A Semiblind Regularization Algorithm for Inverse Problems with Application to Image Deblurring

Abstract

In many inverse problems the operator to be inverted depends on parameters which are not known precisely. In this article we propose a functional that involves as variables both the solution of the problem and the parameters on which the operator depends. We first prove that the functional, even if it is non-convex, admits a global minimum and that its minimization naturally leads to a regularization method. Then, using the popular Alternating Direction Multiplier Method (ADMM), we describe an algorithm to identify a stationary point of the functional. The introduction of the ADMM algorithm lets us easily introduce some constraints on the reconstructions like non-negativity and flux conservation. Since the functional is non-convex a proof of convergence of the method is given. Numerical examples prove the validity of the proposed approach.

Solving inverse problems using data-driven models

“…One further extension of the R-TLS is to include a regularization with respect to parameters determining the operator (operator deviation). This was considered in a general Hilbert space setting (Bleyer and Ramlau 2013) for image deblurring (see Buccini, Donatelli and Ramlau 2018, who call it ‘semi-blind’).…”

Fast Alternating Direction Multipliers Method by Generalized Krylov Subspaces

“…Moreover, our method did not introduce any additional parameter as the dimension of the projection subspace is automatically determined by the algorithm itself. Matters of future research include the application of this method to non-linear problems and to bi-linear functions; see, e.g., [10] and computationally ecient strategy for the determination of the parameter µ.…”

Bilevel inverse problems in neuromorphic imaging

“…In the blind case, no information about the blur kernel is available, making the deblurring problem more challenging as the blur kernel must be estimated from the degraded image in addition to the original image [16,21]. The semi-blind case is a particular instance of a blind image deblurring problem in that some but not all information about the blur kernel is available [6,13,15]. The problems under consideration in this project are of semi-blind type.…”