在本論文中，用於線性回歸問題的Wilcoxon 方法將和碎形影像壓縮結合成為一個新穎的Wilcoxon 碎形影像壓縮。當原始影像被雜訊所污染時，我們認為碎形影像壓縮的架構應對出現於被污染影像上的離群值具免疫之特性，即較不受這些離群值所影響。而這引導出強韌碎形影像壓縮的新概念。本論文所提出之Wilcoxon 碎形影像壓縮是強韌碎形影像壓縮的設計所跨出的第一步。本文將提出四個數值方法用來解決相關的線性Wilcoxon 回歸問題，即最小梯度法、基於二次內插法的線性最小化、基於三次內插法的線性最小化和最小絕對離差法。藉由模擬結果我們可知，相較於傳統碎形影像壓縮， Wilcoxon 碎形影像壓縮對抗由胡椒鹽雜訊導致的離群值之強韌度極佳，然而在對抗高斯雜訊所導致的離群值方面，強韌度並沒有顯著地改善。

In the thesis, the Wilcoxon approach to linear regression problems is combined with the fractal image compression to form a novel Wilcoxon fractal image compression. When the original image is corrupted by noise, we argue that the fractal image compression scheme should be insensitive to those outliers present in the corrupted image. This leads to the new concept of robust fractal image compression. The proposed Wilcoxon fractal image compression is the first attempt toward the design of robust fractal image compression. Four different numerical methods, i.e., steepest decent, line minimization based on quadratic interpolation, line minimization based on cubic interpolation, and least absolute deviation, will be proposed to solve the associated linear Wilcoxon regression problem. From the simulation results, it will be seen that, compared with the traditional fractal image compression, Wilcoxon fractal image compression has very good robustness against outliers caused by salt-and-pepper noise. However, it does not show great improvement of the robustness against outliers caused by Gaussian noise.

誌謝 i
摘要 ii
Abstract iii
List of Figures and Tables iv
Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Brief Sketch of the Contents 4
Chapter 2 Fractal Image Compression 6
2.1 Preliminaries 6
2.2 Introduction to Fractal Image Compression 9
Chapter 3 Wilcoxon Fractal Image Compression 13
3.1 Introduction 13
3.2 Wilcoxon Norm 15
3.3 Linear Wilcoxon Regressor 17
3.4 Robustness Indicator 19
3.5 Combination of FIC and Linear Wilcoxon Regressor 21
Chapter 4 Numerical Methods for Wilcoxon Fractal Image Compression 23
4.1 Steepest Decent with Armijo Rule 24
4.2 Line Search Based on Quadratic Interpolation 25
4.3 Line Search Based on Cubic Interpolation 28
4.4 Least Absolute Deviation Method 29
Chapter 5 Illustrative Examples 34
5.1 Comparison of Numerical Methods. 34
5.2 Comparison of FIC and WFIC 41
Chapter 6 Conclusion and Discussion 51
Appendix 53
References 55

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