Optical tomography on graphs

TL;DR

This paper introduces a novel algorithm for solving inverse problems on graphs, inspired by diffuse optical tomography, with theoretical analysis and modifications for broader applicability.

Contribution

It formulates and analyzes a discrete inverse Born series for graphs, providing convergence, error, and stability estimates, and extends the method to incorporate structural information.

Findings

Proves convergence and stability of the discrete inverse Born series.

Provides error estimates for the approximation.

Demonstrates improved recovery with structural modifications.

Abstract

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.