Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing

Lenglet, Christophe; Rousson, Mikaël; Deriche, Rachid; Faugeras, Olivier

doi:10.1007/s10851-006-6897-z

Cited by 242 publications

(265 citation statements)

References 37 publications

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“…It has been introduced in statistics to model the geometry of the multivariate normal family (the Fisher information metric) [18,64,19] and in simultaneously by many teams in medical image analysis to deal with DTI [33,52,9,42,58]. In [58], we showed that this metric could be used not only to compute distances between tensors, but also as the basis of a complete computational framework on manifoldvalued images as will be detailed in Section 3.…”

Section: Example Of Metrics On Covariance Matricesmentioning

confidence: 99%

Statistical Computing on Non-Linear Spaces for Computational Anatomy

Pennec

Fillard

2015

Handbook of Biomedical Imaging

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Computational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings.

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Section: Example Of Metrics On Covariance Matricesmentioning

confidence: 99%

Statistical Computing on Non-Linear Spaces for Computational Anatomy

Pennec

Fillard

2015

Handbook of Biomedical Imaging

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“…As the spectral decomposition is not unique, a preprocess step, where the eigenvectors are reoriented, is needed and is not trivial. More recently, differential geometric approaches have been developed to generalize the PCA to tensor data [7], for statistical segmentation of tensor images [8], for computing a geometric mean and an intrinsic anisotropy index [9], or as the basis of a full framework for Riemannian tensor calculus [10]. In this last work, we endow the space of tensors with an affine-invariant Riemannian metric to obtain results that are independent of the choice of the spatial coordinate system.…”

Section: A Riemannian Framework For Tensor Calculusmentioning

confidence: 99%

A Riemannian Framework for the Processing of Tensor-Valued Images

Fillard

Arsigny

Ayache

et al. 2005

Lecture Notes in Computer Science

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Abstract. 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 believe that the choice of a more adapted metric could significantly improve the robustness of the result.

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“…Clinical use of dwMRI is hampered by the fact that dwMRI analysis requires radically new approaches, based on abstract representations, a development still in its infancy. Examples are rank-2 symmetric positive-definite tensor representations in diffusion tensor imaging (DTI), pioneered by Basser, Mattiello and Le Bihan et al [1,2] and explored by many others [3,4,5,6,7,8,9,10,11,12,13,14], higher order symmetric positive-definite tensor representations [15,16,17,18,19,20,21], spherical harmonic representations in high angular resolution diffusion imaging (HARDI) [22,23,24,25,26], and SE(3) Lie group representations [27,28,29]. The latter type of representation, developed by Duits et al, appears to bear a particularly close relationship to the theory outlined below.…”

Section: Introductionmentioning

confidence: 99%