影像系統通常被視為非因果性的系統。而對於信號估測來說卡門濾波
器與 Wiener 濾波器是兩種重要的線性濾波器，它們分別用來處理因果信
信號與非因果信信號。然而，卡門濾波器利用改寫信號的動態方程式之後
也可以應用在非因果性系統上。在本篇論文是對於Wiener 濾波器和卡門
濾波器在影像還原這個應用上效能的研究。
由我們的實驗可以瞭解到對於還原誤差上的比較，最佳的是全階數的
Wiener 濾波器，再來依序為卡門濾波器，減少更新卡門濾波器，三階Wiener
濾波器。誤差表現上的比較與濾波器做線性估測時使用到的資料數目有
關。另一方面，如果我們對濾波器的運算量進行比較，最佳的則是減少更
新卡門濾波器，再來依序為三階Wiener 濾波器，卡門濾波器，全階數的
Wiener 濾波器。減少更新卡門濾波器運算量最低是由於在更新部分的計算
量被大量的降低。在這裡需要特別注意的是我們利用矩陣的特殊性讓標準
的卡門濾波器的運算量變的更少，更有效率
除了以上的非因果性影像模型，也可以建立現在點由左點及上點影響
的因果性影像模型。雖然因果性的影像模型看起來不自然，但是它相對於
非因果性的影像模型有一個明顯的好處，就是計算量較低，計算上更有效
率。然而，非因果性影像模型在誤差的表現上較好，因此自然影像應該適
用於非因果性影像模型。

Image generating system is usually considered as a noncausal system. The
Kalman filter and the Wiener filter are two important linear filters for signal
estimation. They are developed for the causal signal and noncausal signal respectively.
However, the Kalman filter can also be applied to the noncausal system by rewriting
the signal generating equation. In this thesis, we study the performance of the Wiener
filter and the Kalman filter applied to image restoration.
Our experiments have demonstrated that the rank of list for error performance is:
the full order Winner filter, the Kalman filter, the reduced Kalman filter, the
three-order Wiener filter. This performance is consisted with the amount of data used
in the linear estimation. On the other hand the list for computation performance is as
following: the reduced Kalman filter, the three-order Wiener filter, the Kalman filter,
the full order Wiener filter. The efficiency of the reduced Kalman filter can be
understood by the computation saving of huge updating procedures. It should be
noted that the efficiency of applying the regular Kalman filter in this thesis is
achieved by fully employed the special form of system matrix involved.
In addition to the above noncausal image model, a causal image model can also
be built if the central pixel is assumed to be affected only by the left and the upper
pixels. The second model is not natural but is obviously advantageous in computation
efficiency compared to the first model. However, the first model is much better than
the second model error performance. Therefore, it is suggested that the natural image
should be modeled as a noncausal model.

第一章 序論……………………………………………………………… 1
第二章 一維線性濾波器與卡門濾波器………………………………… 3
2-1 信號製作…………………………………………………… 3
2-2 線性濾波器………………………………………………… 6
2-3 相關性計算………………………………………………… 8
2-4 卡門濾波器………………………………………………… 12
2-5 在非因果性模型下的卡門濾波…………………………… 15
第三章 一維情況下實驗結果比較……………………………………… 19
3-1 信號製作…………………………………………………… 19
3-2 參數設定…………………………………………………… 19
3-3 實驗結果…………………………………………………… 20
3-4 實驗結果探討……………………………………………… 22
第四章非因果性動態方程式的應用：影像還原……………………… 23
4-1 影像模型簡介……………………………………………… 23
4-2 複合高斯馬可夫隨機場理論回顧………………………… 25
4-3 基於非因果性模型之影像還原…………………………… 28
4-3-1 影像分割…………………………………………… 28
4-3-2 取得增益矩陣以及生成雜訊變異數… 31
4-4 二維線性濾波器…………………………………………… 35
4-5 二維卡門濾波器…………………………………………… 37
4-6 減少更新卡門濾波器……………………………………… 49
第五章 二維情況下實驗結果比較…………………………………………52
第六章 結論……………………………………………………………… 61

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