Interpolation in special orthogonal groups

Shingel, Tatiana

doi:10.1093/imanum/drn033

Cited by 25 publications

(15 citation statements)

References 17 publications

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“…This approach works when all the data points (y i ) i are close enough, but fails in a fundamental way when this assumption is violated. This is by and large due to the typical nontrivial topology of G, see [9] for a detailed study of the case G = SO (3).…”

Section: Introductionmentioning

confidence: 99%

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Xie

2009

IMA Journal of Numerical Analysis

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Abstract:We prove that any interpolatory Lie group subdivision scheme based on combining a linear interpolatory subdivision scheme S with the log-exp adaption to Lie group valued data in [8] produces parametrized curves on the Lie group which are as smooth as the smoothness of S -no matter how smooth S is. We present both an extrinsic proof and an intrinsic proof.We discuss two variations of our main result. (i) We illustrate how smoothness equivalence can break down in a variant of the original log-exp scheme. (ii) We show that the main result of this paper can be easily extended to a multivariate setting.Acknowledgments.

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Section: Introductionmentioning

confidence: 99%

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Xie

2009

IMA Journal of Numerical Analysis

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“…In order to have a curve at least C 1 , and also because the distance between the W i matrices can be quite high, we interpolate first. There are many ways to interpolate on SO(n), see [45] for example, but one of the simplest is to interpolate in the tangent space, which is Euclidean, and to go back to the manifold via the exponential map [24,45].…”

Section: The Geodesic Equation In the Lie Group Casementioning

confidence: 99%

Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

Dugast¹,

Bouleux²,

Marcon³

2018

Preprint

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We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes, this enables a comparison between curves’ shapes and allows then the classification of processes’ measure.

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“…There exists different approaches to tackle the curve fitting problem. Among others, we name here the subdivision schemes approach [Dyn09;WNG07] or the Lie-algebraic methods [Shi08]. However, the most popular approach nowadays is probably to encapsulate the two above-mentioned constraints into an optimization problem…”

Section: Introductionmentioning

confidence: 99%

A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve

Bergmann

Gousenbourger

2018

Front. Appl. Math. Stat.

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We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.

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Interpolation in special orthogonal groups

Cited by 25 publications

References 17 publications

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve

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