Blind deconvolution using bilateral total variation regularization: a theoretical study and application

Blind image deconvolution recovers a deblurred image and the blur kernel from a blurred image. From a mathematical point of view, this is a strongly ill-posed problem and several works have been proposed to address it. One successful approach proposed by Chan and Wong consists in using the total variation (TV) as a regularization for both the image and the kernel. These authors also introduced an Alternating Minimization (AM) algorithm in order to compute a physical solution. Unfortunately, Chan's approach suffers in particular from the ringing and staircasing effects produced by the TV regularization. To address these problems, we propose a new model based on Bilateral Total Variation (BTV) regularization of the image keeping the same regularization for the kernel. We prove the existence of a minimizer of a proposed variational problem in a suitable space using a relaxation process. We also propose an AM algorithm based on our model. The efficiency and robustness of our model are illustrated and compared with the TV method through numerical simulations.