Weighted inhomogeneous regularization for inverse problems with indirect and incomplete measurement data

TL;DR

This paper introduces a weighted inhomogeneous regularization method with a novel exponent design and spatial weights, improving inverse problem solutions with incomplete data by better capturing spatially varying features.

Contribution

It proposes a new weighted inhomogeneous regularization framework with a novel exponent design and spatial weights, enhancing recovery of spatially varying features in inverse problems.

Findings

Improved reconstruction quality in synthetic image experiments.

Enhanced recovery of sea ice data from incomplete measurements.

Demonstrated robustness against spatial feature misclassification.

Abstract

Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's characteristics, such as sparsity or smoothness. Inhomogeneous regularization, which incorporates a spatially varying exponent $p$ in the standard $\ell_p$-norm-based framework, has been used to recover signals with spatially varying features. This study introduces weighted inhomogeneous regularization, an extension of the standard approach incorporating a novel exponent design and spatially varying weights. The proposed exponent design mitigates misclassification when distinct characteristics are spatially close, while the weights address challenges in recovering regions with small-scale features that are inadequately captured by traditional $\ell_p$-norm regularization. Numerical experiments, including synthetic image reconstruction and the recovery of sea ice data from incomplete wave measurements, demonstrate the effectiveness of the proposed method.