Hyperspectral Diffuse Optical Tomography Using the Reduced Basis Method and Sparsity Constraints

Diffuse Optical Tomography (DOT) has long been investigated as an effective imaging technique for soft tissue imaging, such as breast cancer detection. DOT has many benefits, including its use of non-ionizing light and its ability to produce high contrast images, but it also has low resolution. In recent years hyperspectral DOT (hyDOT) has been proposed, in an effort to improve that resolution by adding more information in the spectral domain. In this imaging modality, hundreds or even thousands of different wavelengths in the visible to near infrared range are used in the imaging process. Since tissue absorbs and scatters light differently at different wavelengths, it has been conjectured that this increase of information should provide images that give a better overall idea of the complete spatial reconstruction of the optical parameters. Although hyDOT has been investigated experimentally, a formal theoretical investigation into its mathematical foundations has not been thoroughly performed. This dissertation seeks to lay the groundwork for the mathematical formulation of this imaging modality. First, the forward problem for hyDOT is formulated and the spectral regularity of the solution investigated. We demonstrate that the solution to the governing PDE is very smooth with respect to wavelength. This spectral regularity allows for the application of a model reduction technique to the forward problem known as the Reduced Basis Method. Several proofs are given for the hyDOT forward solution and the spectral regularity term, including existence and uniqueness proofs and proofs showing the continuity of the solution with respect to the diffusion and absorption coefficients and the wavelength. The appropriate function spaces for the optical coefficients with respect to their dependence on the wavelength are explored and a new norm is proposed. Additionally, the hyDOT inverse problem is formulated. New cost functionals are proposed to solve the inverse problem that incorporate the spatial sparsity of the optical parameters and their spectral regularity. Finally, a gradient-based reconstruction algorithm that enforces the spatial sparsity with respect to wavelength, is shown to be very effective and robust in solving the hyDOT inverse problem when used on simulations with a simple geometry