Notes by Shufang Wu http://www.sfu.ca/~vswu Embedded Block Coding with Optimized Truncation - An Image Compression Algorithm Notes by Shufang Wu http://www.sfu.ca/~vswu

Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides ) Optimal Truncation ( 3 slides ) Embedded Block Coding ( 6 slides ) Abstract Quality Layers ( 1 slide ) Conclusion References Q & A

Overview (1-1) Abstract Quality Layers Division of subbands into code-blocks with the same size Subband samples for 3 wavelet decomposition levels What is Embedded Block Coding with Optimized Truncation (EBCOT) ? Highly scalable bit-stream Independent embedded blocks Post-compression rate-distortion (PCRD) optimization Layered bit-stream organization Input of EBCOT Subband samples generated using a wavelet transform, like EZW and SPIHT  Scalability: A key requirement driving the JPEG2000 standardization process.    qewra qewra qewra

Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides ) Optimal Truncation ( 3 slides ) Embedded Block Coding ( 6 slides ) Abstract Quality Layers ( 3 slides ) Conclusion References Q & A

Highly Scalable Compression (2-1) A Scalable Bit-stream A Highly Scalable Bit-stream Highly Scalable Compression A Scalable Bit-stream: One that may be partially discarded to obtain an efficient representation or a lower resolution version of the original image at a different bit-rate. A Highly Scalable Bit-stream: One that may be decompressed in many different ways with different results, depending on what information has been discarded. Compress Once: Decompress Many Ways Refers to the generation of a highly scalable bit-stream

Highly Scalable Compression (2-2) A Resolution Scalable Bit-stream A Distortion Scalable Bit-stream A Spatial Scalable Bit-stream (random access) Elements of a distortion & resolution scalable bit-stream Quality Progression Resolution Progression Resolution Scalable: Containing distinct subsets representing successive resolution levels. Wavelet transform is an important tool in the construction of resolution scalable bit-streams. Distortion Scalable: Containing distinct subsets successively augmenting the quality (reducing the distortion). Embedded quantization is central to the construction of distortion scalable bit-streams. Random Access: Identifying the region within each subband and hence the code-blocks which are required to correctly reconstruct the region of interest. (Not in detail)

Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides ) Optimal Truncation ( 3 slides ) Embedded Block Coding ( 6 slides ) Abstract Quality Layers ( 3 slides ) Conclusion References Q & A

Optimal Truncation (3-1) Post-Compression Rate-Distortion (PCRD) Optimization Rate distortion (R-D) optimization PCRD Problem Description Truncate each code-block bit-stream in an optimal way so as to minimize distortion subject to the bit-rate constraint. The rate-distortion algorithm is applied after all the subband samples have been compressed. Optimal selection of the truncation points, { ni }, so as to minimize overall distortion, D, subject to an overall length constraint, Lmax . D =  Dini, Lmax  L =  Lini Select ni truncated to Lini with distortion Dini for code-block Bi

Optimal Truncation (3-2) Optimal Selection of Truncation Points Simple Algorithm to Find ni in Each Code-block for Given  { ni }, which minimizes (D() + L()) for some  is optimal.(*) D cannot be reduced without also increasing L and vice-versa.  s.t. (*) and L() = Lmax, then { ni } must be an optimal solution. Discrete,  in practice the smallest  s.t. L()  Lmax. Initialization: ni = 0; Loop: For j = 1,2, , Zi; (Zi is the last truncation point) Set L = Lij  Li ni and D = Di ni  Dij ; IF D/L > , THEN ni = j. Guarantees that: Dini + Lini  Diz + Liz for all z  j. The amount by which the length is increased. The amount by which the distortion is decreased.

Optimal Truncation (3-3) Set of Feasible Truncation Points, Pi , for code-block, Bi Distortion-rate slope of a truncation point Property of the slopes Reduced Algorithm Convex Hull Interpretation Distortion Truncation Points An enumeration of points in Pi, 0 = hi0 < hi1 <  < hi|Pi-1|. Feasible Truncation Points i(hin) = (Di hin-1  Di hin)/(Li hin  Li hin-1),  n  1. i(hi0) > i(hi1) > i(hi2) >  > i(hi|Pi-1|). (Strictly decreasing.) ni = max{ hik  Pi | i(hik) >  }. Length

Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides ) Optimal Truncation ( 3 slides ) Embedded Block Coding ( 6 slides ) Abstract Quality Layers ( 3 slides ) Conclusion References Q & A

Embedded Block Coding (6-1) Bit-plane coding Encode first the most significant bits for all samples. Basics Classical context adaptive arithmetic coding (18) Bit-plane coding Fractional bit-plane coding (Passes) Sub-blocks with explicit sub-block significance coding Purpose: Ensure a sufficiently fine embedding. Assumption: Significant samples tend to be clustered.

Embedded Block Coding (6-2) Quantization Scalar quantizer (SQ) Deadzone uniform scalar quantizer Embedded deadzone quantizers Significance of A Sample A state variable, initialized to 0, transitioned to 1 when its first nonzero bit-plane is encoded A function that maps each element in a subset of the real line to a particular value in that subset. 1 x 0.5 1.5 -1 -0.5 -1.5 SQ with the interval about 0 widened ( twice as large ). Family of embedded deadzone quantizers qewra qewra qewra

Embedded Block Coding (6-3) Sub-Block Significance Coding ( 16  16 ) For each bit-plane, first encode information to identify those sub-blocks that contain one or more significant samples One way to encode is quad-tree coding Bit-Plane Coding Primitives 4 primitives: Zero coding (ZC), Run-length coding (RLC), Sign coding (SC) and Magnitude refinement (MR) How to use these primitives ? In every case, the symbol must be coded using arithmetic coder with 18 probability models Embedded Quad-tree Structure Identifying Significance Constructing Tree Entire Code-block IF the sample is not yet significant, ZC or RLC is used; IF it is significant, SC is used; ELSE (the sample is already significant) MR is used. qewra qewra 18 Probability Models: 9 for ZC; 1 for RLC; 5 for SC; 3 for MR.

Embedded Block Coding (6-4) Zero Coding (ZC)  9 To code pth bit of the quantized magnitude M, given that the sample is not yet significant ( M < 2p+1 ) Dependency ( empirical studies ) Run-Length Coding (RLC)  1 To reduce the number of symbols which must be coded Invoked in ZC when sample and its neighbors are all insignificant A group of four samples satisfying 4 conditions is encoded as a single symbol to identify whether any is significant 1) Four consecutive samples must all be insignificant; 2) Their neighbors must all be insignificant; 3) They must reside within the same sub-block; 4) The horizontal index of the first sample must be even. Formation of Significance Coding Contexts horizontal diagonal vertical Note: Neighbors outside the code-block are insignificant. qewra

Embedded Block Coding (6-5) Sign Coding (SC)  5 (concerning the signs of 4 neighbors) Used at most once when found significant in ZC or RLC Redundancy exploited ( 2 symmetry properties ) Magnitude Refinement (MR)  3 To code pth bit of the quantized magnitude M, given that M  2p+1 Dependency ( empirical studies ) 0: Both neighbors are insignificant or significant with different signs; 1: At least one neighbor is positive; -1: At least one neighbor is negative. qewra Context 0: S = 0 and 4 neighbors are insignificant; Context 1: S = 0 and at least one neighbor is significant; Context 2: S = 1. (S is a state variable that transitions from 0 to 1 after MR is first applied to the sample.)

Embedded Block Coding (6-6) Fractional Bit-Planes (4) For each bit-plane, the coding proceeds in some passes and each sample appears in exactly one of the passes Forward Significance Propagation Pass ( 1p for bit-plane p ) Skipping over all samples either insignificant or that do not have a preferred neighborhood(PN) Reverse Significance Propagation Pass ( 2p ) Identical to 1p except two points Magnitude Refinement Pass ( 3p ) Already significant and no information has been coded Normalization Pass ( 4p )  all sample not considered before qewra qewra Appearance of coding passes and quad-tree codes in each block’s embedded bit-stream qewra LL & LH: At least one horizontal neighbor is significant; HL: At least one vertical neighbor is significant; HH: At least one diagonal neighbor is significant. 1) Samples are visited in the reverse order; 2) PN is expanded: At least one of 8 neighbors is significant.

Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides ) Optimal Truncation ( 3 slides ) Embedded Block Coding ( 6 slides ) Abstract Quality Layers ( 3 slides ) Conclusion References Q & A

Abstract Quality Layers (1-1) qewra Tier 1 Embedded block coding Operates on block samples Tier 2 Coding of block contributions to each quality layer Operates on block summary information Code-block samples Compressed image Embedded block bit-streams Second Tier of Two-tiered Coding Structure Abstract Quality Layers Information Must Be Explicitly Identified Length of the segment Number of new coding passes pimax, most significant bit-plane of code-block, Bi qi , index of first layer Bi makes nonempty contribution Substantial inter-block redundancy

Conclusion State-of-the-art Compression Performance Embedded Block Coding High Scalability (3) Resolution scalability Distortion/SNR/quality/rate scalability Spatial scalability (Random access capability) Enabling Use of Post-Compression Rate-Distortion Optimization Introducing the Concept of Abstract Quality Layers

References 1. D. S. Taubman, “High performance scalable image compression with EBCOT,” IEEE Trans. Image Proc., vol. 9, pp. 1158-1170, July 2000 2. D. S. Taubman and M. W. Marcellin, JPEG2000: Image Compression Fundamentals, Standards and Practice, Kluwer Academic Publishers, 2002 3. J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Processing., vol. 41, pp. 3445-3462, December 1993 4. A. Said and W. Pearlman, “A new, fast and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, pp. 243-250, June 1996 5. S. Wu, “Embedded zerotree wavelet: an image coding algorithm”, http://www.sfu.ca/~vswu/courses/CMPT820/presentations/1/present1.ppt, June 2002