Sparsity Regularization in Diffuse Optical Tomography

The purpose of this dissertation is to improve image reconstruction in Diffuse Optical Tomography (DOT), a high contrast imaging modality that uses a near infrared light source. Because the scattering and absorption of a tumor varies significantly from healthy tissue, a reconstructed spatial representation of these parameters serves as tomographic image of a medium. However, the high scatter and absorption of the optical source also causes the inverse problem to be severely ill posed, and currently only low resolution reconstructions are possible, particularly when using an unmodulated direct current (DC) source. In this work, the well posedness of the forward problem and possible function space choices are evaluated, and the ill posed nature of the inverse problem is investigated along with the uniqueness issues stemming from using a DC source. Then, to combat the ill posed nature of the problem, a physically motivated additional assumption is made that the target reconstructions have sparse solutions away from simple backgrounds. Because of this, and success with a similar implementation in Electrical Impedance Tomography, a sparsity regularization framework is applied to the DOT inverse problem. The well posedness of this set up is rigorously proved through new regularization theory results and the application of a Hilbert space framework similar to recent work. With the sparsity framework justified in the DOT setting, the inverse problem is solved through a novel smoothed gradient and soft shrinkage algorithm. The effectiveness of the algorithm, and the sparsity regularization of DOT, is evaluated through several numerical simulations using a DC source with comparison to a Levenberg Marquardt implementation and published error results