Log-Euclidean metrics for fast and simple calculus on diffusion tensors
Open accessAbstract
Diffusion tensor imaging (DT‐MRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positive‐definite matrices), called “tensors” here, has proved difficult in recent years. Usual Euclidean operations on matrices suffer from many defects on tensors, which have led to the use of many ad hoc methods. Recently, affine‐invariant Riemannian metrics have been proposed as a rigorous and general framework in which these defects are corrected. These metrics have excellent theoretical properties and provide powerful processing tools, but also lead in practice to complex and slow algorithms. To remedy this limitation, a new family of Riemannian metrics called Log‐Euclidean is proposed in this article. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. This new approach is based on a novel vector space structure for tensors. In this framework, Riemannian computations can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms. Theoretical aspects are presented and the Euclidean, affine‐invariant, and Log‐Euclidean frameworks are compared experimentally. The comparison is carried out on interpolation and regularization tasks on synthetic and clinical 3D DTI data. Magn Reson Med 56, 2006. © 2006 Wiley‐Liss, Inc.
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The log-Euclidean distance addresses several issues with using Frobenius distances for positive semidefinite matrices.
“We choose to compute dissimilarities between functional connectivity matrices using the log-Euclidean distance (Arsigny et al. 2006), which addresses several issues with using Frobenius distances for positive semidefinite matrices.”
From Similarity-based multimodal regression (2024) · cites this paper as Arsigny et al. 2006
“The Defects of Euclidean Calculus” identifies problems with Frobenius-distance-based tensor calculations, and says log-Euclidean metrics preserve positive-definiteness. But the Conclusions state that the framework assumes positive-definite tensors; the claim’s broader reference to positive semidefinite matrices includes matrices with zero eigenvalues, which the paper does not cover. (LLM, conf=0.88)
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- Authors
- Arsigny, V., Fillard, P., Pennec, X., & Ayache, N.
- Affiliations
- Institut national de recherche en sciences et technologies du numérique (FR)
- Corresponding author
- Arsigny, Vincent
- Venue
- Magnetic Resonance in Medicine
- Publisher
- Wiley
- Published
- 2006-06-20
- Volume
- 56
- Issue
- 2
- Pages
- 411-421
- ISSN
- 0740-3194 (ISSN-L)
- 1522-2594
- OpenAlex
- W2048192550
- Cited by
- 1,113
- References
- 35
- Citations by year
- 2026: 46 · 2025: 34 · 2024: 35 · 2023: 42 · 2022: 44 · 2021: 62 · 2020: 55 · 2019: 64 · 2018: 64 · 2017: 63 · 2016: 78 · 2015: 89 · 2014: 62 · 2013: 69 · 2012: 75
- FWCI
- 34.38
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Magnetic Resonance in Medicine (publishedVersion)
- PubMed (publishedVersion)
- HAL (submittedVersion)
- Topics
- Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Advanced Neuroimaging Techniques and Applications
- Physical Sciences › Mathematics › Computational Mathematics › Tensor decomposition and applications
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Inequalities and Applications
- Keywords
- diffusion tensor imaging, Log-Euclidean metric, Riemannian metrics, symmetric positive definite matrices, matrix logarithm, affine invariance, tensor interpolation, regularization
- MeSH terms
- Algorithms, Brain Mapping, Humans, Image Processing, Computer-Assisted, Mathematics, Diffusion Magnetic Resonance Imaging
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