XVI. Applications of Wavelet Transforms Wavelet 所適用的 applications，通常有以下兩大特點： (1) 信號的頻率分佈，會隨著不同的時間(或地點)有較大變異 (2) Multiscale 的分析扮演重要的角色 Larger sampling interval  ignoring the detail Smaller sampling interval  requiring a lot of data Wavelet transforms compromise them. 目前，文獻上，80% 以上的應用和 image processing 有關

(1) Image Compression (JPEG 2000) DCT AC係數 Zigzag Scan Huffman Coding RGB to YCbCr 量子化 JPEG file 8 × 8 4:2:0 DC係數 差分 編碼 Huffman Coding 量子化表 檔頭 問題：由於 8  8 的切割，在高壓縮率時會造成 blocking effect

Discrete Wavelet Transform JPEG 2000 架構 Quantization Table Discrete Wavelet Transform Bit Plane Conversion RGB to YCbCr Image 4:2:0 Quantization Binary arithmetic coding Fractional Bit-plane coding JPEG 2000 file Tier 2 Encoder (Tier 1) 檔頭 Tier 1: zero coding, sign coding, magnitude refinement coding, run length coding Tier 2: 用以控制檔案大小 (例如只取比較重要的地方編碼) 註：感謝 2010年修課的潘冠臣同學幫忙整理

CR: compression ratio 註：感謝 2006年修課的黃俊德同學 DCT-based image compression Wavelet-based image compression Original image CR = 51.3806 CR = 53.4333 CR: compression ratio 註：感謝 2006年修課的黃俊德同學

bpp: bit per pixel (每一點平均需要多少個 bits) PSNR: peak signal to noise ratio (PSNR), see page 432

使用 JPEG 2000 做影像壓縮的優點： (1) (2) (3) 所以，在高壓縮率之下，重建的影像仍有不錯的品質 Question: Why JPEG 2000 has not replaced the status of JPEG now? 參考資料 C. Christopoulos, A. Skodras, and T. Ebrahimi, “The JPEG2000 still image coding system: An overview,” IEEE Trans. Consumer Electronics, vol. 46, no. 4, pp.1103-1127, Nov. 2000.

Another Compression Algorithm: SPIHT Using the correlation among high frequency parts in different layers B.J. Kim, Z. Xiong, and W.A. Pearlman. “Low bit-rate scalable video coding with 3-D set partitioning in hierarchical trees (3-D SPIHT),” IEEE Trans. Circuits Syst. Video Technol., vol. 10, pp. 1374-1387, 2000.

(2) Edge and Corner Detection (3) Pattern recognition (a) Feature extraction (Using the wavelet features) (b) Computation Time 和縮小的 pattern 互相比較 (節省運算) (4) 強調前景，壓縮背景

(5) Filter Design 如何不傷到 edge，又能夠將 noise 去除掉？

One-stage wavelet filter a1[n] analysis synthesis g[n]  2 x1,L[n]  2 h1[n] x[n] x0[n] a2[n] g1[n] h[n]  2 x1,H[n]  2 做 filter design 時，可以令 a1[n] = 1, a2[n] = 0 for non-edge region a2[n] = 1 for edge region 以 x1,H[n] 的 amplitude 來區分 必要時可使用 two-stage 以上的 wavelet filter

x2,L[n] (2nd stage, lowpass) x2,H[n] (2nd stage, highpass) x1,H[n] (1st stage, highpass)

原信號 使用one-stage 的 wavelet filter 使用two-stage 的 wavelet filter

音樂當中，音每高一個音階，頻率就增為二倍 (6) Music 音樂當中，音每高一個音階，頻率就增為二倍 音樂 每一音階有12個半音，增加一個半音，頻率增加 21/12 倍 (等比級數) Do 升Do Re 升Re Me Fa 升Fa So 升So La 升La Si Hz 270 286 303 321 340 360 382 405 429 454 481 510 540 572 606 642 680 721 764 810 857 908 962 1019 (7) Acoustics

(8) Analyzing the Electrocardiogram (ECG) Is the rhythm of the cardiac valve in synchronization with that of the heart muscle? Does the heart muscle relax between beats? From: A. K. Louis, P. Maab, and A. Rieder, “Wavelets Theory and Applications”, John Wiley & Sons, Chichester, 1997.

(9) 「短期因素」和「長期因素」的分析 population economical data temperature (10) 其他奇奇怪怪的應用 指紋的辦識 羊毛質料的辦識

Time-frequency Analysis 和Wavelet 在應用上的異同處 相同：都能夠處理一個信號的頻率分佈會隨時間而改變的情形 不同：Time frequency analysis 對於瞬間頻率的分析比較精確 Wavelet 可作「巨觀」和「微觀」的分析

附錄十六 Generalization for the Wavelet Transforms 1. Directional Form 2-D Wavelet Transforms 一般的 2-D wavelet transform，其實可分解成沿著 x-axis 以及沿著 y-axis 的 1-D wavelet transforms 的組合 其實，2-D wavelet transform 不一定要沿著 x-axis ， y-axis 來做 Directional 2-D wavelet transforms:  curvelet contourlet bandlet shearlet  Fresnelet  wedgelet  brushlet

 Curvelet (ridgelet) rotation 比較：原本的 1-D wavelet E. Candès and D. Donoho, "Curvelets – a surprisingly effective nonadaptive representation for objects with edges." In: A. Cohen, C. Rabut and L. Schumaker, Editors, Curves and Surface Fitting: Saint-Malo 1999, Vanderbilt University Press, Nashville (2000), pp. 105–120.

the curvelet transform of the input results with different  (four direction for the high-frequency part)

masks in the frequency domain  Contourlet masks in the frequency domain 低頻部分 沒有分成不同的方向 高頻部分 分成各種不同的方向 M. Do and M. Vetterli, "The contourlet transform: An efficient directional multiresolution image representation," IEEE Trans. Image Processing, vol.14, no.12, pp.2091–2106, Dec. 2005.

 Bandlet 根據物體的紋理或邊界，來調整 wavelet transforms 的方向 Stephane Mallet and Gabriel Peyre, "A review of Bandlet methods for geometrical image representation," Numerical Algorithms, Apr. 2002.

2. Stationary Wavelet Transforms ….. g3[n] x3,L[n] ….. g2[n] x2,L[n] g1[n] x1,L[n] h3[n] x3,H[n] x[n] h2[n] x2,H[n] h1[n] x1,H[n] 其中 gj[n] ↑2 gj+1[n] hj[n] ↑2 hj+1[n] Q: 和原本 discrete wavelet transform 不一樣的地方在哪裡？ G. P. Nason and B. W. Silverman, “The stationary wavelet transform and some statistical applications,” Lecture Notes in Statistics, available in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.2662&rep=rep1&type=pdf

3. Bandwidth Form Wavelet Transforms A little modification for g[n] and h[n] 4. Multi-Band Wavelet Transforms Instead of only two outputs

Happy New Year! 祝各位期末考順利，寒假愉快！