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Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics

Zhou, Diwei; Dryden, Ian L.; Koloydenko, Alexey A.; Audenaert, Koenraad M.R.; Bai, Li

Authors

Diwei Zhou

IAN DRYDEN IAN.DRYDEN@NOTTINGHAM.AC.UK
Professor of Statistics

Alexey A. Koloydenko

Koenraad M.R. Audenaert

Li Bai



Abstract

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical considerations are discussed, with some examples. Second we discuss weighted Procrustes methods for diffusion tensor interpolation and smoothing, and we compare methods based on different metrics on a set of examples as well as analytically. We establish a key relationship between the principal-square-root-Euclidean metric and the size-and-shape Procrustes metric on the space of symmetric positive semi-definite tensors. We explain, both analytically and by experiments, why the size-and-shape Procrustes metric may be preferred in practical tasks of interpolation, extrapolation, and smoothing, especially when observed tensors are degenerate or when a moderate degree of tensor swelling is desirable. Third we introduce regularisation methodology, which is demonstrated to be useful for highlighting features of prior interest and potentially for segmentation. Finally, we compare several metrics in a dataset of human brain diffusion-weighted MRI, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric.

Citation

Zhou, D., Dryden, I. L., Koloydenko, A. A., Audenaert, K. M., & Bai, L. (2016). Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics. Journal of Applied Statistics, 43(5), 943-978. https://doi.org/10.1080/02664763.2015.1080671

Journal Article Type Article
Acceptance Date Aug 4, 2015
Online Publication Date Sep 23, 2015
Publication Date Jan 1, 2016
Deposit Date Mar 9, 2017
Publicly Available Date Mar 29, 2024
Journal Journal of Applied Statistics
Print ISSN 0266-4763
Electronic ISSN 1360-0532
Publisher Routledge
Peer Reviewed Peer Reviewed
Volume 43
Issue 5
Pages 943-978
DOI https://doi.org/10.1080/02664763.2015.1080671
Keywords Anisotropy; Metric; Positive definite; Power; Procrustes; Riemannian; Smoothing; Weighted Frechet mean
Public URL https://nottingham-repository.worktribe.com/output/980460
Publisher URL http://www.tandfonline.com/doi/full/10.1080/02664763.2015.1080671
Additional Information This is an Accepted Manuscript of an article published by Taylor & Francis Group in Journal of Applied Statistics on 23/09/2015, available online: http://www.tandfonline.com/doi/full/10.1080/02664763.2015.1080671

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