Recovering the Initial Data of the Wave Equation from Neumann Traces

TL;DR

This paper develops explicit formulas to recover initial data of the wave equation from boundary Neumann traces, with special solutions for ellipsoidal domains, relevant for imaging techniques like photoacoustic tomography.

Contribution

It introduces explicit back-projection formulas for initial data recovery from Neumann traces, including an analytic solution for ellipsoids where the smoothing operator vanishes.

Findings

Explicit inversion formulas for convex domains.

Analytic solution for ellipsoids.

Applicable to tomographic imaging like photoacoustic tomography.

Abstract

We study the problem of recovering the initial data (f, 0) of the standard wave equation from the Neumann trace (the normal derivative) of the solution on the boundary of convex domains in arbitrary spatial dimension. Among others, this problem is relevant for tomographic image reconstruction including photoacoustic tomography. We establish explicit inversion formulas of the back-projection type that recover the initial data up to an additive term defined by a smoothing integral operator. In the case that the boundary of the domain is an ellipsoid, the integral operator vanishes, and hence we obtain an analytic formula for recovering the initial data from Neumann traces of the wave equation on ellipsoids.