SCAIM Seminar: Reconstruction Techniques for Inverse Problems in Microscopy and Tomography

Abstract: We address the task of reconstructing images corrupted by Poisson noise, which is important in various applications, such as fluorescence microscopy, positronemission- tomography (PET) or astronomical imaging. We focus on reconstruction strategies, combining Bregman concepts, expectation maximization (EM) and total variation (TV) based regularization, and present analytical as well as numerical achievements. 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 produce images suffering from blurred edges due to lagged diffusivity, and neither can guarantee positivity nor provide analytical investigations including convergence. The first goal of this talk is to provide an accurate, robust and fast EM-TV method for computing cartoon reconstructions facilitating post-segmentation. The method can be reinterpreted as a modified forward-backward (FB) splitting strategy known from convex optimization. We establish the well-posedness of the basic variational problem and can prove the positivity preserving property of our method. A damped variant of the FB-EM-TV algorithm with modified time steps, is the key step towards convergence. Typically, TV-based reconstruction methods deliver reconstructions suffering from contrast reduction. Hence, as the second goal of this talk, we propose extensions to EM-TV, based on Bregman iterations and inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. We illustrate the performance of our techniques by 2D and 3D synthetic and real-world results in microscopy and tomography. Proceeding to 4D video reconstruction yields interesting challenges. Due to natural patient motion in medical imaging (e.g. heart or lung) or cell migration in microscopy, naive reconstructions can suffer from undesired blurring effects at object boundaries. Finally, we touch on combinations of reconstruction techniques and optimal transport strategies.