algorithm and Example
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
ps – positive significance
ns – negative significance
iz – isolated zero
ztr – zero-tree root
Significant coeff.
move to subordinate list
put zero in dominant list
Subordinate Pass
Output one bit for subordinate list
According to position in up/low half of quantization interval
Repeat with half threshold
Until bit budget achieved
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)

12. UMCP ENEE631 Slides (created by M.Wu © 2001)
12/9/2018
After 1st Pass
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
UMCP ENEE631 Slides (created by M.Wu © 2001)
Divide quantization interval of [32,64) by half: [32,48) => 40 and [48, 64) => 56
ENEE631 Digital Image Processing (Spring'04)

13. UMCP ENEE631 Slides (created by M.Wu © 2001)
12/9/2018
After 2nd Pass
UMCP ENEE631 Slides (created by M.Wu © 2001)
[16,24) => 20, [24, 32) => 28; [32,40) => 36, [40,48) => 44; [48, 56) => 52, [56, 64) => 60
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
ENEE631 Digital Image Processing (Spring'04)

14. 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
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
ENEE631 Digital Image Processing (Spring'04)

15. JPEG 2000 Image Compression Standard
UMCP ENEE631 Slides (created by M.Wu © 2004)
ENEE631 Digital Image Processing (Spring'04)

16. UMCP ENEE631 Slides (created by M.Wu © 2001)
From Christopoulos (IEEE Trans. on CE 11/00)
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

17. JPEG 2000: A Wavelet-Based New Standard
12/9/2018
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 IEEE Sig. Proc Magazine 9/2001
Taubman’s book on JPEG 2000 (on library reserve)
Links and
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
Ref: see also JPEG 2000 Tutorial by Christopolous et IEEE Trans. on consumer electronics 11/00
ENEE631 Digital Image Processing (Spring'04)

18. UMCP ENEE631 Slides (created by M.Wu © 2001)
Examples JPEG2K vs. JPEG
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Christopoulos (IEEE Trans. on CE 11/00)
ENEE631 Digital Image Processing (Spring'04)

19. DCT vs. Wavelet: Which is Better?
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, pp
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

20. A Close Look at Wavelet Transform
UMCP ENEE631 Slides (created by M.Wu © 2004)
ENEE631 Digital Image Processing (Spring'04)

21. Construction of Haar functions
Unique decomposition of integer k  (p, q)
k = 0, …, N-1 with N = 2n, 0 <= p <= n-1
q = 0, 1 (for p=0); 1 <= q <= 2p (for p>0)
e.g., k= k= k= k= k=4 …
(0,0) (0,1) (1,1) (1,2) (2,1) …
hk(x) = h p,q(x) for x  [0,1]
k = 2p + q – 1
“reminder”
power of 2
UMCP ENEE631 Slides (created by M.Wu © 2001/2004) 1 x
ENEE631 Digital Image Processing (Spring'04)

22. 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
Basis images of 2-D (separable) Haar transform
Outer product of two basis vectors
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

23. Compare Basis Images of DCT and Haar
UMCP ENEE631 Slides (created by M.Wu © 2001)
See also: Jain’s Fig.5.2 pp136
ENEE631 Digital Image Processing (Spring'04)

24. Summary on Haar Transform
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
UMCP ENEE631 Slides (created by M.Wu © 2001) 1 x
ENEE631 Digital Image Processing (Spring'04)

25. 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
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

26. Solutions to Coefficient Expansion
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)
Problem: output at each stage may not be symmetric
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

27. 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
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

28. Biorthogonal Wavelets
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
Biorthogonal Wavelets
“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)
UMCP ENEE631 Slides (created by M.Wu © 2001)
ENEE631 Digital Image Processing (Spring'04)

29. 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)

30. 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?
UMCP ENEE408G Slides (created by M.Wu & R.Liu © 2002)
ENEE631 Digital Image Processing (Spring'04)