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A Riemannian Framework for Tensor Computing

This paper proposes to endow the tensor space with an affine-invariant Riemannian metric and demonstrates that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries, the geodesic between two tensors and the mean of a set of tensors are uniquely defined.
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Log‐Euclidean metrics for fast and simple calculus on diffusion tensors

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.
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Diffeomorphic demons: Efficient non-parametric image registration

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Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices

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.
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Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

  • X. Pennec
  • Mathematics
    Journal of Mathematical Imaging and Vision
  • 1 July 2006
This paper provides a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them and develops the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and χ2 law.
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Deep Learning Techniques for Automatic MRI Cardiac Multi-Structures Segmentation and Diagnosis: Is the Problem Solved?

How far state-of-the-art deep learning methods can go at assessing CMRI, i.e., segmenting the myocardium and the two ventricles as well as classifying pathologies is measured, to open the door to highly accurate and fully automatic analysis of cardiac CMRI.
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Multi-scale EM-ICP: A Fast and Robust Approach for Surface Registration

It is shown that EMICP robustly aligns the barycenters and inertia moments with a high variance, while it tends toward the accurate ICP for a small variance, and is used in a multi-scale approach using an annealing scheme on this parameter to combine robustness and accuracy.
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Symmetric Log-Domain Diffeomorphic Registration: A Demons-Based Approach

This work proposes a non-linear registration algorithm perfectly fit for log-Euclidean statistics on diffeomorphisms that outperforms both the demons algorithm and the recently proposed diffeomorphic demons algorithm in terms of accuracy of the transformation while remaining computationally efficient.
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A Log-Euclidean Framework for Statistics on Diffeomorphisms

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.
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Reconstructing a 3D structure from serial histological sections

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