A Riemannian Framework for Tensor Computing

By Xavier Pennec, Pierre Fillard, Nicholas Ayache. International Journal of Computer Vision, 2005

1,787 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/0f445efe6f4dd1037ef9ab757e26030b4fbaed7b

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In Computational Mechanics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Online action recognition using covariance of shape and motion (2014, methodology): “In this case, the distances between a pair of points C1 and C2 on the manifold is defined as the geodesic distance between these points, and is computed using the Affine-Invariant distance metric [26]:”
Pedestrian Detection via Classification on Riemannian Manifolds (2008, background): “…Symþd can be formulated as a connected Riemannian manifold, and an invariant Riemannian metric on the tangent space of Symþd is given by [35] < y; z >X¼ trace X 1 2yX 1zX 1 2 : ð13Þ The exponential map associated to the Riemannian metric expXðyÞ ¼ X 1 2 exp X 1 2yX 1 2 X 1 2 ð14Þ is a global…”
From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices (2014, methodology): “One of the motivations for projecting a higher-dimensional SPD manifold to a lower-dimensional one is to preserve the properties of δ(2) g and δ 2 S [16,19].”
Generalized BackPropagation, Étude De Cas: Orthogonality (2016, background): “The p.d. constraint can be added to the layers of a network by exploiting the geometry of the Symmetric Positive Deﬁnite (SPD) manifolds Pennec et al. (2006) in gBP.”
Distances and inference for covariance operators (2014, background): “3 Interpolation and extrapolation Interpolation and extrapolation in non-linear spaces has been studied in depth for the case of positive definite matrices, for which it has been shown that simply using the Euclidean metric can be very problematic (see, e.g, Arsigny et al., 2006; Pennec et al., 2006”
Clinical DT-MRI estimation, smoothing and fiber tracking with Log-Euclidean metrics (2006, background): “Af ﬁ ne-invariant Riemannian metrics [6, 7, 8] overcome the limitations of the Euclidean calculus on tensors, like the appearance of negative eigenvalues and the swelling effect as described in [5].”
Elastic Functional Coding of Riemannian Trajectories (2016, background): “As described in [28], for a rectangular region R, let {zk}k=1.”
Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces (2014, background): “The first approach exploits the affine-invariant metric, which is the classical Riemannian metric on Sym++(n) [18, 16, 3, 19, 4, 24].”
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices (2007, methodology): “The framework of Riemannian geometry [8] is particularly adapted to this task, since many statistical tools [18] and PDEs can be generalized to this framework.”
Rate-Invariant Analysis of Trajectories on Riemannian Manifolds with Application in Visual Speech Recognition (2014, background): “and others [9, 10] formulated this manifold as the space of nonsingular covariance matrices, i.”
A Riemannian Network for SPD Matrix Learning (2016, methodology): “They are Covariance Discriminative Learning (CDL) [36], CDL on approximate infinitedimensional region covariance descriptors obtained by random Fourier feature (CDL-F) [13], Log-Euclidean Metric Learning (LEML) [17] and SPD Manifold Learning (SPDML) [15] that uses affine-invariant metric (AIM) [1] a”
Monotonic properties of the least squares mean (2010, background): “) Recent contributions that have advocated the use of this metric in applications include [13, 26, 30] for tensor computing in medical imaging and [4] for radar processing.”