基于图像处理和重建的Radon型广义变换及其关键技术研究

项目名称： 基于图像处理和重建的Radon型广义变换及其关键技术研究

项目编号： No.60872095

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 轻工业、手工业

项目作者： 王金平

作者单位： 宁波大学

项目金额： 26万元

中文摘要： 本项目对与图像处理和重建紧相连的Radon 型广义变换中的几个重要问题和关键调和技术, 包括图像重建问题、数值解法以及解析解的存在性和迭代逼近性、基本性质和渐近性态刻划；以及与调和分析、不动点理论、微分方程的相关联系等作出了深入细致的系统的研究。自2008年以来，本项目在Radon 型广义变换及其应用研究方面有若干进展，在SCI、EI学术期刊发表论文18余篇。在调和分析技术应用中非线性算子不动点理论方面，研究了有限族增生算子和非膨胀算子不动点迭代的强收敛性，建立了图像重建中新的调和分析处理方法和理论及相应可靠、高效的计算机算法, 从而深刻全面解释实际问题, 并延拓了Radon型广义变换的研究；利用全连续算子的不动点理论和函数的可积性，结合Green函数有关性质研究了具有奇异的高阶微分方程正解的存在性，给出了此类微分方程奇性的一种新的处理技巧，并利用所得的结果研究了计算机图像重建的数学模型。

中文关键词： Radon 型广义变换；调和分析；不动点理论；图像重建；迭代逼近性

英文摘要： This project studied in detail the generalized transforms of Radon type, which are closely connected to image processing and reconstruction, and their some important theoretic issues and key harmonic techniques, including image reconstruction, numerical solution methods, existence and iterated approximation of analytic solution, basic properties, asymptotic property, their relations to harmonic analysis、fixed point theory、differential equation. Since 2008, the project has made several progresses in the generalized transforms of Radon type and its applications, and has published more than 18 SCI、EI papers. In harmonic techniques of nonlinear operators fixed point theory and its application, this project studied strong convergence of a new iteration for a finite family of accretive operators and iterative algorithm for fixed points for a finite family of nonexpansive mappings. This project also established a new harmonic analysis method and a high-efficient computer algorithm for image reconstruction so as to approach numerical solutions to practical problems and enrich the study of the generalized transforms of Radon type. By means of the fixed point theory of compact operator and integrability of function, combined with some properties of Green function, This project concerned with existence of positive solutions of the singular eigenvalue problem for a higher-order differential equation, and obtained a new method for studying the singularity of this kind of differential equations, furthermore the results have been applied to mathematical model of computed image reconstruction.

英文关键词： the generalized transforms of Radon type; harmonic analysis；fixed point theory;image reconstruction;iterated approximation;