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

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 (TV) 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 of these two closed and convex sets can be used as a part of any blind deconvolution algorithm. Simulation examples are presented.Comment: Submitted to IEEE Selected Topics in Signal Processin

Cell Detection by Functional Inverse Diffusion and Non-negative Group Sparsity−-Part II: Proximal Optimization and Performance Evaluation

In this two-part paper, we present a novel framework and methodology to analyze data from certain image-based biochemical assays, e.g., ELISPOT and Fluorospot assays. In this second part, we focus on our algorithmic contributions. We provide an algorithm for functional inverse diffusion that solves the variational problem we posed in Part I. As part of the derivation of this algorithm, we present the proximal operator for the non-negative group-sparsity regularizer, which is a novel result that is of interest in itself, also in comparison to previous results on the proximal operator of a sum of functions. We then present a discretized approximated implementation of our algorithm and evaluate it both in terms of operational cell-detection metrics and in terms of distributional optimal-transport metrics.Comment: published, 16 page

Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors

In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms such as wavelets or curvelets. Both analysis and synthesis-type sparsity priors are considered. Piecing together the data fidelity and the prior terms, the deconvolution problem boils down to the minimization of non-smooth convex functionals (for each prior). We establish the well-posedness of each optimization problem, characterize the corresponding minimizers, and solve them by means of proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Experimental results are conducted to demonstrate the potential applicability of the proposed algorithms to astronomical imaging datasets