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



Alfred S. Carasso


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.
Siam Journal on Applied Mathematics


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


Carasso, A. (2009), False Characteristic Functions and Other Pathologies in Variational Blind Deconvolution. A Method of Recovery., Siam Journal on Applied Mathematics, [online], (Accessed May 25, 2024)


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Created September 9, 2009, Updated June 2, 2021