Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings

TL;DR

This paper addresses the inverse problem of reconstructing the initial function in photoacoustic tomography using solutions of the fractional wave equation measured on specific geometric surfaces, advancing mathematical techniques in medical imaging.

Contribution

It introduces methods to recover the initial function from fractional wave equation data measured on a sphere and hyperplane, expanding the mathematical framework for PAT.

Findings

Reconstruction formulas for data on a sphere

Reconstruction formulas for data on a hyperplane

Enhanced understanding of fractional wave equations in PAT

Abstract

Photoacoustic tomography (PAT) is a novel and rapidly promising technique in the field of medical imaging, based on the generation of acoustic waves inside an object of interest by stimulating non-ionizing laser pulses. This acoustic wave is measured using the detector on the outside of the object and converted into an image of the human body by several inversions. Thus, one of mathematical problems in PAT is how to recover the initial function from the solution of the wave equation on the outside of the object. In this study we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We provide how to recover the initial function from the data, the solution of the fractional wave equation, measured on the sphere and hyperplane.