影像模型是各種影像處理應用的基礎。複合高斯馬可夫影像模型相對於一般馬可夫隨機場，將適用影像的範圍從紋理圖拓展到自然圖，也將影像分割的應用拓展到影像還原。影像還原是基於以下兩個步驟遞迴計算：根據影像場來建立邊緣場，再以剛建立的邊緣場來還原影像場。
對於一個成功的複合高斯馬可夫模型而言，邊緣場是相當重要的。而好的邊緣場必需具備直邊及橫邊的對稱，在整個遞迴的運算過程中，工作的順序及更新的時機對於結果會有很大的影響，這兩個技巧就是這次為複合高斯馬可夫尋找最佳影像模式的基礎。此外我們還加入了相鄰像素點的亮度差來判斷邊是否存在。
我們將用視覺及數值來驗證我們得到最佳的影像模式，此外我們會以製作人工圖來證明我們得到的影像模式是正確的。

Image modeling is a foundation for many image processing applications. The compound Gauss-Markov (CGM) image model has been proven useful in picture restoration for natural images. In contrast, other Markov Random Fields (MRF) such as Gaussian MRF models are specialized on segmentation for texture image. The CGM image is restored in two steps iteratively: restoring the line field by the assumed image field and restoring the image field by the just computed line field.
The line fields are most important for a successful CGM modeling. A convincing line fields should be fair on both fields: horizontal and vertical lines. The working order and update occasions have great effects on the results of line fields in iterative computation procedures. The above two techniques are the basic for our research in finding the best modeling for CGM. Besides, we impose an extra condition for a line to exist to compensate the bias of line fields. This condition is based upon a requirement of a brightness contrast on the line field.
Our best modeling is verified by the effect of image restoration in visual quality and numerical values for natural images. Furthermore, an artificial image generated by CGM is tested to prove that our best modeling is correct.

第１章 緒論 1
第２章 複合高斯馬可夫隨機場理論之回顧 4
２.１ 簡介 4
２.１.１ 影像模式的介紹 4
２.１.２ 像素間的一些基本關係 5
２.２複合高斯馬可夫隨機場（Compound Gauss-Markov Random Fields） 6
２.２.１ 高斯馬可夫隨機場 6
２.２.２ 複合高斯馬可夫隨機場 8
２.３ 聯合最大後置機率（MAP）的估測 11
２.３.１ 決策搜尋法（Deterministic Search） 15
２.４ 複合高斯馬可夫模型之參數 16
第３章 複合高斯馬可夫影像模式之觀察 19
３.１ 簡介 19
３.１.１ 如何找最佳化的影像模式 19
３.１.２ 如何檢驗最佳化的影像模式 19
３.２ 利用技巧以遞回的方式找出最佳化的影像模式 20
３.２.１ 複合高斯馬可夫影像場的特徵 20
３.２.２ 五大技巧：邊緣場處理順序、掃圖順序、更新、亮度差、乘係數 21
３.２.３ 更新與順序觀念的對稱性（服務與被服務） 29
３.３ 以能量的觀點來檢驗最佳化的影像模式 31
３.３.１ 如何計算一張影像的能量 31
３.３.２ 計算像素點的能量和邊緣場的能量 32
３.３.３ 計算生成雜訊的能量和邊緣場的能量 34
３.３.４ 分別以兩種能量的方式檢驗影像模式 34
３.４ 以圖的好壞來檢驗最佳化的影像模式 35
３.４.１ 如何用圖來判別影像模式的好壞 35
３.４.２ 製做人工圖的方式 35
３.５ 以抗雜訊的能力來檢驗最佳化的影像模式 40
３.５.１ 在影像還原上面抗雜訊能力的比較 40
３.５.２ 人工圖抗雜訊能力的比較 40
３.６ 以自我支持度來檢驗最佳化的影像模式 43
第４章 實驗結果與分析 44
４.１ 簡介 44
４.２ 影像模式的最佳化 44
４.３ 能量的分析比較 51
４.４ 不同模像模下的人工圖 58
４.４.１

[1]F.C. Jeng, J.W. Woods, ”Image Estimation by Stochastic Relaxation in the Compound Gaussian Case,” Proceedings ICASSP 1988（New York,1988）pp.1016-1019
[2]F.C. Jeng, J.W. Woods, ”Compound Gauss-Markov Random Fields for Image Estimation,” IEEE Transactions, Acoust., Speech and Signal Proc., vol.39, pp.683-697, 1991
[3]F.C. Jeng, J.W. Woods, ”Simulated Annealing in Compound Gauss Markov Random Fields,” IEEE Trans. Inform. Theory IT-36, pp.94-101(1990)
[4]S. Geman, D. Geman, ”Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”, IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, 721-741(1988)
[5]T. Pavlidis, Structure Pattern Recognition, Springer, New York, 1977
[6]R.M. Haralick and Linda. G. Shapiro, Computer and Robot Vision, Vol.1, Addison-Wesley Pub. Co. 1992
[7]S.L. Horowitz and Y. Pavlidis, “Picture Segmentation by a Directed Split-and-Merge Procedure,” Proc. 2nd Int Joint Conf. Pattern Recognition, pp.424-433, 1974
[8]S.W. Zucker, “Region Growing：Childhood and adolescence,” Computer Graphics and Image Processing, 5, pp.382-399, 1976
[9]J.W. Woods, ”Two-dimensional Discrete Markovian Fields,” IEEE Trans. Inform. Theory IT-18,232-240(1972)
[10]R. Kashyap and R. Chellappa, “Estiamtion and choice of neighbors in spatial-interaction models of images,” IEEE Trans. Inform. Theroy, vol. IT-29, pp60-72, Jan. 1983
[11]P.J. Green and D.M. Titterington, “Recursive Method in Image Processing,” Bulletin of the International Statist. Institute, pp.51-67, 1987
[12]R. Kindermann and J.L. Snell, “Markov Random Field and Their Application,” Providence, RI, American Mathematical Society, 1980
[13]T. Pavlidis, Structure Pattern Recognition, Springer, New York, 1977
[14]R.M. Haralick and Linda. G. Shapiro, Computer and Robot Vision, Vol.1, Addison-Wesley Pub. Co. 1992
[15]S.L. Horowitz and Y. Pavlidis, “Picture Segmentation by a Directed Split-and-Merge Procedure,” Proc. 2nd Int Joint Conf. Pattern Recognition, pp.424-433, 1974
[16]S.W. Zucker, “Region Growing：Childhood and adolescence,” Computer Graphics and Image Processing, 5, pp.382-399, 1976
[17]R.L. Kashyap,R. Chellappa and A. Khotanzad, Texture Classification Using Features Derived from Random Field Models, Patt. Recognition Letters, vol. 1, Oct. 1982, 43-50.
[18]R. Chellappa and S. Chatterjee, “Classification of Texture Using Gaussian Markov Random Field Models,” IEEE Trans. Acoust., Speech and Signal Proc., vol. 33, Aug. 1985, pp.959-963.