Bochner Subordination, Logarithmic Diffusion Equations, and Blind Deconvolution of Hubble Space Telescope Imagery and Other Scientific Data

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 Lévy 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 paper 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 Lévy stable characteristic functions. Such improvement is validated by the substantially smaller image Lipschitz exponents that ensue, confirming increased fine structure recovery. These results resolve the unexplained appearance of exceptionally low Lévy stable exponents in previous work on the same class of images. The paper is illustrated with striking enhancements of gray scale and colored images.