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.

翻译：反问题中的泊松分布测量通常源于通过离散化或有限分辨率探测器观测的泊松点过程，最典型的例子是正电子发射断层扫描（PET）。这类反问题通常通过贝叶斯方法进行重建。一个自然的问题是：当信噪比趋于无穷大时，重建结果是否以及如何收敛，且这种收敛如何与探测器尺寸等其他参数相互作用。本文针对[arXiv:2311.17784, arXiv:1902.07521]中的示例性贝叶斯重建泛函进行了相应的变分分析，该泛函考虑了动态PET成像（即待重建物体随时间变化）并采用了最优传输正则化。