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

Published

Author(s)

Alfred S. Carasso

Abstract

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, generalized Linnik characteristic functions can have almost Gaussian behavior near the origin, and low exponent Levy stable behavior 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 Levy stable characteristic functions. Such improvement is validated by the substantially smaller image Lipschitz exponents that ensue, confirming increased fine structure recovery. The paper is illustrated with several striking enhancements of gray scale and colored imagery.
Citation
Siam Journal on Applied Mathematics
Volume
3

Keywords

blind deconvolution, Bochner subordination, generalized Linnik laws, image deblurring, Levy stable laws, low exponent fractional and logarithmic diffusion equations, Lipschitz exponents, Hubble space telescope, scanning electron microscope, Whirlpool galaxy, Starburst galaxy

Citation

Carasso, A. (2010), Bochner Subordination, Logarithmic Diffusion Equations, and Blind Deconvolution of Hubble Space Telescope Imagery and Other Scientific Data, Siam Journal on Applied Mathematics, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=903667 (Accessed December 10, 2023)
Created November 30, 2010, Updated June 2, 2021