A spline-based forward model for Optical Diffuse Tomography

Reconstruction algorithms for Optical Diffuse Tomography (ODT) rely heavily on fast and accurate forward models. Arbitrary geometries and boundary conditions need to be handled rigorously since they are the only input to the inverse problem. From this perspective, Finite Element Methods (FEM) are good candidates to implement a forward model. However, these methods require to mesh the domain of interest, which is impractical on a routine basis. The other downside of the FEM is that the basis functions are often not compatible with the ones used for solving the inverse problem, which typically have less degrees of freedom. In this work, we tackle the 2D problem, and propose a forward model that uses a mesh-free discretization based on linear B-Splines. It combines the advantages of the FEM, while offering a fast and much simpler way of handling complex geometries. Another motivation for this work is that the underlying B-spline model is equally suitable for the subsequent reconstruction part of the process (solving the inverse problem). In particular, it is compatible with wavelets and multiresolution-type signal representations.