Presenter : r98942058余芝融 Discrete Wavelet Transform on image compression EE lab.530
Overview • Introduction to image compression • Wavelet transform concepts • Subband Coding • Haar Wavelet • Embedded Zerotree Coder • References EE lab.530
Introduction to image compression • Why image compression? • Ex: 3504X2336 (full color) image : 3504X2336 x24/8 = 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
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
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
Quick Review • The time-frequency plane for STFT is uniform Constant resolution at all frequencies 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 computedfor each segment. • The width changes with each spectral component 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 waveletΨ(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
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
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
window a • Low freq. large • High freq. small EE lab.530
Gaussian Window for S-Transform High Frequency Time Shifted Low Frequency SKC-2009 EE lab.530
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 a0=2,b0=1, the set of the wavelet EE lab.530
Discrete Wavelet Transform • The DWT coefficient • We can reconstruct f(t) with the wavelet coefficient by EE lab.530
Subband Coding EE lab.530
WT compression EE lab.530
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 ½ ½ ½ g[n] h[n] n n -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -½ then (difference of 2-point) (Average of 2-point) EE lab.530
Haar Transform • 2-steps 1.Separate Horizontally 2. Separate Vertically EE lab.530
2-Dimension(analysis) Approximation Horizontal Edge Vertical Edge Diagonal EE lab.530
Haar Transform Step 1: EE lab.530
Haar Transform • Step 2: LL HL LH HH L H EE lab.530
First level Second level Most important part of the image Third level EE lab.530
Example: 1st horizontal separation Original image O 2nd level DWT result 1st vertical separation EE lab.530
Original Image LL HL LH HH EE lab.530
LL2 HL2 HL LH2 HH2 LH HH HL2 HL LL3 HL3 LH3 HH3 LH2 HH2 LH HH EE lab.530
Embedded Zerotree Wavelet Coder EE lab.530
Structure of EZW • Root: a • Descendants: a1, a2, a3 … EE lab.530
3-level Quantizer(Dominant) sp sn EE lab.530
EZW Scanning Order scan order of the transmission band EE lab.530
EZW Scanning Order scan order of the transmission coefficient EE lab.530
Scanning Order sp: significant positive sn: significant negative zr: zerotree root is: isolated zero EE lab.530
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
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: 1.5T0=24 • Difference:26-24=2 • Threshold for the 2-level quantizer: • The new reconstructed value: 24+4=28 EE lab.530
2-level Quantizer(For Refinement) EE lab.530
New Threshold: T1=8 • iz zr zr sp sp iz iz→14-bit EE lab.530
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
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] NancyA. 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 Processing1, 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
