Explicit Inversion of the Attenuated Photoacoustic Operator in General Observation Geometries

TL;DR

This paper derives explicit inversion formulas for the attenuated photoacoustic operator in planar and spherical geometries, enabling stable and unique image reconstruction in photoacoustic imaging.

Contribution

It provides the first explicit formulas for inverting the attenuated photoacoustic operator in general geometries, including a stable filtered backprojection formula for spherical observation.

Findings

Explicit inversion formulas for planar and spherical geometries.

Range characterization of the forward operator in distributional spaces.

Stable reconstruction formula of the filtered backprojection type.

Abstract

In this paper, we derive explicit reconstruction formulas for two common measurement geometries: a plane and a sphere. The problem is formulated as inverting the forward operator $R^a$, which maps the initial source to the measured wave data. Our first result pertains to planar observation surfaces. By extending the domain of $R^a$ to tempered distributions, we provide a complete characterization of its range and establish that the inverse operator $(R^a)^{-1}$ is uniquely defined and "almost" continuous in the distributional topology. Our second result addresses the case of a spherical observation geometry. Here, with the operator acting on $L^2$ spaces, we derive a stable reconstruction formula of the filtered backprojection type.