Abstract:

This talk will highlight some interesting questions arising in following areas of application:

1. Fractional diffusion and image denoising: Helium Ion Microscopes (HIM) are capable of acquiring images with better than one nanometer resolution, and HIM images are particularly rich in morphological surface detail. However, such images are generally quite noisy. A major challenge is to denoise these images while preserving delicate surface information. A highly effective slow motion denoising technique, based on solving fractional diffusion equations, will be discussed.

2. Plausible, but false, backward parabolic reconstructions: Identifying sources of groundwater pollution, and deblurring galaxy images, are two important applications requiring numerical computation of time-reversed parabolic equations. However, while backward uniqueness generally holds in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This talk will exhibit previously unsuspected examples of physically meaningful, well-behaved, yet completely false reconstructions from slightly imprecise data.

3. Logarithmic diffusion and deblurring of Hubble imagery. Generalized Linnik processes and associated logarithmic diffusion equations can be constructed by appropriate Bochner randomization of the time variable in Brownian motion and the related heat conduction equation. Remarkably, over a large frequency range, behavior in Linnik characteristic functions matches Fourier domain behavior in a large class of Hubble telescope images. A powerful blind deconvolution procedure, based on postulating system optical transfer functions in the form of Linnik characteristic functions, will be discussed.