A Mathematical Framework for Deep Learning in Elastic Source Imaging

TL;DR

This paper introduces a mathematical framework for elastic source imaging that integrates low-dimensional manifold regularization, improving reconstruction performance with sparse data, and links it to deep learning techniques.

Contribution

It presents a novel framework combining manifold regularization with inverse problem algorithms, connecting it to deep convolutional framelet expansion in machine learning.

Findings

Enhanced source reconstruction with sparse measurements.

The framework is mathematically equivalent to deep convolutional framelet expansion.

Numerical examples demonstrate improved performance.

Abstract

An inverse elastic source problem with sparse measurements is of concern. A generic mathematical framework is proposed which incorporates a low- dimensional manifold regularization in the conventional source reconstruction algorithms thereby enhancing their performance with sparse datasets. It is rigorously established that the proposed framework is equivalent to the so-called \emph{deep convolutional framelet expansion} in machine learning literature for inverse problems. Apposite numerical examples are furnished to substantiate the efficacy of the proposed framework.