Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

False Characteristic Functions and Other Pathologies in Variational Blind Deconvolution. A Method of Recovery.

Published

Author(s)

Alfred S. Carasso

Abstract

Given a blurred image g(x, y), variational blind deconvolution seeks to reconstruct both the unknown blur k(x, y), and the unknown sharp image f(x, y), by minimizing an appropriate cost functional. This paper restricts attention to a rich and significant class of zero phase isotropic blurs that includes Gaussians, Lorentzians, and other heavy-tailed densities, together with their con- volutions. A recently developed highly efficient nonlinear variational approach is found to produce inadmissible reconstructions, consisting of partially deblurred images f†(x, y), associated with physically impossible blurs k†(x, y). Three basic flaws in this variational procedure are identified, and shown to be the cause of this phenomenon. A method is then developed that can recover useful information from k†(x, y), by constructing a physically valid rectified blur h#(x, y), based on the low frequency part of k†(x, y). A crucial step involves interpreting h#(x, y) as the pth convolution root of the true blur k(x, y), for some postulated p ≥ 2. Deconvolution is performed in slow motion, by solving an associated parabolic pseudo-differential equation backwards in time, with the blurred image g(x, y) as data at t = 1. Behavior in the evolution as t 0 can be monitored, and used to readjust the value of p. Previously developed APEX/SECB methodologies make such ill-posed continuation feasible. This recovery method is highly effective in several instructive examples involving synthetically blurred images.
Citation
Siam Journal on Applied Mathematics
Volume
70

Keywords

image deblurring, variational blind deconvolution, minimum norm solutions, false characteristic functions, infinitely divisible blurs, fractional diffusion equations, ill-posed continuation, APEX method, SECB method.

Citation

Carasso, A. (2009), False Characteristic Functions and Other Pathologies in Variational Blind Deconvolution. A Method of Recovery., Siam Journal on Applied Mathematics, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=890089 (Accessed July 31, 2021)
Created September 9, 2009, Updated June 2, 2021