On an inverse problem in photoacoustic

TL;DR

This work addresses the inverse problem of reconstructing the Cauchy data $a(x)$ and $b(x)$ for the wave equation from boundary measurements $F=u|_{S\times[0,T]}$ on the unit sphere, a setting relevant to photoacoustic tomography. It develops a hybrid methodology: in 3D an exterior forward problem combined with time reversal and an interior reverse-time formulation, realized via residues or Volterra integral equations, to algebraically recover the initial data; in 2D an iterative recursive scheme using angular decompositions and singular integral equations handles the lack of trailing edge. The paper provides constructive procedures, explicit representations, and convergence rationale for recovering $a$ and $b$ from boundary data, with clear connections to known filtered backprojection ideas. The results pave the way for stable, implementable IP solvers in both 3D and 2D and set the stage for extending to variable sound speed in future work.

Abstract

We consider the problem of reconstruction of the Cauchy data for the wave equation in $\mathbb{R}^3$ and $\mathbb{R}^2$ by the measurements of its solution on the boundary of the unit ball.

Theorems & Definitions (13)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• Proposition 5
• Remark 1
• Proposition 6