Spherical averages and applications to spherical splines and interpolation
ACM Transactions on Graphics (TOG)
Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Statistical variability in nonlinear spaces: application to shape analysis and dt-mri
Statistical variability in nonlinear spaces: application to shape analysis and dt-mri
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision
Statistics of shape via principal geodesic analysis on lie groups
CVPR'03 Proceedings of the 2003 IEEE computer society conference on Computer vision and pattern recognition
Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis
MICCAI'05 Proceedings of the 8th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part I
Unsupervised Riemannian Clustering of Probability Density Functions
ECML PKDD '08 Proceedings of the 2008 European Conference on Machine Learning and Knowledge Discovery in Databases - Part I
Speech Emotion Classification on a Riemannian Manifold
PCM '08 Proceedings of the 9th Pacific Rim Conference on Multimedia: Advances in Multimedia Information Processing
Fast GL(n)-Invariant Framework for Tensors Regularization
International Journal of Computer Vision
4th order diffusion tensor interpolation with divergence and curl constrained Bézier patches
ISBI'09 Proceedings of the Sixth IEEE international conference on Symposium on Biomedical Imaging: From Nano to Macro
Learning averages over the lie group of symmetric positive-definite matrices
IJCNN'09 Proceedings of the 2009 international joint conference on Neural Networks
An algorithm to compute averages on matrix Lie groups
IEEE Transactions on Signal Processing
Visual Tracking via Particle Filtering on the Affine Group
International Journal of Robotics Research
Geodesic-loxodromes for diffusion tensor interpolation and difference measurement
MICCAI'07 Proceedings of the 10th international conference on Medical image computing and computer-assisted intervention - Volume Part I
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part VI
Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank
SIAM Journal on Matrix Analysis and Applications
Free form face recognition using kernel sparse representation
Proceedings of the Seventh Indian Conference on Computer Vision, Graphics and Image Processing
MICCAI'10 Proceedings of the 13th international conference on Medical image computing and computer-assisted intervention: Part I
Statistical analysis of tensor fields
MICCAI'10 Proceedings of the 13th international conference on Medical image computing and computer-assisted intervention: Part I
Interpolating 3D diffusion tensors in 2D planar domain by locating degenerate lines
ISVC'10 Proceedings of the 6th international conference on Advances in visual computing - Volume Part I
International Journal of Computer Vision
Diffeomorphism invariant riemannian framework for ensemble average propagator computing
MICCAI'11 Proceedings of the 14th international conference on Medical image computing and computer-assisted intervention - Volume Part II
Image and Vision Computing
Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds
Image and Vision Computing
Manifold statistics for essential matrices
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part II
Linear invariant tensor interpolation applied to cardiac diffusion tensor MRI
MICCAI'12 Proceedings of the 15th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part II
Unscented Kalman Filtering on Riemannian Manifolds
Journal of Mathematical Imaging and Vision
IPMI'13 Proceedings of the 23rd international conference on Information Processing in Medical Imaging
Anisotropy Preserving DTI Processing
International Journal of Computer Vision
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The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positive-definite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision.