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 talk 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.

Speaker Bio: Alfred S. Carasso received the Ph.D degree in mathematics at the University of Wisconsin in 1968. He was a professor of mathematics at the University of New Mexico, and a visiting staff member at the Los Alamos National Laboratory, prior to joining NIST in 1982. His major research interests lie in the theoretical and computational analysis of ill-posed continuation problems in partial differential equations, together with their application in inverse heat transfer, system identification, and image reconstruction and computer vision. He pioneered the use of time-reversed fractional and logarithmic diffusion equations in blind deconvolution of wide classes of images. He is the author of seminal theoretical papers, is a patentee in the field of image analysis, and is an active speaker at national and international conferences in applied mathematics.

Presentation Slides: PDF

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