A semiblind regularization algorithm for inverse problems with application to image deblurring

In many inverse problems the operator to be inverted is not known precisely, but only a noisy version of it is available; we refer to this kind of inverse problem as semiblind. In this article, we propose a functional which involves as variables both the solution of the problem and the operator itself. We first prove that the functional, even if it is nonconvex, admits a global minimum and that its minimization naturally leads to a regularization method. Later, 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 allows us to easily impose some constraints on the computed solutions like nonnegativity and flux conservation. Since the functional is nonconvex a proof of convergence of the method is given. Numerical examples prove the validity of the proposed approach.