DISCRETE WAVELET TRANSFORM ON IMAGE COMPRESSION Presenter r

DISCRETE WAVELET TRANSFORM ON IMAGE COMPRESSION Presenter : r 98942058 余芝融 EE lab. 530 1

Overview Introduction to image compression Wavelet transform concepts Subband Coding Haar Wavelet Embedded Zerotree Coder References EE lab. 530 2

Introduction to image compression Why image compression? Ex: 3504 X 2336 (full color) image : 3504 X 2336 x 24/8 = 24, 556, 032 Byte = 23. 418 Mbyte Objective Reduce the redundancy of the image data in order to be able to store or transmit data in an efficient form. EE lab. 530 3

Introduction to image compression For human eyes, the image will still seems to be the same even when the Compression ratio is equal 10 Human eyes are less sensitive to those high frequency signals Our eyes will average fine details within the small area and record only the overall intensity of the area, which is regarded as a lowpass filter. EE lab. 530 4

Quick Review Fourier Transform Does not give access to the signal’s spectral variations To circumvent the lack of locality in time →STFT EE lab. 530 5

Quick Review The time-frequency plane for STFT is uniform Constant resolution at all frequencies EE lab. 530 6

Continuous Wavelet Transform FT &STFT use “wave” to analyze signal WT use “wavelet of finite energy” to analyze signal Signal to be analyzed is multiplied to a wavelet function, the transform is computed for each segment. The width changes with each spectral component EE lab. 530 7

Continuous Wavelet Transform Wavelet: finite interval function with zero mean(suited to analysis transient signals) Utilize the combination of wavelets(basis func. ) to analyze arbitrary function Mother wavelet Ψ(t): by scaling and translating the mother wavelet, we can obtain the rest of the function for the transformation(child wavelet, Ψa, b(t)) EE lab. 530 8

Continuous Wavelet Transform Performing the inner product of the child wavelet and f(t), we can attain the wavelet coefficient We can reconstruct f(t) with the wavelet coefficient by EE lab. 530 9

Continuous Wavelet Transform Adaptive signal analysis -At higher frequency , the window is narrow, value of a must be small The time-frequency plane for WT(Heisenberg) multi-resolution diff. freq. analyze with diff. resolution EE lab. 530 10

window Low freq. High freq. a large small EE lab. 530 11

Gaussian Window for S-Transform High Frequency Time Shifted Low Frequency EE lab. 530 SKC-2009 12

Discrete Wavelet Transform Advantage over CWT: reduce the computational complexity(separate into H & L freq. ) Inner product of f(t)and discrete parameters a & b If a 0=2, b 0=1, the set of the wavelet EE lab. 530 13

Discrete Wavelet Transform The DWT coefficient We can reconstruct f(t) with the wavelet coefficient by EE lab. 530 14

Subband Coding EE lab. 530 15

WT compression EE lab. 530 16

2 -point Haar Wavelet(oldest & simplest) g[n] = 1/2 for n = − 1, 0 h[0] = 1/2, h[− 1] = − 1/2, g[n] = 0 otherwise h[n] = 0 otherwise g[n] ½ ½ -3 -1 0 -2 ½ h[n] 1 2 3 n -3 -2 -1 0 1 2 3 n -½ then (Average of 2 -point) (difference of 2 -point) EE lab. 530 17

Haar Transform 2 -steps 1. Separate Horizontally 2. Separate Vertically EE lab. 530 18

2 -Dimension(analysis) Approximatio n Horizontal Edge Vertical Edge Diagonal EE lab. 530 19

Haar Transform Step 1: A B C D A+B C+D L A-B C-D H (0, 0) (0, 1) (0, 2) (0, 3) (1, 0) (1, 1) (1, 2) (1, 3) (2, 0) (2, 1) (2, 2) (2, 3) (3, 0) (3, 1) (3, 2) (3, 3) EE lab. 530 20

Haar Transform Step 2: A C B D A+B C+D LL A-B L H HL C-D LH HH LL HL (0, 0) (0, 1) (0, 2) (0, 3) (1, 0) (1, 1) (1, 2) (1, 3) (2, 0) (2, 1) (2, 2) (2, 3) (3, 0) (3, 1) (3, 2) (3, 3) L H LH EE lab. 530 HH 21

LL 1 HL 1 LH 1 HH 1 LL 2 LH 2 HH 2 LH 1 First level Most important part of the image HL 2 HL 1 HH 1 Second level LL 3 HL 3 LH 3 HH 3 LH 2 HL 1 HH 2 LH 1 HH 1 Third level EE lab. 530 22

Example: 20 15 30 20 35 50 5 10 17 16 31 22 33 53 1 9 15 18 17 25 33 42 -3 -8 21 22 19 18 43 37 -1 1 1 st horizontal separation Original image O 68 103 6 19 326 -38 6 19 76 79 -4 -7 16 -32 2 -7 2 -3 4 1 -10 5 -2 -9 1 st vertical separation 2 nd level DWT result EE lab. 530 23

Original Image LL HL LH HH EE lab. 530 24

LL 2 HL LH 2 HH 2 LH LL 3 HL 3 LH 3 HH HL 2 HL LH 2 HH 2 LH EE lab. 530 HH 25

Embedded Zerotree Wavelet Coder EE lab. 530 26

Structure of EZW Root: a Descendants: a 1, a 2, a 3 … EE lab. 530 27

3 -level Quantizer(Dominant) sp sn EE lab. 530 28

EZW Scanning Order LL 3 LH 3 HL 2 HH 3 HL 1 LH 2 HH 2 LH 1 HH 1 scan order of the transmission band EE lab. 530 29

EZW Scanning Order scan order of the transmission coefficient EE lab. 530 30

Scanning Order sp: significant positive sn: significant negative zr: zerotree root is: isolated zero EE lab. 530 31

Example: Get the maximum coefficient=26 Initial threshold : 1. 26>16 →sp 2. 6<16 & 13, 10, 6, 4 all less than 16→zr 3. -7<16 & 4, -4, 2, -2 all less than 16→zr 4. 7<16 & 4, -3, 2, 0 all less than 16→zr EE lab. 530 32

Each symbol needs 2 -bit: 8 bits The significant coefficient is 26, thus put it into the refinement label : Ls= {26} To reconstruct the coefficient: 1. 5 T 0=24 Difference: 26 -24=2 Threshold for the 2 -level quantizer: The new reconstructed value: 24+4=28 EE lab. 530 33

2 -level Quantizer(For Refinement) EE lab. 530 34

New Threshold: T 1=8 iz zr zr sp sp iz iz→ 14 -bit EE lab. 530 35

Important feature of EZW It’s possible to stop the compression algorithm at any time and obtain an approximate of the original image The compression is a series of decision, the same algorithm can be run at the decoder to reconstruct the coefficients, but according to the incoming but stream. EE lab. 530 36

References [1] C. Gargour, M. Gabrea, V. Ramachandran, J. M. Lina, ”A short introduction to wavelets and their applications, ” Circuits and Systems Magazine, IEEE, Vol. 9, No. 2. (05 June 2009), pp. 57 -68. [2] R. C. Gonzales and R. E. Woods, Digital Image Processing. Reading, MA, Addison-Wesley, 1992. [3] Nancy. A. Breaux and Chee-Hung Henry Chu, ” Wavelet methods for compression, rendering, and descreening in digital halftoning, ” SPIE proceedings series, vol. 3078, pp. 656 -667, 1997. [4] M. Barlaud et al. , "Image Coding Using Wavelet Transform" IEEE Trans. on Image Processing 1, No. 2, 205 -220 (April, 1992). [5] J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients, ” IEEE Trans. Acous. , Speech, Signal Processing, vol. 41, no. 12, pp. 3445 -3462, Dec. 1993. EE lab. 530 37