Riemannian center of mass and mollifier smoothing

Karcher, Hermann

doi:10.1002/cpa.3160300502

Cited by 935 publications

(820 citation statements)

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“…No further sphere or circle can be used to reduce the dimensionality. Instead, we find the Fréchet mean (Fréchet, 1944(Fréchet, , 1948Karcher, 1977;Bhattacharya & Patrangenaru, 2003)…”

Section: ·1 Geometry Of Nested Spheresmentioning

confidence: 65%

“…The Fréchet mean is unique when the support of x † i is a proper subset of a half-circle in S 1 (Karcher, 1977), which is often satisfied in practice. If there are multiple Fréchet means, then the data must be carefully inspected.…”

Section: ·1 Geometry Of Nested Spheresmentioning

confidence: 99%

“…Other candidates for w are the full Procrustes mean preshape and the geodesic mean of the preshapes. These are relevant to the Fréchet mean (Fréchet, 1948;Karcher, 1977;Bhattacharya & Patrangenaru, 2003), where the geodesic mean is the Fréchet mean with the Riemannian distance and the Procrustes mean is using the full Procrustes distance, which is extrinsic to S d . Recently, it has been observed that the curvature of the manifold S d sometimes makes the Fréchet mean inadequate, since it can lie outside the main support of the data.…”

Section: ·2 Principal Nested Spheres For Planar Shapesmentioning

confidence: 99%

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Analysis of principal nested spheres

Jung¹,

Dryden²,

Marron³

2012

Biometrika

130

135

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SUMMARYA general framework for a novel non-geodesic decomposition of high-dimensional spheres or high-dimensional shape spaces for planar landmarks is discussed. The decomposition, principal nested spheres, leads to a sequence of submanifolds with decreasing intrinsic dimensions, which can be interpreted as an analogue of principal component analysis. In a number of real datasets, an apparent one-dimensional mode of variation curving through more than one geodesic component is captured in the one-dimensional component of principal nested spheres. While analysis of principal nested spheres provides an intuitive and flexible decomposition of the high-dimensional sphere, an interesting special case of the analysis results in finding principal geodesics, similar to those from previous approaches to manifold principal component analysis. An adaptation of our method to Kendall's shape space is discussed, and a computational algorithm for fitting principal nested spheres is proposed. The result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation, as exemplified by several datasets.

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“…No further sphere or circle can be used to reduce the dimensionality. Instead, we find the Fréchet mean (Fréchet, 1944(Fréchet, , 1948Karcher, 1977;Bhattacharya & Patrangenaru, 2003)…”

Section: ·1 Geometry Of Nested Spheresmentioning

confidence: 65%

Section: ·1 Geometry Of Nested Spheresmentioning

confidence: 99%

Section: ·2 Principal Nested Spheres For Planar Shapesmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of principal nested spheres

Jung¹,

Dryden²,

Marron³

2012

Biometrika

130

135

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“…To define such a quantity, we can use the generalization of the arithmetic mean to manifolds. To be more specific, the mean or center of a set C of points in the metric space S (with respect to a distance D) has been given by Karcher in [28] as the element m C A S that minimizes the sum of square distances D's to the points x in the set, i.e.,…”

Section: Preliminariesmentioning

confidence: 99%

Tangent-based manifold approximation with locally linear models

Karygianni¹,

Frossard²

2014

Signal Processing

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a b s t r a c tIn this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represent manifold data accurately while preserving the manifold's structure. For this purpose, we employ a greedy technique that partitions manifold samples into groups, which are approximated by low dimensional subspaces. We start by considering each manifold sample as a different group and we use the difference of local tangents to determine appropriate group mergings. We repeat this procedure until we reach the desired number of sample groups. The best low dimensional affine subspaces corresponding to the final groups constitute our approximate manifold representation. Our experiments verify the effectiveness of the proposed scheme and show its superior performance compared to state-of-the-art methods for manifold approximation.

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“…The existence and uniqueness of these means on Riemannian manifolds has been studied first by Karcher (who relax the definition to local minima) [37] and then in [38,44,45,3,68,69]. Thus, it seems natural to investigate if we can define a Riemannian metric compatible with the Lie group operations.…”

Section: Bi-invariant Means In Lie Groupsmentioning

confidence: 99%

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

Pennec

Arsigny

2012

Matrix Information Geometry

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Abstract. When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this paper, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance.

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Riemannian center of mass and mollifier smoothing

Cited by 935 publications

References 9 publications

Analysis of principal nested spheres

Analysis of principal nested spheres

Tangent-based manifold approximation with locally linear models

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

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