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The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data
Journal of Mathematical Imaging and Vision (2011)
  • Maher Moakher, University of Tunis El Manar
  • Mourad Zéraï, University of Tunis El Manar
Abstract
In this paper we present a Riemannian framework for smoothing data that are constrained to live in P(n) , the space of symmetric positive-definite matrices of order n. We start by giving the differential geometry of P(n) , with a special emphasis on 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.
Disciplines
Publication Date
2011
Citation Information
Maher Moakher and Mourad Zéraï. "The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data" Journal of Mathematical Imaging and Vision Vol. 40 Iss. 2 (2011)
Available at: http://works.bepress.com/maher_moakher/8/