A Riemannian Framework for the Processing of Tensor-Valued Images

In this paper, we present a novel framework to carry out computations on tensors, i.e. symmetric positive definite matrices. We endow the space of tensors with an affine-invariant Riemannian metric, which leads to strong theoretical properties: The space of positive definite symmetric matrices is replaced by a regular and geodesically complete manifold without boundaries. Thus, tensors with non-positive eigenvalues are at an infinite distance of any positive definite matrix. Moreover, the tools of differential geometry apply and we generalize to tensors numerous algorithms that were reserved to vector spaces. The application of this framework to the processing of diffusion tensor images shows very promising results. We apply this framework to the processing of structure tensor images and show that it could help to extract low-level features thanks to the affine-invariance of our metric. However, the same affine-invariance causes the whole framework to be noise sensitive and we be...

• Type: Conference
• Year: 2005
• Where: DSSCV
• Authors: Pierre Fillard, Vincent Arsigny, Nicholas Ayache, Xavier Pennec
• Keywords: Computer Vision, DSSCV 2005, Positive Definite, Symmetric Positive Definite, Tensor