Self-supervised learning for a nonlinear inverse problem with forward operator involving an unknown function arising in photoacoustic tomography

TL;DR

This paper introduces a self-supervised learning approach to solve a nonlinear inverse problem in photoacoustic tomography, where both the initial pressure and the unknown sound speed are recovered simultaneously, addressing challenges of ill-posedness and nonlinearity.

Contribution

It proposes a novel self-supervised learning framework for nonlinear inverse problems involving unknown functions, specifically applied to photoacoustic tomography.

Findings

Framework successfully recovers initial pressure and sound speed.

Demonstrates effectiveness on experimental data.

Addresses ill-posedness and nonlinearity in inverse problems.

Abstract

In this article, we are concerned with a nonlinear inverse problem with a forward operator involving an unknown function. The problem arises in diverse applications and is challenging in the presence of an unknown function, which makes it ill-posed. Additionally, the nonlinear nature of the problem makes it difficult to use traditional methods, and thus, the study addresses a simplified version of the problem by either linearizing it or assuming knowledge of the unknown function. Here, we propose self-supervised learning to directly tackle a nonlinear inverse problem involving an unknown function. In particular, we focus on an inverse problem derived in photoacoustic tomograpy (PAT), which is a hybrid medical imaging with high resolution and contrast. PAT can be modeled based on the wave equation. The measured data provide the solution to an equation restricted to surface and initial pressure of an equation that contains biological information on the object of interest. The speed of a sound wave in the equation is unknown. Our goal is to determine the initial pressure and the speed of the sound wave simultaneously. Under a simple assumption that sound speed is a function of the initial pressure, the problem becomes a nonlinear inverse problem involving an unknown function. The experimental results demonstrate that the proposed framework performs successfully.