The DOI matches its Crossref record; 13 of the 13 references it cites were found in a registry or an index.
Geometric Means in a Novel Vector Space Structure on Symmetric Positive‐Definite Matrices
Abstract
In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive‐definite matrices, called Log‐Euclidean. The approach is based on two novel algebraic structures on symmetric positive‐definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. From bi‐invariant metrics on the Lie group structure, we define the Log‐Euclidean mean from a Riemannian point of view. This notion coincides with the usual Euclidean mean associated with the novel vector space structure. Furthermore, this means corresponds to an arithmetic mean in the domain of matrix logarithms. We detail the invariance properties of this novel geometric mean and compare it to the recently introduced affine‐invariant mean. The two means have the same determinant and are equal in a number of cases, yet they are not identical in general. Indeed, the Log‐Euclidean mean has a larger trace whenever they are not equal. Last but not least, the Log‐Euclidean mean is much easier to compute.
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Cites
A Riemannian Framework for Tensor Computing
Pennec, X., Fillard, P., & Ayache, N. (2006) · International Journal of Computer Vision · doi:10.1007/s11263-005-3222-z · Cited by 1,550
Self-citationLog-Euclidean metrics for fast and simple calculus on diffusion tensors
Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006) · Magnetic Resonance in Medicine · doi:10.1002/mrm.20965 · Cited by 1,113
Self-citationIntrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements
Pennec, X. (2006) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-006-6228-4 · Cited by 751
Self-citationA Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
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Lie Groups, Lie Algebras, and Representations
Hall, B. C. (2003) · Graduate Texts in Mathematics · doi:10.1007/978-0-387-21554-9 · Cited by 691
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Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics
Fillard, P., Pennec, X., Arsigny, V., & Ayache, N. (2007) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2007.899173 · Cited by 249
Self-citationApproximating the Logarithm of a Matrix to Specified Accuracy
Cheng, S. H., Higham, N. J., Kenney, C., & Laub, A. J. (2001) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479899364015 · Cited by 132
A Fast Spectral Method for Active 3D Shape Reconstruction
Li, J., & Hero, A. O. (2004) · Journal of Mathematical Imaging and Vision · doi:10.1023/b:jmiv.0000011324.14508.fb · Cited by 117
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Barachant, A., Bonnet, S., Congedo, M., & Jutten, C. (2012) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2011.2172210 · Cited by 878
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Semantic Segmentation with Second-Order Pooling
Carreira, J., Caseiro, R., Batista, J., & Sminchisescu, C. (2012) · Lecture Notes in Computer Science · doi:10.1007/978-3-642-33786-4_32 · Cited by 458
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Metadata
- Authors
- Arsigny, V., Fillard, P., Pennec, X., & Ayache, N.
- Venue
- SIAM Journal on Matrix Analysis and Applications
- Published
- 2007-01
- Volume
- 29
- Issue
- 1
- Pages
- 328-347
- ISSN
- 0895-4798 (ISSN-L)
- 1095-7162
- OpenAlex
- W1986964250
- Cited by
- 819
- References
- 13
- FWCI
- 28.25
- Type
- journal article
- Language
- en
- Open access
- closed
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