Coordinates-Based Diffusion Over the Space of Symmetric Positive-Definite Matrices

Abstract

In some recent applications, the objects of interest are matrices. Diffusion Tensor MRI or in short DTI is an example of such an application. In this case, the image is divided into voxels where each voxel is described by a 3 × 3 symmetric positive-definite (SPD) matrix. In this chapter, we present an intrinsic approach for diffusion over the space of n × n symmetric positive-definite matrices, denoted by Pn. The basis of this framework is the description of Pn as a Riemannian manifold by means of the local coordinates and a natural Riemannian metric. One may choose various coordinate systems to cover the Pn manifold. We show that the analytical calculations, as well as the numerical implementation, become simple by choosing Iwasawa coordinates. Then, we define a G L(n)-invariant metric over Pn with respect to these coordinates. The metric is defined by means of the scalar product on Symn, the space of n × n symmetric matrices. The “image” is described here as a section of a fiber bundle. Then, the metric over Pn is combined with the metric over the twoor three-dimensional image domain in order to form the metric over the section. By means of the Beltrami framework, we define a functional over the space of sections. Variation of this functional leads to a set of ½n(n + 1) coupled equations of motion with respect to the local coordinates on Pn. The solution of these equations defines a structure-preserving flow on this manifold. Finally, we demonstrate this framework on the case of P3 by smoothing of real DTI datasets.