The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data

In this paper we present a Riemannian framework for smoothing data that are constrained to live in P(n)\mathcal{P}(n), the space of symmetric positive-definite matrices of order n. We start by giving the differential geometry of P(n)\mathcal{P}(n), with a special emphasis on P(3)\mathcal{P}(3), considered at a level of detail far greater than heretofore. We then use the harmonic map and minimal immersion theories to construct three flows that drive a noisy field of symmetric positive-definite data into a smooth one. The harmonic map flow is equivalent to the heat flow or isotropic linear diffusion which smooths data everywhere. A modification of the harmonic flow leads to a Perona-Malik like flow which is a selective smoother that preserves edges. The minimal immersion flow gives rise to a nonlinear system of coupled diffusion equations with anisotropic diffusivity. Some preliminary numerical results are presented for synthetic DT-MRI data.