文档介绍：摘要
图像复原问题是光学成像研究的一个反问题,又是图像处理领域中最为活跃的
研究课题之一,而反问题常常具有不适定性,正则化方法是解决该类问题的有效工
具。
本文主要介绍了图像复原的基本概念和理论,给出了成像系统的基本模型,然
后通过求解逆问题获得图像的复原模型,对原始图像进行合理估计,并对不适定问
题的正则化方法的理论及应用进行了探讨与分析,主要工作如下:
首先,从正则化方法的数学理论入手,分析了图像的退化模型和图像复原的病
态特征。介绍了一些比较常见的正则化方法,以及正则参数选取的几种算法,研究
了正则化算子和滤波函数的关系。在数值实验方面,主要采用偏差原理和 Newton 迭
代法求解正则化参数。其次,给出了用 Tikhonov 正则化方法求解第一类算子积分方
程的算例,采用迭代的 Lavrentiev 正则化方法对加噪的二值图像进行复原,编程实现
了算法,分析了不同参数的选取对图像复原效果的影响。再次,对于 PET 医学图像,
采用 CD-FBP 方法和传统的 FBP 方法进行图像重建,编程给出了具体的数值实现,
并将两种重建方法进行了比较。最后,对全文进行了总结,指出了目前研究工作的
不足和要进一步开展的工作。

关键词:图像复原正则化不适定问题迭代方法 FBP CD-FBP 滤波函数
I
Abstract
The image restoration is an inverse problem in the optical imaging system. It is also
one of the most active research field in image process at present. Inverse problems
commonly are ill-posed. Regularization techniques are effective tools in solving ill-poesd
problems.
In this thesis, the basic concept and theory are introduced and the ordinary model of
imaging system is depicted. The theory and applications of regularization method are
introduced and analyzed in detail. The main works of this thesis are listed as follows:
Firstly, the degraded image model and the ill-posed character of image restoration are
analyzed based on the regularization technique for dealing with ill-posed problems. Some
regularization techniques and selection method of regularization parameters are introduced.
The discrepancy principle and Newton iterative method are used to obtain the
regularization parameter in the numerical experiments. The relationship between
regularization operator and filter function has studied.
Secondly, Tikhonov regularization method is used to solve the first kind operator
equation. Iterative Lavrentiev regularization method is used in restoration of a noised
binary image, and the program implementation pleted. The restoration effects are
analyzed by selection of different regularization parameters.
Thirdly, for the reconstruction of PET in