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

Pennec, Xavier; Arsigny, Vincent

doi:10.1007/978-3-642-30232-9_7

Cited by 36 publications

(58 citation statements)

References 60 publications

(84 reference statements)

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“…Moreover, the connections of this family are also invariant by right translation [32], thus invariant by inversion also since they are already left invariant. This make them particularly interesting since they are fully compatible with all the group operations.…”

Section: Cartan-schouten Connectionsmentioning

confidence: 95%

“…More generally, the condition ad * (X, X) = 0 for all X ∈ g turns out to be a necessary and sufficient condition to have a bi-invariant metric [33]. It is important to notice that geodesics of the left-and right-invariant metrics differ in general as there do not exists bi-invariant metrics even for simple groups like the Euclidean motions [32]. However, right invariant geodesics can be easily obtained from the left invariant one through inversion: if φ(t) is a left invariant geodesic joining identity to the transformation φ 1 , then φ −1 (t) is a right-invariant geodesic joining identity to φ −1 1 .…”

Section: Riemannian Setting: Levi Civita Connectionmentioning

confidence: 99%

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Discrete Ladders for Parallel Transport in Transformation Groups with an Affine Connection Structure

Lorenzi

Pennec

2014

Geometric Theory of Information

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International audienceModeling the temporal evolution of the tissues of the body is an important goal of medical image analysis, for instance for understanding the structural changes of organs affected by a pathology, or for studying the physiological growth during the life span. For such purposes we need to analyze and compare the observed anatomical differences between follow-up sequences of anatomical images of different subjects. Non-rigid registration is one of the main instruments for modeling anatomical differences from images. The aim of non-rigid registration is to encode the observed structural changes as deformation fields of the image space, which represent the warping required to match observed differences. This way, anatomical changes can be modeled and quantified by analyzing the associated deformations. The comparison of temporal evolutions thus requires the transport (or "normalization") of longitudinal deformations in a common reference frame. Normalization of longitudinal deformations can be done in different ways, depending on the feature of interest. For instance, local volume changes encoded by the scalar Jacobian determinant of longitudinal deformations can be compared by scalar resampling in a common reference frame via inter-subject registration. However, if we consider vector-valued deformation trajectories instead of scalar quantities, the transport is not uniquely defined anymore. Among the different normalization methods for deformation trajectories, the parallel transport is a powerful and promising tool which can be used within the ''diffeomorphic registration'' setting. Mathematically, parallel transporting a vector along a curve consists in translating it across the tangent spaces to the curve by preserving its parallelism according to a given derivative operation called (affine) connection. This chapter focuses on explicitly discrete algorithms for parallel transporting diffeomorphic deformations. Schild's ladder is an efficient and simple method proposed in theoretical Physics for the parallel transport of vectors along geodesics paths by iterative construction of infinitesimal geodesics parallelograms on the manifold. The base vertices of the parallelogram are given by the initial tangent vector to be transported. By iteratively building geodesic diagonals along the path, Schild's Ladder computes the missing vertex which corresponds to the transported vector. In this chapter we first show that the Schild ladder can lead to an effective computational scheme for the parallel transport of diffeomorphic deformations parameterized by tangent velocity fields. Schild's ladder may be however inefficient for transporting longitudinal deformations from image time series of multiple time points, in which the computation of the geodesic diagonals is required several times. We propose therefore a new parallel transport method based on the Schild's ladder, the "pole ladder", in which the computation of geodesics diagonals is minimized. Differently from the Schild's ladder, the pole ladder is...

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Section: Cartan-schouten Connectionsmentioning

confidence: 95%

Section: Riemannian Setting: Levi Civita Connectionmentioning

confidence: 99%

Discrete Ladders for Parallel Transport in Transformation Groups with an Affine Connection Structure

Lorenzi

Pennec

2014

Geometric Theory of Information

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“…A naturally bi-invariant candidate for the mean on Lie groups is the group exponential barycenter [2] defined as follows. A group exponential barycenter m of the dataset {g i } i=1,..,N is a solution, if there are some, of the following group barycenter equation:…”

Section: Statistics On Lie Groupsmentioning

confidence: 99%

“…of upper triangular matrices of size n × n and the Lie group SE(n) of rotations and translations of R n do not have any bi-invariant metric, while they admit a locally unique bi-invariant mean [2]. Therefore, if we want to characterize the bi-invariant mean with an additional geometric structure on Lie groups, we have to consider a structure that is more general than the Riemannian one.…”

Section: Using Riemannian and Pseudo-riemannian Structures For Statismentioning

confidence: 99%

Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics

Miolane

Pennec

2015

Entropy

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In computational anatomy, organ's shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemannian framework on Lie groups, one needs to define a Riemannian metric compatible with the group structure: a bi-invariant metric. However, it is known that Lie groups, which are not a direct product of compact and abelian groups, have no bi-invariant metric. However, what about bi-invariant pseudo-metrics? In other words: could we remove the assumption of the positivity of the metric and obtain consistent statistics on Lie groups through the pseudo-Riemannian framework? Our contribution is two-fold. First, we present an algorithm that constructs bi-invariant pseudo-metrics on a given Lie group, in the case of existence. Then, by running the algorithm on commonly-used Lie groups, we show that most of them do not admit any bi-invariant (pseudo-) metric. We thus conclude that the (pseudo-) Riemannian setting is too limited for the definition of consistent statistics on general Lie groups.

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“…However, the above barycentric iteration continues to make sense. The key idea developed in [24] is to consider Eq. (1) as an exponential barycenter of the canonical Cartan connection.…”

Section: Bi-invariant Means As Exponential Barycentersmentioning

confidence: 99%