Study of four regularization methods for the inverse problem in bioluminescence tomography

As a promising tool for in-vivo molecular imaging of small animals, Bioluminescence Tomography (BLT) aims at the quantitative reconstruction of the bioluminescent source distribution from the detected optical signals on the body surface. Mathematically, BLT is a highly ill-posed inverse problem per se. Most existing works are based on Tikhonov regularization in which the selection of the proper regular parameter is quite difficult. In this paper, two direct regularization methods, truncated singular value decomposition (TSVD) and truncated total least squares (TTLS), as well as two iterative regularization methods, conjugate gradient least squares (CGLS) and least squares QR decomposition (LSQR), are applied to the inverse problem in BLT, with the finite element method solving the diffusion equation. In the numerical simulation, a heterogeneous phantom is designed to compare and evaluate the four methods. The results show that all the four methods can reconstruct the position of bioluminescence sources accurately and are more convenient in the determination of regularization parameter than Tikhonov method. In addition, with a priori knowledge of the source permissible region employed in the reconstruction, the iterative methods are faster than the two direct methods. Among the four methods, LSQR performs quite stably when both model noise and measure noise are considered.