Total variation (TV) is a powerful regularization method that has been widely
applied in different imaging applications, but is difficult to apply to diffuse
optical tomography (DOT) image reconstruction (inverse problem) due to complex
and unstructured geometries, non-linearity of the data fitting and
regularization terms, and non-differentiability of the regularization term. We
develop several approaches to overcome these difficulties by: i) defining
discrete differential operators for unstructured geometries using both finite
element and graph representations; ii) developing an optimization algorithm
based on the alternating direction method of multipliers (ADMM) for the
non-differentiable and non-linear minimization problem; iii) investigating
isotropic and anisotropic variants of TV regularization, and comparing their
finite element- and graph-based implementations. These approaches are evaluated
on experiments on simulated data and real data acquired from a tissue phantom.
Our results show that both FEM and graph-based TV regularization is able to
accurately reconstruct both sparse and non-sparse distributions without the
over-smoothing effect of Tikhonov regularization and the over-sparsifying
effect of L1 regularization. The graph representation was found to
out-perform the FEM method for low-resolution meshes, and the FEM method was
found to be more accurate for high-resolution meshes.Comment: 24 pages, 11 figures. Reviced version includes revised figures and
improved clarit