Fall Seminar Series - October 20, 2005
University of Minnesota
School of Statistics
College of Liberal Arts

Apex Blind Deconvolution of Hubble Telescope Imagery and the Use of Levy Stable Laws

Alfred S. Carasso
National Institute of Standards and Technology

Thursday, October 20, 2005
3:30 PM, 115 Ford Hall
Minneapolis, East Bank Campus
Social at 3:00 PM, 300 Ford Hall

Abstract

Blind deconvolution seeks to deblur an image without knowing the cause of the blur. This problem is fraught with mathematical difficulties, as severe ill-conditioning is compounded with non-uniqueness of solutions. Few reliable algorithms are known, and most approaches involve time-consuming iterative procedures.

Recently, a new non-iterative approach to blind deconvolution was formulated, based on suitably restricting the allowable class of blurs, and using Fast Fourier Transform techniques. This so-called APEX method can process high resolution 1024 x 1024 imagery in quasi real-time, a monumental breakthrough for the class of images for which the method is applicable. However, not all images can be usefully enhanced with the APEX method.

In constructing the APEX method, selection of the allowable class of blurs was guided primarily by considerations of pure mathematics. A class of point spread functions was sought, that generalized the Gaussian density, possessed appropriate semigroup properties, and a relatively simple Fourier space representation. These properties result in a tractable deconvolution problem that can be solved in slow motion. The class of radially symmetric L\'{e}vy stable characteristic functions

\hat{h}(\xi,\eta)= exp\{-\alpha (\xi2+\eta2)^{\beta}\}, \qquad \alpha > 0,~~0 < \beta \leq 1,

is the simplest example of a broad class of blurs with the requisite mathematical properties. That class is currently the basis for the APEX method.

The APEX method has been found surprisingly effective in quite diverse applications, including MRI and PET brain scans, scanning electron microscopy, and most recently, Hubble space telescope color imagery. In most cases, the detected point spread functions that successfully enhance these images have very low exponent $\beta$, and are very far from Gaussian. The reasons behind these successful applications remain an open problem.