Convergence of Poisson point processes and of optimal transport regularization with application in variational analysis of PET reconstruction

TL;DR

This paper analyzes the convergence properties of Bayesian PET reconstruction methods that incorporate Poisson point process models and optimal transport regularization, focusing on how these methods behave as noise diminishes and parameters vary.

Contribution

It provides a variational analysis of the convergence of PET reconstruction algorithms using optimal transport regularization in the context of Poisson-distributed measurements.

Findings

Convergence of the reconstruction functional as signal-to-noise ratio increases.

Interaction between detector resolution and reconstruction accuracy.

Insights into dynamic PET imaging reconstruction stability.

Abstract

Poisson distributed measurements in inverse problems often stem from Poisson point processes that are observed through discretized or finite-resolution detectors, one of the most prominent examples being positron emission tomography (PET). These inverse problems are typically reconstructed via Bayesian methods. A natural question then is whether and how the reconstruction converges as the signal-to-noise ratio tends to infinity and how this convergence interacts with other parameters such as the detector size. In this article we carry out a corresponding variational analysis for the exemplary Bayesian reconstruction functional from [arXiv:2311.17784,arXiv:1902.07521], which considers dynamic PET imaging (i.e.\ the object to be reconstructed changes over time) and uses an optimal transport regularization.