Riemannian curvature-driven flows for tensor-valued data

Authors:
Mourad Zéraï;Maher Moakher
Affiliations:
Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at Tunis, ENIT, LAMSIN, Tunis Belvédère, Tunisia;Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at Tunis, ENIT, LAMSIN, Tunis Belvédère, Tunisia
Venue:
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Year:
2007

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Abstract

We present a novel approach for the derivation of PDE modeling curvature-driven flows for matrix-valued data. This approach is based on the Riemannian geometry of the manifold of symmetric positive-definite matrices P(n). The differential geometric attributes of P(n) -such as the bi-invariant metric, the covariant derivative and the Christoffel symbols- allow us to extend scalar-valued mean curvature and snakes methods to the tensor data setting. Since the data live on P(n), these methods have the natural property of preserving positive definiteness of the initial data. Experiments on three-dimensional real DT-MRI data show that the proposed methods are highly robust.