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

2024/07/24 by Mauritz, Marco, Wirth, Benedikt
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2407.17135

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.

Related

• A Bayesian model for dynamic mass reconstruction from PET listmode data
• Total Variation Regularisation in Measurement and Image space for PET reconstruction
• Posterior Estimation for Dynamic PET imaging using Conditional Variational Inference
• Nonparametric posterior learning for emission tomography with multimodal data
• Off-the-grid regularisation for Poisson inverse problems