Optical tomography as a PDE-constrained optimization problem

We report on the implementation of an augmented Lagrangian approach for solving the inverse problems in diffuse optical tomography (DOT). The forward model of light propagation is the radiative transport equation (RTE). The inverse problem is formulated as a minimization problem with the RTE being considered as an equality constraint on the set of 'optical properties-radiance' pairs. This approach allows the incorporation of the recently developed technique of PDE-constrained optimization, which has shown great promise in many applications that can be formulated as infinite-dimensional optimization problems. Compared to the traditional unconstrained optimization approaches for optical tomographic imaging where one solves several forward and adjoint problems at each optimization iteration, the method proposed in this work solves the forward and inverse problems simultaneously. We found in initial studies, using synthetic data, that the image reconstruction time can typically be reduced by a factor of 10 to 30, which depends on a combination of noise level, regularization parameter, mesh size, initial guess, optical properties and system geometry.