| Optical diffusion tomography is a new imaging technique in which a near infrared laser source is used to obtain measured data for reconstructing the absorption and/or scattering property of a medium. In the tissue imaging application, the low energy optical radiation presents significantly lower health risks than x-ray radiation. However, the photon path must be described by a partial differential equation, which presents a mathematically challenging nonlinear inverse problem. In order to achieve high resolution and stable image reconstructions, this dissertation deals with various issues for iterative regularized inversion algorithms. A Bayesian framework with shot noise detection statistics and the generalized Gaussian Markov random field prior model is introduced as a reasonable estimation criterion for the inverse problem. The feasibility of standard nonlinear optimization algorithms for the resultant maximum a posteriori estimation problem is proven by showing the Fréchet differentiability of the forward scattering operator. Based on the analysis of the Fréchet derivative, a computationally efficient optimization method, called iterative coordinate descent Born (ICD/Born), is developed by combining the coordinate-wise update technique and successive Born approximations, thereby providing high resolution reconstructions. Finally, a fast implementation of the ICD/Born approach is derived based on nonlinear multi-grid algorithms, resulting in an order of magnitude speed-up over the fixed grid ICD/Born algorithm. |
