| Abstract: |
| Quantitative photoacoustic tomography (QPAT) seeks to recover optical parameters of a medium from internal absorbed-energy data generated by optical illumination and measured through acoustic waves. In practice, these data are most reliable only near the illuminated boundary, since attenuation and limited detector sensitivity reduce their usefulness at greater depth. This motivates a local inverse problem: reconstructing the absorption and scattering coefficients only in a boundary layer where the signal remains informative. We consider the stationary radiative transport equation in a smooth bounded domain, with absorption coefficient $\mu_a$, scattering coefficient $\mu_s$, and a prescribed scattering kernel. For a given inflow illumination $g$, the internal data take the form
\[
H(x,g)=\mu_a(x)\int_{\mathbb S^{n-1}}u(x,d)\,ds(d),
\]
where $u$ denotes the photon density, solution to the radiative transport equation. We show that with a finite number of suitably chosen illuminations, one obtains a local linear system whose invertibility guarantees unique recovery of the coefficients up to higher-order depth errors. The analysis provides a rigorous explanation of depth-limited resolution in (QPAT) and suggests a practical reconstruction strategy in strongly attenuating media. Numerical experiments support the asymptotic model and illustrate stable coefficient recovery in shallow subregions near the boundary. |
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