Gradient Flows on a Riemannian Submanifold for Discrete Tomography

We present a smooth geometric approach to discrete tomography that jointly performs tomographic reconstruction and label assignment. The flow evolves on a submanifold equipped with a Hessian Riemannian metric and properly takes into account given projection constraints. The metric naturally extends the Fisher-Rao metric from labeling problems with directly observed data to the inverse problem of discrete tomography where projection data only is available. The flow simultaneously performs reconstruction and label assignment. We show that it can be numerically integrated by an implicit scheme based on a Bregman proximal point iteration. A numerical evaluation on standard test-datasets in the few angles scenario demonstrates an improvement of the reconstruction quality compared to competitive methods.