False Characteristic Functions and Other Pathologies in Variational Blind Deconvolution. A Method of Recovery
Alfred S. Carasso · SIAM Journal on Applied Mathematics · 2009
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 its attention to a rich and significant class of infinitely divisible isotropic blurs that includes Gaussians, Lorentzians, and other heavy-tailed densities, together with their convolutions. A recently developed highly efficient nonlinear variational approach is found to produce inadmissible reconstructions, consisting of partially deblurred images $f^{\dagger}(x,y)$, associated with physically impossible blurs $k^{\dagger}(x,y)$. 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 $k^{\dagger}(x,y)$, by constructing a physically valid rectified blur $h^{#}(x,y)$, based on the low frequency part of $k^{\dagger}(x,y)$. A crucial step involves interpreting $h^{#}(x,y)$ as the $p^{th}$ convolution root of the true blur $k(x,y)$, for some postulated real number $p\geq2$. 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 of the evolution as $t\downarrow0$ 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 found highly effective in several instructive examples involving synthetically blurred images.