# Wavelet Transform (WT) & JPEG-2000

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1 Chapter 8 Wavelet Transform (WT) & JPEG A Review of WT Wave vs. Wavelet [castleman] Figure 8.1 Sinusoidal waves (top two) and wavelets (bottom two) (horizontal axis: time; vertical axis: magnitude) 1

2 Wave: Does not have compact support (extends to infinity) Transient signal: Have compact support (non-zero only in a short interval) Many image features (e.g., edges) highly localized in spatial position Trends vs. Anomaly [shapiro 1993] Anomaly, burst, transient signal Non-Statioanry Signal Analysis Stationary signal: properties not evolve in time FT is suitable Non-Stationary signal: Time-Frequency Analysis 2

3 Time-Frequency Analysis 2-D time-frequency space Can be derived from Figure 8.1 Preceding WT Started with Gabor s windowed FT short-time Fourier transform (STFT) Another approach: WT STFT vs. WT [rioul 1991] STFT: Resolution in time and frequency cannot be arbitrarily small Due to Heisenberg inequality: t f 1/(4π ) Once window is chosen: t and f fixed Figure 8.2 (left) 3

4 Meaning anomaly (burst) and trend cannot be analyzed with good resolution ( t or f ) simultaneously Frequency Frequency Time Time STFT Wavelet Transform Figure 8.2 Comparison of STFT and WT in the time-frequency plane WT: Constant relative bandwidth (const. Q): f / f = constant 4

5 Meaning: t as f ( f ), and f as f as f, obtain high time resolution as f, obtain high freq. resolution Figure 8.2 (right) Constant Bandwidth Relative Constant Bandwidth F 2F 3F 4F F 2F 4F Figure 8.3 Constant bandwidth analysis (for FT) and relative constant bandwidth analysis (for WT) Example [castleman] 5

6 A two tone burst corrupted by random noise (Figure 8.4(a)) castlman figure (a) FT: not easy to be interpreted (Figure 8.4(b)) In particular, phase spectrum castlman figure (b) 6

7 Filter Bank Theory (WT) castleman Figure 4-12 Figure 8.5 Implementation of a bandpass filter bank H ( S) = 1 x) f ( x) i i g = i i 7

8 fig8.6 castleman Figure Figure 8.6 Smooth bandpass filters: (a) impulse responses; (b) transfer functions castleman Figure Fig.8.7 Smooth bandpass filter bank output 8

9 8.1.6 Unification of Several Techniques Filter Bank Analysis Pyramid Coding Subband Coding Three Types of WT DWT CWT (Continuous WT) Wavelet series expansion DWT (Discrete WT) Most closely resembles unitary transforms Most useful in image compression Given a set of orthonormal basis fct. s DWT just like unitary transform 9 Orthonormal wavelets with compact support (by Daubechies):

10 { ( )} {2 j/2 (2 j rψ x = rψ x k)} j,k: integers compact support: [0,2r-1] shift: k dilation (scaling): 2 j a= a= Figure 8.8: Scaling (A Morlet wavelet with a=1 (top) and a=6 (bottom)) N-point signal N coefficients N N image N 2 coefficients 8.2 DWT for Image Compression 10

11 8.2.1 Block Diagram Similar to the other TC Input image 2-D Wavelet transform for image decomposition Quantization Coding of quantized coefficients Bitstream Figure 8.9: Block diagram of image coding with DWT Image Decomposition Scale 1 (Figure 8.10 (a)): 4 subbands: LL, HL, LH, HH 1 1 Each coeff. a 2 2 area in the original image Low frequencies: 0< ω < π / 2 High frequencies: 11 Scale 2 (Figure 8.8 (b)): 1 1 π / 2< ω < π 4 subbands: LL, HL, LH, HH

12 Each coeff. a 2 2 area in scale 1 image Low Frequency: 0< ω < π / 4 High frequencies: π / 4< ω < π / 2 LL 1 HL 1 LL 2 LH 2 HL 2 HH 2 HL 1 LH 1 HH 1 LH 1 HH 1 Figure D DWT: (left) scale 1, and (right) scale 2 decomposition. At a coarser scale, coefficients represent a larger spatial area of the image but a narrow band of frequencies. Parent Children 12

13 Descendants: corresponding coeff. at finer scales Ancestors: corresponding coeff. at coarser scales LL 3 HL 3 LL 3 HL 3 LH 3 HH 3 HL 2 HL 1 LH 3 HH 3 HL 2 HL 1 LH 2 HH 2 LH 2 HH 2 LH 1 HH 1 LH 1 HH 1 Figure 8.11 (a) Parent-children dependencies of subbands, arrow points from the subband of parents to the subband of children. (b) The scanning order of the subbands for encoding the significance map. 13 Feature 1: Energy distribution similar to other TC: Concentrated in low frequencies

14 Feature 2: Spatial self-similarity across subbands DWT Image Compression Thresholding (deterministic) many zeros, data compression Differences from DCT Technique In conventional TC Anomaly (edge) many nonzero coeff. TC allocates too many bits to trend Insignificant energy few bits left to anomalies Problem at Very Low Bit-rate Coding: block artifacts 14 DWT Trends & anomalies infor. available

15 Major difficulty: fine detail coefficients associated with anomalies the largest no. of coeff. Problem: how to efficiently represent position information? Successful techniques Embedded zerotree wavelet (EZW) [shapiro 1993] Set partitioning in hierarchical trees (SPIHT) [said 1996] 8.3 EZW Image Coding [shapiro 1993] Embedded Coding

16 Having all lower bit rate codes of the same image embedded at the beginning of the bit stream. Bits are generated in order of importance. Encoder can terminate encoding at any point, allowing a target rate to be met exactly Suitable for scalability applications Zerotree of DWT Coefficients Significance map: Binary decision as to a pixel = 0 or not Total encoding cost = cost of encoding singnificance map + cost of encoding nonzero values 16 Target rate more zero coeff. i.e., Cost of encoding position of nonzero coeff. An element of zerotree:

17 A coeff.: itself and all of its descendants are insignificant w.r.t. thereshold T Zerotree root: An element of zerotree, & not a descendant of a zero element at a coarser scale Isolated zero: Insignificant, but has some significant descendant Significance map can be efficiently represented as a string of four symbols: Zerotree root Isolated zero Positive significant coeff. Negative significant coeff. 17 Figure 8.12 Flow chart for encoding a coefficient of the significance map Shapiro Figure 6

18 8.3.3 Comparison DCT coding Run-length (RLC) [within the same scale] End of block (EOB) [within the same scale] EZW coding 18 More efficient due to: Using self-similarity across different scales [e.g., zerotree root] Higher quality of reconstructed image:

19 Due to more efficient in position encoding No possibility that a significant coeff. be obscured by a statistical energy measure Experimental results reported Barbara image: 2.4 db better for same bit rate 0.12 bpp savings for the same PSNR Wavelet Transform for JPEG-2000 DCT: core technique of image decomposition for most standards Things change recently:

20 MPEG-4: adopt WT for still image coding [mpeg4]. JPEG-2000: considering using WT as its core technique [jpeg2000]. Reason: WT provides: excellent coding efficiency, especially for VLBC wonderful spatial and quality scalable functionality 20 JPEG-2000 also aims at: accessing, manipulating, editing in compressed domain. Requirements of JPEG-2000 include: Low bit-rate compression performance:

21 Lower than 0.25 bit/pixel for highly detailed gray level images JPEG fails for the case Primary feature of JPEG Lossless and lossy compression Large images: JPEG does not allow for images greater than 64K by 64K without tiling. 21 Single decomposition architecture: JPEG has 44 modes. Desired to have a single decomposition architecture that can encompass interchange among applications. Transmission in noisy environments: error robustness, important for wireless communication. Computer generated imagery

22 Compound documents both continuous-tone and bi-level images compress images from 1-bit to 16-bit for each color component JPEG does not work well for binary image Progressive transmission Allowing images be transmitted with increasing pixel resolution e.g., World Wide Web, image archiving. 22 Real-time encoding and decoding Fixed-rate, fixed-size and limited workspace memory Other requirements: e.g., backwards compatibility with JPEG, interface with MPEG-4.

23 All these requirements are seriously considered during the development of JPEG Too early, however, to comment whether all targets can be reached. No doubt: The core requirement on coding efficiency at VLBC will be achieved by using WT Seoul meeting [March 1999]: Lots of changes EBCOT (embedded block coding with optimized truncation) References 23 [cohen 1989] L. Cohen, "Time-Frequency Distributions - A Review," Proceeding IEEE, Vol. 77, No. 7, July 1989, pp [daubechies 1992] I. Daubechies, Ten Lectures on Wavelets, CBMS series, Philadelphia, SIAM, [grossman 1984] A. Grossman and J. Morlet, "Decompositions of hardy functions into square integrable wavelets of constant shape," SIAM Journal of Mathematical Analysis, July 1984, 15(4), pp

24 [jpeg2000 vm] ISO/IEC JTC 1/SC 29/WG 1, JPEG 2000 Verification Model Version 1.2, August 14, [mpeg4] ISO/IEC , Coding of audio-visual objects, Nov [rioul 1991] O. Rioul and M. Vetterli, Wavelets and signal processing, IEEE Signal Processing Magazine, pp , October [said 1996] A. Said and W. A. Pearlman, "A new fast and efficient image codec based on set partitioning in hierarchical trees," IEEE Trans. on Circuits and Systems for Video Technology, June 1996, pp [shapiro 1993] J. Shapiro, "Embedded image coding using zerotrees of wavelet coefficients," IEEE Trans. on Signal Processing, Dec. 1993, pp [vetterli 1995] M. Vetterli and J. Kovacevic, "Wavelets and subband coding," Englewood Cliffs, NJ: Prentice-Hall,

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