Tensor-based morphometry (TBM) studies encode the anatomical information in spatial deformations which are locally characterized by Jacobian matrices. Current methods perform voxel-wise statistical analysis on some features, such as the Jacobian determinant or the Cauchy–Green deformation tensor, which are not complete descriptors of the local deformation. This article introduces a right-invariant Riemannian distance on the GL+ (n) group of Jacobian matrices making use of the complete geometrical information of the local deformation. A numerical recipe for the computation of the proposed distance is given. Additionally, experiments are performed on both a synthetic deformation study and a cross-sectional brain MRI study.
Zacur, E, Bossa, M & Olmos, S 2014, 'Multivariate tensor-based morphometry with a right-invariant riemannian distance on GL+(n)', Journal of Mathematical Imaging and Vision, vol. 50, no. 1, pp. 18-31. https://doi.org/10.1007/s10851-013-0479-7
Zacur, E., Bossa, M., & Olmos, S. (2014). Multivariate tensor-based morphometry with a right-invariant riemannian distance on GL+(n). Journal of Mathematical Imaging and Vision, 50(1), 18-31. https://doi.org/10.1007/s10851-013-0479-7
@article{8fe0e06932724edda58c674429d23475,
title = "Multivariate tensor-based morphometry with a right-invariant riemannian distance on GL+(n)",
abstract = "Tensor-based morphometry (TBM) studies encode the anatomical information in spatial deformations which are locally characterized by Jacobian matrices. Current methods perform voxel-wise statistical analysis on some features, such as the Jacobian determinant or the Cauchy–Green deformation tensor, which are not complete descriptors of the local deformation. This article introduces a right-invariant Riemannian distance on the GL+ (n) group of Jacobian matrices making use of the complete geometrical information of the local deformation. A numerical recipe for the computation of the proposed distance is given. Additionally, experiments are performed on both a synthetic deformation study and a cross-sectional brain MRI study.",
keywords = "Jacobian matrices, Right-invariant Riemannian metric, Statistics on manifolds, Tensor-based morphometry",
author = "Ernesto Zacur and Matias Bossa and Salvador Olmos",
year = "2014",
month = jan,
day = "1",
doi = "10.1007/s10851-013-0479-7",
language = "English",
volume = "50",
pages = "18--31",
journal = "Journal of Mathematical Imaging and Vision",
issn = "0924-9907",
publisher = "Springer",
number = "1",
}