Untitled

Abstract

This paper is devoted to blind deconvolution and blind separation problems. Blind deconvolution is the identi®cation of a point spread function and an input signal from an observation of their convolution. Blind source separation is the recovery of a vector of input signals from a vector of observed signals, which are mixed by a linear (unknown) operator. We show that both problems are paradigms of nonlinear ill-posed problems. Consequently, regularization techniques have to be used for stable numerical reconstructions. In this paper we develop a rigorous convergence analysis for regularization techniques for the solution of blind deconvolution and blind separation problems. Convergence of regularized point spread functions and signals to a solution is established and a convergence rate result in dependence of the noise level is presented. Moreover, we prove convergence of the alternating minimization algorithm for the numerical solution of regularized blind deconvolution problems and present some numerical examples. Moreover, we show that many neural network approaches for blind inversion can be considered in the framework of regularization theory.

Bayesian inverse problems with unknown operators

“…Since the convolution operator K ϑ is smoothing, the inverse problem is ill-posed. If the kernel g ϑ is unknown, the problem is called blind deconvolution occurring in many applications [4,20,32]. In a density estimation setting this problem as already been intensively investigated, see [10,17,18,26] among others.…”

“…Examples include semi-blind and blind deconvolution for image analysis. Therein, the operator is given by K ϑ f = g ϑ (• − y) f (y)dy with some unknown convolution kernel g ϑ [4,20,32]. More general integral operators such as singular layer potential operators appear in the context of partial differential equations, see examples in [7,15].…”

“…There are different approaches in the inverse problem literature to deal with this identifiability problem, particularly in the context of semi-blind or blind deconvolution. One approach is to find the so called minimum norm solution which has a minimal distance to some a priori estimates for ϑ and f, see [4,20]. Another idea is to assume that some approximation of the unknown operator is available for the reconstruction of f, see [18,32].…”

Estimating the Size of the Point Spread Function Using Fuzzy Logic for Blind Restoration of Satellite Image

“…as given by equation (1), it is therefore an inverse problem. Because of the noisy data, this problem is part of the ill-posed inverse problems class [14]. According to Hadamard, the problem of type !…”

Bilevel optimization for calibrating point spread functions in blind deconvolution

“…We also mention that nonconvex image priors are considered for blind deconvolution in the work [1], which are favorable for certain sparse images [14,28,29]. The convergence analysis of an alternating minimization scheme for such double-regularization based variational approaches in appropriately chosen function spaces is carried out in [4,31]. An exception of variational approaches to blind deconvolution is [32], where the optimality condition is "diagonalized" by Fourier transform and thus can be solved by some non-iterative root-finding algorithm.…”