Inhomogenous Regularization with Limited and Indirect Data

TL;DR

This paper introduces a method to adaptively select spatially varying regularization parameters in inverse problems with limited data, improving reconstruction quality by classifying image patches without needing true derivatives.

Contribution

It proposes a novel strategy to determine inhomogeneous p in lp regularization using statistical and patch-wise info from a single measurement, applicable to indirect and limited data.

Findings

The method is robust across different reconstructions.

It effectively classifies patches to assign appropriate p values.

Numerical tests demonstrate improved image recovery quality.

Abstract

For an ill-posed inverse problem, particularly with incomplete and limited measurement data, regularization is an essential tool for stabilizing the inverse problem. Among various forms of regularization, the lp penalty term provides a suite of regularization of various characteristics depending on the value of p. When there are no explicit features to determine p, a spatially varying inhomogeneous p can be incorporated to apply different regularization characteristics that change over the domain. This study proposes a strategy to design the exponent p when the first and second derivatives of the true signal are not available, such as in the case of indirect and limited measurement data. The proposed method extracts statistical and patch-wise information using multiple reconstructions from a single measurement, which assists in classifying each patch to predefined features with corresponding p values. We validate the robustness and effectiveness of the proposed approach through a suite of numerical tests in 1D and 2D, including a sea ice image recovery from partial Fourier measurement data. Numerical tests show that the exponent distribution is insensitive to the choice of multiple reconstructions.