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Log‐Euclidean metrics for fast and simple calculus on diffusion tensors
TLDR
A new family of Riemannian metrics called Log‐Euclidean is proposed, based on a novel vector space structure for tensors, which can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms.
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices
TLDR
This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure.
A Log-Euclidean Framework for Statistics on Diffeomorphisms
TLDR
This article focuses on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm, which is a simple 3D vector field and well-defined for diffe morphisms close enough to the identity.
Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics
TLDR
It is shown that Riemannian metrics for tensors, and more specifically the log-Euclidean metrics, are a good candidate and that this criterion can be efficiently optimized and that the positive definiteness of tensors is always ensured.
A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration
TLDR
The results presented here on real 3D locally affine registration suggest that the novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated.
Fast and Simple Calculus on Tensors in the Log-Euclidean Framework
TLDR
New metrics called Log-Euclidean are proposed, which have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations.
Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics
TLDR
It is shown that Riemannian metrics for tensors, and more specifically the log-Euclidean metrics, are a good candidate and that this criterion can be efficiently optimized and that the positive definiteness of tensors is always ensured.
Fast and Simple Computations on Tensors with Log-Euclidean Metrics.
TLDR
This article presents a new familly of metrics, called Log-Euclidean, which have the same excellent theoretical properties as affine-invariant metrics and yield very similar results in practice, and presents experimental results for multilinear interpolation, dense extrapolation of tensors and anisotropic diffusion of tensor fields.
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