Explicit Inversion of the Attenuated Photoacoustic Operator in General Observation Geometries

Cong Shi

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arXiv · 2508.19064

Explicit Inversion of the Attenuated Photoacoustic Operator in General Observation Geometries

Cong Shi

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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.

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Cong Shi. 2025-08-26. Explicit Inversion of the Attenuated Photoacoustic Operator in General Observation Geometries. https://arxiv.org/abs/2508.19064

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