Abstract

In Fluorescence Microscopy, a small and almost transparent sample containing a distribution of fluorophore is illuminated, e.g. with a laser, to activate the fluorescence. A camera outside the sample detects the activated fluorescence light and the fluorophore distribution inside the sample is estimated from these exterior measurements. In Light Sheet Fluorescence Microscopy (LSFM), the strategy is to optically section the sample by illuminating a single plane perpendicular to the camera at a time, which produces a good reconstruction of the fluorophore distribution by direct imaging, but blurring artifacts are observed on the fluorophore reconstruction as you move further away from the illumination side. This motivated us to consider a mathematical model for LSFM that includes a small diffusion effect in the illumination stage of LSFM . In the model we considered, the reconstruction of the fluorophore distribution can be recast as a backwards heat equation inverse problem, where the goal is to reconstruct the initial condition in the heat equation from measurements of the solution in non-cylindrical space-time surface observation set. For this specific family of backwards heat equation problems, we obtain uniqueness and logarithmic stability results, which then translate into corresponding uniqueness and stability results for the LSFM inverse problem of reconstructing the fluorophore distribution. This is a joint work with Pablo Arratia, Victor Casta\~neda, Evelyn Cueva, Steffen H\”artel, Axel Osses and Benjamin Palacios.