A Riemannian approach to anisotropic filtering of tensor fields

C. A. Castaño-Moraga; C. Lenglet; R. Deriche; J. Ruiz-Alzola

doi:10.1016/j.sigpro.2006.02.049

A Riemannian approach to anisotropic filtering of tensor fields

C. A. Castaño-Moraga, C. Lenglet, R. Deriche, J. Ruiz-Alzola

Radiology

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Abstract

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.

Original language	English (US)
Pages (from-to)	263-276
Number of pages	14
Journal	Signal Processing
Volume	87
Issue number	2
DOIs	https://doi.org/10.1016/j.sigpro.2006.02.049
State	Published - Feb 2007

Bibliographical note

Funding 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.

Keywords

Anisotropic filtering
DT-MRI
Differential geometry
Geodesic distance
Nonlinear diffusion
Regularization
Riemannian manifold
Tensor fields

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abstract = "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.",

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author = "Casta{\~n}o-Moraga, {C. A.} and C. Lenglet and R. Deriche and J. Ruiz-Alzola",

note = "Funding 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 R{\'e}gion Provence-Alpes-C{\^o}te d{\textquoteright}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.",

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AU - Deriche, R.

AU - Ruiz-Alzola, J.

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KW - Differential geometry

KW - Geodesic distance

KW - Nonlinear diffusion

KW - Regularization

KW - Riemannian manifold

KW - Tensor fields

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A Riemannian approach to anisotropic filtering of tensor fields

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Keywords

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