Adjoint field methods for non-linear tomographic medical imaging problems

We show results for full three-dimensional non-linear inversion of the parameters of a diffusive partial differential equation, specifically for an optical tomography application. We compute functional derivatives of the parameters with respect to the mean-squared error using the adjoint field method, and implement two forms of regularization. In the first, a penalty term is introduced into the error functional, and in the second, the solution to the inverse problem is assumed to belong to a parametrized class of functions. In the case where this assumption is correct, our results demonstrate that the parameters can recovered with high accuracy, yielding a better inversion result than the traditional Tikhonov-type approach.