A Riemannian Framework for Tensor Computing

Abstract

A preliminary version appeared as INRIA Research Report 5255, July 2004.Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve

Analysis of Distance/Similarity Measures for Diffusion Tensor Imaging

“…Condition (7) is important in applications where you need to take the mean or do interpolation between tensors [2,12].…”

“…Additionally, it is invariant to any linear change of coordinates. Pennec et al [12] introduce a similar framework with the same distance measure, and extend it with methods for filtering and regularization of tensor fields. Arsigny et al [2] introduce a new Log-Euclidian framework.…”

“…Thus they cannot directly be translated into a distance measure that always fulfills metric condition (5 (6). They also fulfill the triangle inequality (7) if the tensors A and B are infinitesimally close [2,12,16]. Therefore, they are Riemannian metrics.…”

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

“…A number of teams in medical image processing proposed independently to endow this space with the affine-invariant metric X , Y Σ = Tr(V.Σ −1 .W.Σ −1 ) which is completely independent of the choice of the coordinate system. This allowed to generalize to SPD-valued images a number of image processing algorithms [57]. The same metric was previously introduced in statistics to model the geometry of the multivariate normal family (the Fisher information metric) [15,61,16].…”

“…Classically, in a manifold endowed with a Riemannian metric, the natural choice of mean is called the Riemannian center of mass or Fréchet mean. The Riemannian structure proves to be a powerful and consistent framework for computing simple statistics [72,50,53,[11][12][13]55] and can be extended to an effective computing framework on manifold-valued images [57]. On a Lie group, this Riemannian approach is consistent with the group operations if a biinvariant metric exists, which is for example the case for compact groups such as rotations [54,49].…”