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

  • Authors:

  • Maher Moakher;Mourad Zéraï

  • Affiliations:

  • Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at Tunis, University of Tunis El Manar, ENIT-LAMSIN, Tunis Belvédère, Tunisia 1002;Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at Tunis, University of Tunis El Manar, ENIT-LAMSIN, Tunis Belvédère, Tunisia 1002 ...

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2011

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Abstract

In this paper we present a Riemannian framework for smoothing data that are constrained to live in $\mathcal{P}(n)$ , the space of symmetric positive-definite matrices of order n. We start by giving the differential geometry of $\mathcal{P}(n)$ , with a special emphasis on $\mathcal{P}(3)$ , 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.