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 $\mathbb{R}\times(0,\infty)$ 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.

中文翻译：

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二维光的数学建模--片状荧光显微镜图像重建

我们研究了光片荧光显微镜 (LSFM) 的逆问题，其中需要重建荧光分子的密度。我们的第一步是提出一个数学模型来描述光学相机在 LSFM 实验期间获得的测量结果。考虑了两个有意义的阶段：激发和荧光。我们提出了一个近轴模型来描述与费米笔形束方程直接相关的激发过程。对于荧光阶段，我们使用传输方程来描述光子向检测相机的传输。对于建模后得到的数学逆问题，我们提出了唯一性结果，将问题重新定义为 $\mathbb{R}\times(0,\infty)$ 中热方程初始条件的恢复。在时空曲线中。此外，我们提出了数值实验，通过离散化所提出的模型并将这个问题作为一个大而稀疏的线性系统的解决方案来恢复荧光分子的密度。一些迭代和正则化的方法被用来实现这个目标。结果表明，与目前使用的直接采集方法相比，求解逆问题可以获得更好的重建效果。