Image Compression-JPEG 2000

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Presentation transcript:

Image Compression-JPEG 2000 ECE 591

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

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.

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, pp692-695.

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

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).

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

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

Discrete Wavelet Transform

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); http://matlab.izmiran.ru/help/toolbox/wavelet/dwt2.html

Discrete Wavelet Transform Using matlab command DWT2, IDWT2

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

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?

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

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

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.

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

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

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

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

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

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

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 Skrodras @ 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 tutorials @ http://www.jpeg.org/JPEG2000.htm

Required Readings Usevitch’s tutorial in IEEE Sig. Proc. Magazine 9/01x http://www.cs.tau.ac.il/~amir1/WAVELET/PAPERS/lossy_wavele t_compress01.pdf  Xiong’s paper on DCT vs. Wavelet on IEEE Trans. CSVT 

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