# Recovery of Pressure and Wave Speed for Photoacoustic Imaging under a Condition of Relative Uncertainty

## TL;DR

This paper investigates the inverse problem in photoacoustic tomography, aiming to recover pressure and wave speed from boundary data, and establishes conditions for uniqueness under relative uncertainty.

## Contribution

It introduces practical assumptions ensuring uniqueness in recovering pressure and wave speed, especially when their relative differences are small.

## Key findings

- Uniqueness holds when wave speed differences are much smaller than pressure differences.
- Provides conditions under which joint reconstruction is theoretically guaranteed.
- Discusses implications for iterative reconstruction algorithms.

## Abstract

In this paper, we study the photoacoustic tomography problem for which we seek to recover both the initial state of the pressure field and the wave speed of the medium from the knowledge of a single boundary measurement. The goal is to propose practical assumptions to define a set of initial conditions and wave speeds over which uniqueness for this inverse problem is guaranteed. The main result of the paper is that given two sets of wave speeds and pressure profiles, they cannot produce the same acoustic measurements if the relative difference between the wave speeds is much smaller than the relative difference between the pressure profiles. Implications for iterative joint-reconstruction algorithms are discussed.