Fast GL(n)-Invariant Framework for Tensors Regularization

AbstractWe propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on a local differential geometric approach. The manifold of symmetric positive-definite (SPD) matrices, Pn, is parameterized via the Iwasawa coordinate system. In this framework distances on Pn are measured in terms of a natural GL(n)-invariant metric. Via the mathematical concept of fiber bundles, we describe the tensor-valued image as a section where the metric over the section is induced by the metric over Pn. Then, a functional over the sections accompanied by a suitable data fitting term is defined. The variation of this functional with respect to the Iwasawa coordinates leads to a set of $\frac{1}{2}n(n+1)$ coupled equations of motion. By means of the gradient descent method, these equations of motion define a Beltrami flow over Pn. It turns out that the local coordinate approach via the Iwasawa coordinate system results in very simple numerics that leads to fast convergence of the algorithm. Regularization results as well as results of fibers tractography for DTI are presented.