On fluorophore imaging by nonlinear diffusion model with dynamical iterative scheme

Fluorescence imaging aims at recovering the absorption coefficient of fluorophore in biological tissues. Due to the nonlinear dependence of excitation and emission fields on the unknown coefficient, such an inverse problem with the boundary measurement as inversion input is nonlinear. We reformulate this inverse problem as an optimization problem for a non-convex cost functional consisting of the unknown absorption coefficient depending both on the excitation field and the emission one with a penalty term. The existence of minimizer of the cost functional is proved rigorously, with explicit expression for the Fréchet derivative of the cost functional. For seeking the local minimizer considered as the approximate solution to the inverse problem efficiently, we develop a dynamical process which makes the non-convex cost functional quadratic at each iteration step, by freezing the unknown coefficient for the excitation field. This new dynamical quadratic optimization process is proven convergent and decreases the amount of computations for the fluorescence imaging, while the reconstruction accuracy keeps almost unchanged. Such advantages are verified by several numerical examples for different configurations of the absorption coefficient to be identified, comparing our proposed scheme with the algorithm for the original non-convex cost functional.