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

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 imaging 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 data set of human brain diffusion-weighted magnetic resonance imaging, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric.

Keywords:

• anisotropy
• metric
• positive definite
• power
• Procrustes
• Riemannian
• smoothing
• weighted Fréchet mean

AMS Subject Classifications:

• 62G05
• 62H11
• 62H35
• 62P10

Acknowledgments

The diffusion MR image data used in this paper are provided by the Division of Academic Radiology, University of Nottingham and Queen's Medical Centre, UK, and the authors would like to thank Dr Paul Morgan and Professor Dorothee Auer for their assistance. The authors are thankful to the editor and anonymous referees for their helpful comments and suggestions on improving the manuscript. The third author thanks God for making this work possible.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental online material for this article contains statements and proofs of the main theoretical results and can be accessed at http://dx.doi.org/10.1080/02664763.2015.1080671.

Notes

1. The reference tensor is already on the real scale that is determined by the b-value of 1000 s/mm${}^2$.

Additional information

Funding

This work was supported by the European Commission FP6 Marie Curie Action Programme under the CMIAG (Collaborative Medical Image Analysis on Grid) project. The second author acknowledges support from a Royal Society Wolfson Research Merit Award and EPSRC grant EP/K022547/1. The third author acknowledges support from the National Institute for Health Research grant i4i FPD2.