Some New Results in Inverse Reconstruction

This talk will discuss two distinct topics involving nonstandard parabolic problems. The first topic deals with the problem of reconstructing the past from imprecise knowledge of the present, which arises in numerous contexts. Currently, identifying sources of ground water pollution, and deblurring astronomical galaxy images, are two important applications generating considerable interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. Recently, an iterative procedure originating in the field of Spectroscopy has been successfully applied to solve nonlinear parabolic equations backward in time. This has led to the discovery of previously unsuspected 1D examples of well-behaved, physically plausible, but COMPLETELY FALSE reconstructions of the initial data at time t=0, given approximate values for the solution at time t=1. More striking examples of false reconstructions are likely in 2D. These examples indicate that highly detailed prior information about the true solution is a necessary ingredient in many backward reconstruction problems.

The second topic represents important collaborative work with Andras Vladar, who leads NIST's Scanning Electron Microscope Metrology Project. Helium ion microscopes (HIM) are capable of acquiring images with better than 1 nm resolution, and HIM images are particularly rich in morphological surface details. However, such images are generally quite noisy. A major challenge is to denoise these images while preserving delicate surface information. This talk will present a powerful SLOW MOTION denoising technique, based on solving linear fractional diffusion equations forward in time. The method is easily implemented computationally, using fast Fourier transform (FFT) algorithms. When applied to actual HIM images, the method is found to reproduce the essential surface morphology of the sample with high fidelity. In contrast, such highly sophisticated methodologies as Curvelet Transform denoising, and Total Variation denoising using split Bregman iterations, are found to eliminate vital fine scale information, along with the noise. Image Lipschitz exponents are a useful image metrology tool for quantifying the fine structure content in an image. This tool is applied to rank order the above three distinct denoising approaches, in terms of their texture preserving properties.

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

Note: Visitors from outside NIST must contact Robin Bickel; (301) 975-3668; at least 24 hours in advance.