Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data

TL;DR

This paper develops a mathematically precise method for reconstructing sources in photoacoustic tomography using limited spatial and temporal data, enabling practical imaging with finite measurement surfaces.

Contribution

It introduces a new exact reconstruction technique from finite open surfaces, reducing data requirements and providing explicit algorithms for circular and spherical geometries.

Findings

Exact reconstruction of Radon projections from finite data.

Algorithms are explicit and asymptotically fast.

Numerical simulations validate the method.

Abstract

We investigate the inverse source problem for the wave equation, arising in photo- and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing solution of this problem in terms of the measured data, under the assumption of constant and known speed of sound. However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yields theoretically exact reconstruction of the standard Radon projections of the source from the data measured on a finite open surface. In addition, this technique reduces the time interval where the data should be known. In general, our method requires a pre-computation of densities of certain single-layer potentials. However, in the case of a truncated circular or spherical acquisition surface, these densities are easily obtained analytically, which leads to fully explicit asymptotically fast algorithms. We test these algorithms in a series of numerical simulations.