Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

TL;DR

This paper analyzes the convergence rates of a Tikhonov-type regularization method for nonlinear inverse problems with Poisson data, relevant in photonic imaging, using a Kullback-Leibler data fidelity and convex penalties.

Contribution

It introduces a convergence analysis for a Tikhonov-type method with Poisson data, providing rates under variational source conditions for both a priori and Lepskii parameter choices.

Findings

Convergence rates of the expected reconstruction error are established as exposure time increases.

The method applies to various photonic imaging applications like PET and microscopy.

The approach accommodates general convex penalty terms in the regularization.

Abstract

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations $\gdag = F(\udag)$ where $\gdag$ is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density $t\gdag$ where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule.