在自然界的系統模型描述一般都以因果性系統居多，就是過去只會對未來造成影響。相對的非因果性系統，就是過去、現在、未來都會互相有關，探討的部分就比較少，本論文即是針對非因果性系統下受到加成雜訊影響之信號還原之探討。
針對線性濾波器的使用我們利用動態方程式的特性將原本需要複雜計算的相關性簡化成前一時刻的相關性可以推出下一時刻所需的相關性，以達到遞迴的效果。我們又發現所有相關性計算植基於系統輸出與輸入之相關性。所以我們利用Mason Rule求出系統所有輸出入相關性的計算，再將動態方程式改寫成因果性形式，如此我們便可以將原本獨立的相關性計算改為遞迴計算。在求取系統輸出入方法上，我們更進一步推導出遞迴的Mason Rule來加快我們計算速度；另外我們也改寫動態方程式以求得適合使用卡門濾波器的參數。
本系統應用到影像還原上，我們使用影像分割以求出相關性高的區域，再取得區域的生成雜訊變異數及像素與生成雜訊之相關性矩陣，最後配合線性濾波器做還原。我們發現本方法與複合高斯馬可夫隨機場的影像還原相比，有較好的數值與視覺效果。

The causal system is more practical then the noncausal system in the world. Causality implies only the past input can effect the future output. As a consequence, noncausal system is seldom investigation. The purpose of this thesis is to study the signal recury for a noncausal system.
The principle of signal estimation is based upon the Wiener-Hopf equation. Therefore, the correlation computation is very important. By transforming the noncausal dynamic equations to a causal equation, we achieve a partial recursive computation structure for correlation computation. However the current input is not independent of the past
signal in the noncausal system. Hence, the Mason Rule is applied to solved this problem to make the above recursive structure complete. Furthermore, a recursive computation of Mason Rule for stage propagation is developed in this thesis to accelerating the processing speed.
Our algorithm is applied to image restoration. We first segment the image to find the required generating input ponen for each correlated region. Secondly, we extend our 1-D algorithms to 2-D algorithm to restore the image. Our method is compared with the method developed base upon the Gaussian Markov model. The experiments results demonstrate the advantage of method in both visual quailty and numerical results.

第一章 序論……………………………………………………………… 1

第二章 線性估測器主體之設計………………………………………… 3

第三章 非因果性下線性濾波器與卡門濾波器………………………… 8
3-1 相關性計算………………………………………………… 8
3-2 Mason Rule………………………………………………… 13
3-2-1 Mason Rule 背景介紹 …………………………… 13
3-2-2 Recursive Mason Rule…………………………… 16
3-3 非因果模型下的卡門濾波………………………………… 25

第四章 線性濾波器和卡門濾波器實驗結果和比較…………………… 26
4-1 信號製作…………………………………………………… 26
4-2 參數設定…………………………………………………… 27
4-3 實驗結果…………………………………………………… 28

第五章 非因果性動態方程式的應用：影像還原……………………… 31
5-1 影像模型簡介……………………………………………… 31
5-2 複合高斯馬可夫隨機場理論回顧………………………… 33
5-2-1 聯合最大後置機率的估測………………………… 35
5-2-2 複合高斯馬可夫模型參數………………………… 38
5-3 基於非因果性模型之影像還原…………………………… 40
5-3-1 影像分割…………………………………………… 40
5-3-2 取得相關性矩陣與生成雜訊……………………… 43
5-3-3 線性濾波器還原…………………………………… 47
5-4 實驗結果與比較…………………………………………… 48
5-5 總結………………………………………………………… 55

第六章 結論……………………………………………………………… 58
參考文獻………………………………………………………………59

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