Convergence of the alternating minimization algorithm for blind deconvolution

Blind deconvolution refers to the image processing task of restoring the original image from a blurred version without the knowledge of the blurring function. One approach that has been proposed recently [T. Chan, C. Wong, IEEE Trans. Image Process. 7 (1998) 370-375; Y. You, M. Kaveh, IEEE Trans. Image Process. 5 (1996) 416-428] is a joint minimization model in which an objective function is set up consisting of three terms: the data fitting term, and the regularization terms for the image and the blur. This model implicitly defines a one-parameter family of blurred images and point spread functions (PSFs), from which the user can decide, usually using additional information, which is the "best" restored image. To find a local minimum of the objective function, we use an alternating minimization (AM) procedure [Y. You, M. Kaveh, IEEE Trans. Image Process. 5 (1996) 416-428] in which we fix either the blur or the image and minimize respect to the other variable, each step of which is a standard non-blind deconvolution problem. While the model is not convex and thus allows multiple solutions, we have found that the AM procedure always converges globally, but with the converged solution depending on the initial guess. In this paper, we give an analysis of the AM procedure which explains the convergence behavior and the observed robustness of the method.