Solving Poisson Inverse Problems in Phase Retrieval and Single Photon Emission Computerized Tomography

Abstract

We live in a world where many objects cannot be imaged directly and hence rely on reconstruction algorithms to solve the corresponding inverse imaging problems. However, lots of information is contaminated or even lost when samples are collected by imaging devices, so that the resulting inverse problem is ill-posed and challenging to solve. As the recorded photon arrivals by the sensor are often assumed to follow Poisson distributions, algorithms for solving Poisson inverse problems are crucial. This thesis tackles two applications where Poisson inverse problems arise: phase retrieval and single photon emission computerized tomography (SPECT). For phase retrieval, we propose novel optimization algorithms working in low-count regimes, including a novel majorize-minimize (MM) algorithm, a modified Wirtinger flow algorithm using the observed Fisher information for step size and a generative image prior based on score matching. Our proposed algorithms lead to faster convergence rate and improved reconstruction quality evaluated both qualitatively and quantitatively. For SPECT imaging, we focus on deep learning (DL) solutions including: 1) We propose end-to-end training of unrolled iterative convolutional neural network (CNN) using our memory efficient Julia toolbox for SPECT image reconstruction. 2) We propose a DL algorithm for joint dosimetry estimation and image deblurring for estimating patient’s absorbed dose-rate distribution in radionuclide therapy. 3) We propose unsupervised coordinate-based learning for predicting missing SPECT projection views.