An adaptive augmented regularization method and its applications

TL;DR

This paper introduces an adaptive augmented regularization model (AARM) that dynamically adjusts to local properties of the estimated function, improving inverse problem solutions in fields like seismic exploration and medical imaging.

Contribution

The paper develops a novel AARM that adapts regularization locally based on data, bridging Bayesian and regularization methods with an efficient iterative algorithm.

Findings

AARM outperforms traditional Tikhonov and Total-Variation models in numerical tests.

The model effectively characterizes local smoothness properties.

Numerical examples demonstrate improved solution quality.

Abstract

Regularization method and Bayesian inverse method are two dominating ways for solving inverse problems generated from various fields, e.g., seismic exploration and medical imaging. The two methods are related with each other by the MAP estimates of posterior probability distributions. Considering this connection, we construct a prior probability distribution with several hyper-parameters and provide the relevant Bayes' formula, then we propose a corresponding adaptive augmented regularization model (AARM). According to the measured data, the proposed AARM can adjust its form to various regularization models at each discrete point of the estimated function, which makes the characterization of local smooth properties of the estimated function possible. By proposing a modified Bregman iterative algorithm, we construct an alternate iterative algorithm to solve the AARM efficiently. In the end, we provide some numerical examples which clearly indicate that the proposed AARM can generates a favorable result for some examples compared with several Tikhonov and Total-Variation regularization models.