The Stability of the Backward Problem for Photoacoustic Imaging in Attenuating Media via Carleman Estimates
For researchers in photoacoustic imaging, this provides a theoretical foundation for stable reconstruction in attenuating media, addressing a previously ill-posed inverse problem.
This paper establishes the first conditional stability estimates for the backward problem in photoacoustic tomography with fractional-order damping using novel Carleman estimates, and proposes a Tikhonov regularization method with proven convergence rates, validated numerically.
This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments.