A fast solver for an inverse problem arising in bioluminescence tomography

Bioluminescence tomography (BLT) is a new method in biomedical imaging, with a promising potential in monitoring non-invasively physiological and pathological processes in vivo at the cellular and molecular levels. The goal of BLT is to quantitatively reconstruct a three dimensional bioluminescent source distribution within a small animal from two dimensional optical signals on the surface of the animal body. Mathematically, BLT is an under-determined inverse source problem and is severely ill-posed, making its numerical treatments very challenging. In this paper, we provide a new Tikhonov regularization framework for the BLT problem. Compared with the existing reconstruction methods about BLT, our new method uses an energy functional defined over the whole problem domain for measuring the data fitting, associated with two related but different boundary value problems. Based on the new formulation, a fast solver is introduced by transforming the proposed optimization model into a system of partial differential equations. Moreover, a finite element method is used to obtain a regularized discrete solution. Finally, numerical results show that the fast solver for BLT is feasible and effective.