Inverse optical tomography through PDE constrained optimisation in $L^\infty$

TL;DR

This paper develops a mathematical framework for Fluorescent Optical Tomography (FOT) using PDE-constrained optimization in $L^ Infinity$, aiming to improve the precision and foundational understanding of this non-invasive biomedical imaging technique.

Contribution

It introduces novel $L^ Infinity$ calculus of variations methods to model FOT as a PDE-constrained minimization problem, establishing new mathematical foundations.

Findings

Formulation of FOT as an $L^ Infinity$ optimization problem.

Development of PDE-constrained minimization framework.

Theoretical groundwork for future numerical methods.

Abstract

Fluorescent Optical Tomography (FOT) is a new bio-medical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses non-ionising red and infrared light. Mathematically, FOT can be modelled as an inverse parameter identification problem, associated with a coupled elliptic system with Robin boundary conditions. Herein we utilise novel methods of Calculus of Variations in $L^\infty$ to lay the mathematical foundations of FOT which we pose as a PDE-constrained minimisation problem in $L^p$ and $L^\infty$.