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Microlocal,FunctionalAnalytic and Numerical Study of Partial Differential Equations and Tomography

偏微分方程和层析成像的微局部、泛函分析和数值研究

基本信息

批准号：
16540140
负责人：
KANEKO Akira
金额：
$ 2.37万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

This is a long lasting research, both theoretical and computer aided practical, on partial differential equations and tomography. For the period of this fund, I executed mainly the following study and got results: (1) Fujita type critical exponent for the blowup of solutions of heat equations with power non-linearity could distinguish the aperture of cones, but for narrower domains such as paraboloidal or cylindrical ones it could not separate them and even bounded ones. I introduced a logarithmic non-linearity to distinguish them and developed a general theory. Especially, I introduced a function logg u interpolating log u and 1/log u, and by taking for the non-linear term the product of its p-th power and u, I succeeded in showing that a critical exponent of blowup is defined for any domain. Concrete determination of this new critical exponent for respective domains is left for future research (2) Reconstruction from the two projections of binary plane figures has almost always many … More olutions, and relations between the solutions or the search for good solutions is not well clarified. In this research, I introduced a structure of digraph to the set of the solutions by means of switching operation. Through the study of the properties of this graph, I developed a new means of approach to this problem. About thus introduced switching graph, many interesting results were obtained on various connectivity especially the Haminlonian property. Many open problems arose, thus presenting a new stimulating research field. (3) I developed techniques to incorporate algebraic-geometric codes to images, and gave its applications to self restoring images, masking, digital footprints etc. (4) Deconvolutions are typical examples of ill-posed problems as well as the tomography. By calculating deconvolution introducing a kind of regularization, I tried sharpening of defocused photos, and verified its practical use. This study will be published after I advance it up to incorporate an automatic choice of appropriate deconvolution parameters. Less

这是一项关于偏微分方程和层析成像的长期研究，无论是理论上的还是计算机辅助的实践。在本基金期间，主要进行了以下几方面的研究：(1)具有幂非线性的热方程解爆破的Fujita型临界指数可以区分锥体的开度，但对于抛物面或柱面等较窄的区域，它不能将它们分开，甚至不能区分有界域。我引入了对数非线性来区分它们，并发展了一个普遍的理论。特别地，我引入了一个函数loggu，它插值logu和1/logu，通过取其p次方与u的乘积作为非线性项，我成功地证明了对于任何区域，爆破的临界指数都是定义的。这一新的临界指数的具体确定留待以后的研究(2)从二值平面图形的两个投影重建几乎总是有许多…更多的解决方案，以及解决方案和寻找好的解决方案之间的关系没有得到很好的澄清。在本研究中，我通过切换运算将有向图的结构引入到解的集合中。通过对这个图的性质的研究，我发展了一种新的方法来解决这个问题。关于这样引入的转换图，在各种连通性，特别是Haminlonian性质方面，得到了许多有趣的结果。出现了许多悬而未决的问题，从而呈现出一个新的振奋人心的研究领域。(3)发展了将代数几何编码融入到图像中的技术，并给出了它在自恢复图像、掩模、数字足迹等方面的应用。(4)反卷积是病态问题和层析成像的典型例子。通过计算解卷积，引入一种正则化方法，尝试对散焦照片进行锐化处理，并验证了其实际应用。这项研究将在我提前将其纳入适当反褶积参数的自动选择之后发表。较少