1. Image Compression-JPEG 2000
ECE 591

2. Examples: JPEG2K vs. JPEG
From Christopoulos (IEEE Trans. on CE 11/00)

3. Secret: Discrete Wavelet Transform
The discrete wavelet transform (DWT) 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.

4. 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
Difference due to DCT vs. Wavelet transform is less than 1dB
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 Technology, v.9, no.5, 8/99, pp

5. 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 called mother wavelet by dilations and shifting:
where a is the scaling parameter and b is the shifting parameter

6. Discrete Wavelet Transform
The signal is also decomposed simultaneously using a high pass filter h. The outputs giving the detail coefficients (from the high-pass filter) and approximation coefficients (from the low-pass).

7. Discrete Wavelet Transform
This decomposition is repeated to further increase the frequency resolution and the approximation coefficients decomposed with high and low pass filters and then down-sampled.
The tree is known as a filter bank

8. Downsampling and Upsampling
Let M denote the downsampling factor.
Reduce the data by picking out every Mth sample: h(k) = g(Mk). Data rate reduction occurs in this step.
Upsampling

9. Discrete Wavelet Transform

10. Matlab Comand X = imread(‘barbarb.bmp’); nbcol = size(X,1);
[cA1,cH1,cV1,cD1] = dwt2(X,'db1');
cod_X = wcodemat(X,nbcol);
cod_cA1 = wcodemat(cA1,nbcol);
cod_cH1 = wcodemat(cH1,nbcol);
cod_cV1 = wcodemat(cV1,nbcol);
cod_cD1 = wcodemat(cD1,nbcol);
dec2d = [cod_cA1, cod_cH1; cod_cV1, cod_cD1];
imagesc(dec2d);

11. Discrete Wavelet Transform
Using matlab command DWT2, IDWT2

12. Discrete Wavelet Transform
Separable transform by successively applying 1-D DWT to rows and columns
From Usevitch’s article in IEEE Sig.Proc. Mag. 9/01

13. 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 coefficients
Significant coeff. tend to cluster at the same spatial location in each frequency subband
time-freq. or space–freq. analysis
Two set of info. to code:
Where are the significant coefficients?
What values are the significant coefficients?

14. 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” zero-trees
Successive approximation quantization
Send most-significant-bits first and gradually refine coefficient value
“Embedded” nature of coded bit-stream
get higher fidelity image by adding extra refining bits

15. EZW
The EZW encoder exploits the zerotree based on the observation that wavelet coefficients decrease with scale. It assumes that there will be a very high probability that all the coefficients in a quad tree will be smaller than a certain threshold if the root is smaller than this threshold

16. EZW
The relations between wavelet coefficients in different subbands (left), how to scan them (upper right) and the result of using zerotree (lower right) symbols (T) in the coding process. An H means that the coefficient is higher than the threshold and an L means that it is below the threshold. The zerotree symbol (T) replaces the four L's in the lower left part and the L in the upper left part.

17. EZW Algorithm and Example
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
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 coefficients
Move to subordinate list
Set them to 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
threshold = initial_threshold;
do { dominant_pass(image);
subordinate_pass(image);
threshold = threshold/2; }
while (threshold > minimum_threshold

18. After 1st Pass
From Usevitch (IEEE Sig.Proc. Mag. 9/01)
Divide quantization interval of [32,64) by half: [32,48) => 40 and [48, 64) => 56

19. After 2nd Pass
From Usevitch (IEEE Sig.Proc. Mag. 9/01)

20. EZW and Beyond
Can apply DWT to entire images or larger blocks than 8x8
Symbol sequence can be entropy encoded
e.g. arithmetic coding
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

21. EZW Results
P (positive significance), N (negative significance), T (isolated zero) or Z (zero-tree root)
D1: pnztpttttztttttttptt
D2: ztnptttttttt
D3: zzzzzppnppnttnnptpttnttttttttptttptttttttttptttttttttttt
D4: zzzzzzztztznzzzzpttptpptpnptntttttptpnpppptttttptptttpnp
D5: zzzzztzzzzztpzzzttpttttnptppttptttnppnttttpnnpttpttppttt
D6: zzzttztttztttttnnttt

22. JPEG 2000 Image Compression Standard
From Christopoulos (IEEE Trans. on CE 11/00)

23. 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
JPEG2000 Tutorial by IEEE Sig. Proc Magazine 9/2001
David Taubman: “High Performance Scalable Image Compression with EBCOT”, IEEE Trans. On Image Proc, vol.9(7), 7/2000.
Taubman’s book on JPEG 2000
Links and

24. Required Readings
Usevitch’s tutorial in IEEE Sig. Proc. Magazine 9/01x
t_compress01.pdf
 Xiong’s paper on DCT vs. Wavelet on IEEE Trans. CSVT 

25. Any Questions?