Phase and TV Based Convex Sets for Blind Deconvolution of Microscopic Images

Abstract—In this article, two closed and convex sets for blind deconvolution problem are proposed. Most blurring functions in microscopy are symmetric with respect to the origin. Therefore, they do not modify the phase of the Fourier transform (FT) of the original image. As a result blurred image and the original image have the same FT phase. Therefore, the set of images with a prescribed FT phase can be used as a constraint set in blind deconvolution problems. Another convex set that can be used during the image reconstruction process is the Epigraph Set of Total Variation (ESTV) function. This set does not need a prescribed upper bound on the total variation of the image. The upper bound is automatically adjusted according to the current image of the restoration process. Both the TV of the image and the point spread function are regularized using the ESTV set. Both the phase information set and the ESTV are closed and convex sets. Therefore they can be used as a part of any blind deconvolution algorithm. Simulation examples are presented. Index Terms—Projection onto Convex Sets, Blind Deconvolu-tion, Inverse Problems, Epigraph Set