非线性抛物方程的逆时间危险延拓及非线性模糊图像去模糊实验。

IF 1.5 4区 工程技术
Alfred S Carasso
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引用次数: 12

摘要

识别地下水污染源和消除纳米尺度图像以及天文星系图像的模糊,是涉及时间倒抛物方程数值计算的两个重要应用。令人惊讶的是,我们对非线性抛物方程的后向延拓知之甚少。本文将20世纪30年代起源于光谱学的迭代法应用于求解一类广义二维非线性后向抛物型方程。此外,以前未预料到的困难被揭示,可能排除有用的向后延拓在抛物线方程偏离太强烈的线性,自治,自伴随,规范模型。本文对选定的二维非线性方程进行了反向延拓的研究,在这些方程中使用若干幅锐化图像作为初始数据,生成虚拟的模糊图像,并在某正时间t处捕获相应的解。从t= t到t= 0的反向延拓成功,将恢复原始锐化图像。视觉识别为重建方案的成功或失败程度提供了有意义的评估。文中还举例说明了某些类型的非线性的意想不到的影响。视觉上和统计上难以区分的模糊图像呈现,具有截然不同的去模糊结果。这些例子表明,一个图像是如何非线性模糊是至关重要的,除了数量的模糊。所研究的方程代表了布朗运动的非线性推广,模糊图像可以被解释为视觉上表达了新的随机过程的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930's, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.

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来源期刊
自引率
33.30%
发文量
10
期刊介绍: The Journal of Research of the National Institute of Standards and Technology is the flagship publication of the National Institute of Standards and Technology. It has been published under various titles and forms since 1904, with its roots as Scientific Papers issued as the Bulletin of the Bureau of Standards. In 1928, the Scientific Papers were combined with Technologic Papers, which reported results of investigations of material and methods of testing. This new publication was titled the Bureau of Standards Journal of Research. The Journal of Research of NIST reports NIST research and development in metrology and related fields of physical science, engineering, applied mathematics, statistics, biotechnology, information technology.
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