Forward-Backward EM-TV methods for inverse problems with Poisson noise

We address the task of reconstructing images corrupted by Poisson noise, which is important in various applications, such as fluorescence microscopy, positron emission tomography (PET), or astronomical imaging. In this work, we focus on reconstruction strategies, combining the expectation-maximization (EM) algorithm and total variation (TV) based regularization, and present a detailed analysis as well as numerical results. Recently extensions of the well known EM/Richardson-Lucy algorithm received increasing attention for inverse problems with Poisson data. However, most algorithms for regularizations like TV lead to convergence problems for large regularization parameters, cannot guarantee positivity, and rely on additional approximations (like smoothed TV). The goal of this work is to provide an accurate, robust and fast FB-EM-TV method for computing cartoon reconstructions facilitating post-segmentation and further image quantifications. Motivated by several applications we provide a statistical modeling of inverse problems with Poisson noise in terms of Bayesian MAP estimation and relate it to the continuous variational setting. We focus on minimizing the energy functional with the Kullback-Leibler divergence as the data fidelity and TV as regularization, subject to non-negativity constraints. Our proposed FB-EM-TV minimization algorithm is a semi-implicit, alternating two step method consisting of an EM step and the solution of a weighted ROF problem. The method can be reinterpreted as a modified forward-backward (FB) splitting strategy known from convex optimization. First of all, we establish the wellposedness of the variational problem under general conditions, in particular we give a proof of existence, uniqueness and stability. Under certain assumptions on the given data, we can prove positivity preservation of our iteration method. A damped variant of the FB-EM-TV algorithm, interpreted as a splitting strategy with modified time steps, is the key step towards global convergence. In addition, we present a Bregman-FB-EM-TV strategy, extending the FB-EM-TV framework, which corrects the natural loss of contrast using TV via iterative Bregman distance regularization. Finally, we illustrate the performance of the proposed algorithms and confirm the analytical concepts by 2D and 3D synthetic and real-world results in optical nanoscopy and positron emission tomography. Forward-Backward EM-TV methods for inverse problems with Poisson noise 2