A New Method of Robust Image Compression Based on Embedded Zerotree Wavelet Algorithm Charles D. Creusere IEEE Transactions on Image Processing, Vol. 6, No. 10, October 1997 學 生 : 戴 錦 輝

OUTLINE 1. Introduction 2. Wavelet Transform 3. EZW Image Compression 4. Conclusions 5. References Ref

1. Introduction The author proposes a wavelet-based image compression algorithm that achieves robustness to transmission errors by partitioning the transform coefficients into groups and independently processing each group using an embedded coder.

2. Wavelet Transform Fig.1(a)An example of Haar wavelet transform using lifting

Fig.1(b)An example of Haar wavelet transform using lifting

離散小波轉換可用以偵測音高週期 Fig. 2 (a)

Fig. 2 (b)

Fig. 2 (c)

Fig. 2 (d)

Fig. 2(e)Application in 1-D Wavelet Transform “ㄚ”音經五次離散小波轉換後的波形，兩高點(peaks)的距離就是音高週期

Fig. 3 (a)

Fig. 3 (b)

Fig. 4

Fig. 4 左上角是原影像在V(x,y)的低解析度影像，右上角是列向量經一次離散小波轉換後的影像，左下角是行向量經一次離散小波轉換後的影像，右下角是經一次離散小波轉換後的影像。

% p.290 凌波初步 load Tiffany.mat Y = dwt(3,Origin,1); [Ya Yb Yc] = split(Y,128,128); Ya = saturate(round(Ya), 1, 256); Yb = 1 + 255* (abs(Yb) <4); Yc = 1 + 255* (abs(Yc) <4); image([Ya Yb; Yc]); colormap(g256); print -deps Tdtwo sum(sum(Yb==1 )) + sum(sum(Yc==1 ))

3. EZW Image Compression EZW這個方法是由Shapiro於1993年發表的，它是一種對離散小波轉換後係數編碼的方法。當影像作離散小波轉換後，高頻部份的係數會小於低頻部份的係數。 係數大的部份是影像低頻的部份，由這部份可得到模糊的影像。低頻的部份比較重要。係數小的部份是影像高頻的部份，它可使影像更加清晰。

影像壓縮編碼程序 步驟一:設定門檻值: N=5 步驟二:計算EZW的重建數值 步驟三:建立重要係數表 步驟四:建構第一次精鍊值 步驟五:先前重要係數的再精鍊 步驟六:重新設定重要係數的係數值 步驟七:(重複步驟三四五六) EZW 重複步驟三四五六，找出每一次切割的重要係數，並精鍊先前取出的重要係數，直到門檻值為0或使用者認為可以停止。

頻帶 係數值 符號 重建數值 LL3 63 POS 48 HL3 -34 NEG -48 LH3 -31 IZ HH3 23 ZTR HL2 表1:第一次切割所建立的重要係數表 頻帶 係數值 符號 重建數值 LL3 63 POS 48 HL3 -34 NEG -48 LH3 -31 IZ HH3 23 ZTR HL2 49 10 14 -13 LH2 15 -9 -7 HL1 7 13 3 4 LH1 -1 47 -3 2

LH2 15 -9 -7 HL1 7 13 3 4 LH1 -1 47 -3 2 頻帶 係數值 符號 重建數值 LL3 63 POS 48 -34 NEG -48 LH3 -31 IZ HH3 23 ZTR HL2 49 10 14 -13 LH2 15 -9 -7 HL1 7 13 3 4 LH1 -1 47 -3 2

表2:第一次切割之精鍊表 係數值 符號 精鍊數值 63 1 56 -34 -40 49 47 40

7 6 -7 10 14 -13 7 13 -12 7 3 4 6 -1 5 -7 3 9 4 -2 3 2 -31 23 14 -9 -7 3 -12 -14 8 -5 9 -1 7 3 0 -3 2 2 -3 6 -4 5 11 5 6 4 6 -2 2 3 -2 0 4 3 6 3 6 0 3 -4 4 Fig. 4:第一次切割結束前的係數重新設定

表3:第二次切割所建立的重要係數表 LL3 7 IZ HL3 6 ZTR LH3 -31 NEG -24 HH3 23 POS 24 LH2 頻帶 係數值 符號 重建數值 LL3 7 IZ HL3 6 ZTR LH3 -31 NEG -24 HH3 23 POS 24 LH2 15 14 -9 -7 HH2 3 -12 -14 8

表4:第二次切割所建構之第一次精鍊數值 係數值 符號 精鍊數值 -31 1 -28 23 20

表5:第二次切割之精鍊值建構 係數值 符號 精鍊數值 63 1 60 -34 -36 49 52 47 44

Fig. 5 “winter”的影像

Fig. 6是編碼後的影像,Bit planes 1-10 during zerotree encoding of the “winter” image, using Haar wavelets.

Fig.6 Original image

Fig. 7 “Lena”的影像

Fig. 8是解碼後的影像, Progressive decoding of the “Lena” image, which was encoded with the zerotree algorithm using Daubechies D6 wavelets.

At bit plane 10, the decoded image has 2. 0% pixel error(31 At bit plane 10, the decoded image has 2.0% pixel error(31.4PSNR) with a compression ratio of 5.16:1(1.5 bits per pixel).

Peak Signal to Noise Ratio（PSNR） MSE和PSNR都是用來檢測兩張圖是否相似。 ， MSE的公式: PSNR 的公式:

“ ”

“ ”

Combining the decimation technique to the wavelet coefficients before the zerotree algorithm, we can achieve additional compression. Using 10% of the wavelet coefficients, each algorithm provides a compression ratio of approximately 2.5: 1 and 31-32 dB PSNR. With 10 bit planes and 10% retained wavelet coefficients, the compression ratio of the zerotree encoded image is 7.35: 1 with a PSNR of 30.3dB.

4. Conclusion The compression performance of this algorithm is competitive with virtually all known techniques. The remarkable performance can be attributed to the use of the following four features: a discrete wavelet transform, which decorrelates most sources fairly well. zerotree coding, which by predicting insignificance across scales using an image model that is easy for most images to satisfy, provide substantial coding gains. successive-approximation, which allows the coding of multiple significance maps using zerotrees, and allows the encoding or decoding to stop at any point. adaptive arithmetic coding, which allows the entropy coder to incorporate learning into the bit stream itself.

5. References 1.J. M. Shapiro, “Embedded Image Coding Using Zerotrees of Wavelet Coefficients,” IEEE Trans. Signal Processing, vol.41, Dec. 1993 pp. 3445-3462. 2.陳璽煌先生成大博士論文: “A study on Speech signal Processing Using Wavelet Transforms”, May,2002 3.單維彰著: “凌波初步”,全華科技 1999 4.Stephen Welstead, “Fractal and Wavelet Image Compression Techniques,” SPIE Publications (December 1999) 5.吳炳飛等著: “JPEG2000影像壓縮技術”, 全華科技, 2003 6.陳同孝、張真誠、黃國峰著: “數位影像處理技術”,旗標 Back