Abstract. In this article we study the reconstruction problem in TAT/PAT on an attenuating media. Namely, we prove a reconstruction procedure of the initial condition for the damped wave equation via Neumann series that works for arbitrary large smooth attenuation coefficients extending the result of Homan in [5]. We also illustrate the theoretical result by including some numerical experiments at the end of the paper.
We study an inverse problem for Light Sheet Fluorescence Microscopy (LSFM), where the density of fluorescent molecules needs to be reconstructed. Our first step is to present a mathematical model to describe the measurements obtained by an optic camera during an LSFM experiment. Two meaningful stages are considered: excitation and fluorescence. We propose a paraxial model to describe the excitation process which is directly related with the Fermi pencil-beam equation. For the fluorescence stage, we use the transport equation to describe the transport of photons towards the detection camera. For the mathematical inverse problem that we obtain after the modeling, we present a uniqueness result, recasting the problem as the recovery of the initial condition for the heat equation in R × (0, ∞) from measurements in a space-time curve. Additionally, we present numerical experiments to recover the density of the fluorescent molecules by discretizing the proposed model and facing this problem as the solution of a large and sparse linear system. Some iterative and regularized methods are used to achieve this objective. The results show that solving the inverse problem achieves better reconstructions than the direct acquisition method that is currently used.
In X-ray CT scan with metallic objects, it is known that direct application of the filtered back-projection (FBP) formula leads to streaking artifacts in the reconstruction. These are characterized mathematically in terms of wave front sets in [13]. In this work, we give a quantitative microlocal analysis of such artifacts. We consider metal regions with strictly convex smooth boundaries and show that the streaking artifacts are conormal distributions to straight lines tangential to at least two boundary curves. For metal regions with piecewise smooth boundaries, we analyze the streaking artifacts especially due to the corner points. Finally, we study the reduction of the artifacts using appropriate filters.
We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.
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