Presenter : r 余芝融 1 EE lab.530

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

Introduction to image compression  Why image compression?  Ex: 3504X2336 (full color) image : 3504X2336 x24/8 = 24,556,032 Byte = Mbyte  Objective Reduce the redundancy of the image data in order to be able to store or transmit data in an efficient form. 3 EE lab.530

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 5 EE lab.530

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

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 7 EE lab.530

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, Ψ a,b (t)) 8 EE lab.530

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 9 EE lab.530

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 10 EE lab.530

window a  Low freq. large  High freq. small EE lab

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

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 13 EE lab.530

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

Subband Coding 15 EE lab.530

16 EE lab.530 WT compression

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

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

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

Haar Transform ABCDA+BC+DA-BC-D LH (0,0)(0,1)(0,2)(0,3)(0,0)(0,1)(0,2)(0,3) (1,0)(1,1)(1,2)(1,3)(1,0)(1,1)(1,2)(1,3) (2,0)(2,1)(2,2)(2,3)(2,0)(2,1)(2,2)(2,3) (3,0)(3,1)(3,2)(3,3)(3,0)(3,1)(3,2)(3,3) Step 1: 20 EE lab.530

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

LL1HL1 LL2HL2 HL1 LH2HH2 LH1HH1LH1HH1 LL3HL3 HL2 HL1 LH3HH3 LH2HH2 LH1HH1 First levelSecond level Third level Most important part of the image 22 EE lab.530

Example: Original image O 1 st horizontal separation 1 st vertical separation 2 nd level DWT result 23 EE lab.530

24 Original Image LH HL HH LL

EE lab LL2HL2 LH2HH2 LH HL HH LH HL HH HL2 LH2HH2 LL3HL3 HH3LH3

Embedded Zerotree Wavelet Coder EE lab

Structure of EZW  Root: a  Descendants: a1, a2, a3 EE lab …

3-level Quantizer(Dominant) EE lab sp sn

EZW Scanning Order EE lab LL3HL3 HL2 HL1 LH3HH3 LH2 HH2 LH1HH1 scan order of the transmission band

EZW Scanning Order EE lab scan order of the transmission coefficient

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

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

 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: 1.5T 0 =24  Difference:26-24=2  Threshold for the 2-level quantizer:  The new reconstructed value: 24+4=28 EE lab

2-level Quantizer(For Refinement) EE lab

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

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

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 [2] R. C. Gonzales and R. E. Woods, Digital Image Processing. Reading, MA, Addison-Wesley, [3] NancyA. Breaux and Chee-Hung Henry Chu,” Wavelet methods for compression, rendering, and descreening in digital halftoning,” SPIE proceedings series, vol. 3078, pp , [4] M. Barlaud et al., "Image Coding Using Wavelet Transform" IEEE Trans. on Image Processing 1, No. 2, (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 , Dec EE lab