Bochner Subordination, Logarithimic Diffusion Equations, and Blind Deconvolution of Hubble Space Tel

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 but finite frequency range, generalized Linnik characteristic functions can exhibit almost Gaussian behavior near the origin, while behaving like low exponent isotropic Levy stable laws away from the origin. Such behavior matches Fourier domain behavior in a large class of real blurred images of considerable scientific interest, including Hubble space telescope imagery and scanning electron micrographs. This talk
develops a powerful blind deconvolution procedure based on postulating system optical transfer functions (otf) in the form of generalized Linnik characteristic functions. The system otf and ‘true’ sharp image are then reconstructed by solving a related logarithmic diffusion equation backwards in time, using the blurred image as data at time t = 1. The present methodology significantly improves upon previous work based on system otfs in the form of isotropic Levy stable characteristic functions. These results resolve the unexplained appearance of exceptionally low Levy exponents in previous work on the same class of images. The talk will be illustrated with striking enhancements of gray scale and colored images