Mathematical Modeling for 2D Light-Sheet Fluorescence Microscopy image reconstruction

TL;DR

This paper develops a mathematical model for 2D Light-Sheet Fluorescence Microscopy, addressing the inverse problem of reconstructing fluorescent molecule density, and demonstrates improved image reconstructions through numerical methods.

Contribution

It introduces a novel paraxial model for LSFM and formulates the inverse problem as a heat equation initial condition recovery, with numerical validation.

Findings

The proposed model accurately describes LSFM measurements.

Numerical methods improve reconstruction quality over direct acquisition.

The inverse problem solution outperforms current direct methods.

Abstract

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.