Tensors are nowadays an increasing research domain in different areas, especially in image processing, motivated for example by diffusion tensor magnetic resonance imaging (DT-MRI). Up to now, algorithms and tools developed to deal with tensors were founded on the assumption of a matrix vector space with the constraint of remaining symmetric positive definite matrices. On the contrary, our approach is grounded on the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions, where it is possible to define an affine-invariant Riemannian metric and express statistics on the manifold of symmetric positive definite matrices. In this paper, we focus on the contribution of these tools to the anisotropic filtering and regularization of tensor fields. To validate our approach we present promising results on both synthetic and real DT-MRI data.
Bibliographical noteFunding Information:
This research was partially supported by the European Project SIMILAR (FP6-507609), the Spanish National Projects USIMAG (TEC2004-06647-C03-02) and ECIM (TIC-2001-38008-C02-01), the French National Project ACI Obs-Cerv, the European Project IMAVIS (HPMT-CT-2000-00040) and the Région Provence-Alpes-Côte d’Azur. The authors would like to thank J.L. Anton, M. Roth (Centre IRMf, CHU la Timone, Marseille, France) and N. Wotawa for the human brain DTI datasets used in this paper.
- Anisotropic filtering
- Differential geometry
- Geodesic distance
- Nonlinear diffusion
- Riemannian manifold
- Tensor fields