IMAGE LUNCH:: SPD Tensors Regularization via Iwasawa Decomposition

In this talk I will present an algorithm for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This algorithm is based on fundamentals of differential geometry, and on unique tensor decomposition, the Iwasawa decomposition. As a result of this decomposition, the original elements of the tensors are replaced by a new set of elements. This new set of elements defines the Iwasawa coordinate system which is used here to parameterize the manifold of SPD tensors. In this framework, the manifold is equipped with the natural GL(n)-invariant Riemannian metric, defined with respect to the new coordinate system. Then, using this metric and a functional over the tensor field, we construct a set of Laplace-Beltrami equations for the Iwasawa coordinates. The numerical solution of this set of equations eventually defines a multi-channel regularization process on the manifold of SPD tensors. To demonstrate this algorithm, at the end of the talk I will present regularization results of real, in vivo, DT-MRI datasets, as well as regularized fiber tractography.