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

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 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 † (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 p th convolution root of the true blur k(x, y), for some postulated real number 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 of 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 found highly effective in several instructive examples involving synthetically blurred images.

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

“…9 psfs. (9). Generalized Linnik (brown) and Lévy stable (red) psfs exhibit heavier tails than the Gaussian psf (blue).…”

“…Such an object has distinct mathematical properties not shared by Fourier transforms of general L 1 functions [4]. Optical transfer functions are examples of characteristic functions [9]. The types of otfs considered here can be viewed as Fourier transforms of Green's functions for generalized diffusion equations, just as a Gaussian distribution is the Green's function for a heat conduction equation.…”

“…The method in [24] requires a prior guess for the system otf. However, even with a good guess, this method often returns a questionable sharp image associated with a new, but physically impossible, otf [9]. Other more promising approaches, such as [2], exploit prior knowledge of the edge map in the unknown image.…”

Extensions of the Justen–Ramlau blind deconvolution method

“…It may in some applications be attractive to use the computed image and PSF as initial approximations for one of the available iterative methods to enhance the quality of the computed image further. Having a fairly accurate initial approximation for the iterative methods reduces the computational effort required by the latter and may also affect the quality of the computed image; see Carasso [4] for a discussion on a particular post-processing method. We are presently investigating this kind of hybrid methods.…”

Bochner subordination, logarithmic diffusion equations, and blind deconvolution of hubble space telescope imagery and other scientific data

“…The method in [24] requires a prior guess for the system otf. However, even with a good guess, this method often returns a questionable sharp image associated with a new, but physically impossible otf [9]. If and when comprehensive blind deconvolution methods become available, the methodology presented here is likely to remain useful.…”