Fast implementation for fluorescence tomography based on coordinate descent with limited measurements

Fluorescence molecular tomography (FMT) can three-dimensionally resolve molecular activities in in vivo small animal through the reconstruction of the distribution of fluorescent probes. Due to large number of unknowns and limited measurements from the surfaces of small animals, the FMT problem is often ill-posed and ill-conditioned. Though various L2-norm regularizations can make the solution stable, they usually make the solution over-smoothed. During the early stages of tumor detection, fluorescent sources that indicate the distribution of tumors are usually small and sparse, which can be regarded as a type of a priori information. L1-norm regularizations have been incorporated to promote the sparsity of optical tomographic problems. In this paper, an efficient method with the L1-norm regularization based on coordinate descent is proposed to solve the FMT problem with extremely limited measurements. The proposed method minimizes the objective by solving a sequence of scalar minimization subproblems in multi-variable minimization. Each subproblem improves the estimate of the solution via minimizing along a determined coordinate with all other coordinates fixed. This algorithm first updates the coordinate that makes the energy decrease the most. Non-existence of matrix-vector multiplication in the iteration process makes the proposed algorithm time-efficient. To evaluate this method, we compare it to the iterated-shrinkage-based algorithm with L1-norm regularization in numerical experiments. The proposed algorithm is able to obtain satisfactory reconstruction results even when the measurements are very limited. Besides, the proposed algorithm is about two orders of magnitude faster than the iterated-shrinkage-based algorithm, which enables the proposed algorithm into practical applications.