Interpolation in Lie Groups

Marthinsen, Arne

doi:10.1137/s0036142998338861

Cited by 28 publications

(20 citation statements)

References 30 publications

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“…See, for example, [7]. 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.…”

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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“…See, for example, [7]. 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.…”

Section: Introductionmentioning

confidence: 99%

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Xie

2009

IMA Journal of Numerical Analysis

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“…Park and Kang [11] derived a rational interpolating scheme for the group of rotations by representing the group with Cayley parameters and using Euclidean methods in this parameter space. Marthinsen [12] suggests the use of Hermite interpolation and the use of truncated inverse of the differential of the exponential mapping and the truncated Baker-Campbell-Hausdorff formula to simplify the constuction of interpolation polynomials. The advantage of these methods is that they produce rational curves.…”

Section: Introductionmentioning

confidence: 99%

An SVD-based projection method for interpolation on SE(3)

Belta

Kumar

2002

IEEE Trans. Robot. Automat.

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This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA + (n), a subgroup of the affine group in R n and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA + (n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/meam_papers/10 334 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 18, NO. 3, JUNE 2002 An SVD-Based Projection Method for Interpolation on SE(3) Calin Belta, Student Member, IEEE, and Vijay Kumar, Senior Member, IEEE Abstract-This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in + ( ), a subgroup of the affine group in IR and 2) the projection of the trajectories onto (3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on + ( ). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on (3). Third, from a computational point of view, the method we propose is less expensive than traditional methods.

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“…A vast amount of literature has been devoted to the problem of transformation interpolation [Barr et al 1992;Juttler 1994;Marthinsen 1999;Belta and Kumar 2002;Hofer and Pottmann 2004;Li and Hao 2006]. This is not surprising, because the construction of interpolation curves for given key transformations (e.g., camera orientations) is a fundamental problem in computer animation.…”

Section: Blending Vs Interpolationmentioning

confidence: 99%

Skinning with dual quaternions

Kavan

Collins

Žára

et al. 2007

Proceedings of the 2007 Symposium on Interactive 3D Graphics and Games

234

165

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Figure 1: A comparison of dual quaternion skinning with previous methods: log-matrix blending [Cordier and Magnenat-Thalmann 2005] and spherical blend skinning [Kavan and Zara 2005]. The proposed approach not only eliminates artifacts, but is also much easier to implement and more than twice as fast. AbstractSkinning of skeletally deformable models is extensively used for real-time animation of characters, creatures and similar objects. The standard solution, linear blend skinning, has some serious drawbacks that require artist intervention. Therefore, a number of alternatives have been proposed in recent years. All of them successfully combat some of the artifacts, but none challenge the simplicity and efficiency of linear blend skinning. As a result, linear blend skinning is still the number one choice for the majority of developers. In this paper, we present a novel GPU-friendly skinning algorithm based on dual quaternions. We show that this approach solves the artifacts of linear blend skinning at minimal additional cost. Upgrading an existing animation system (e.g., in a videogame) from linear to dual quaternion skinning is very easy and has negligible impact on run-time performance.

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Interpolation in Lie Groups

Cited by 28 publications

References 30 publications

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

An SVD-based projection method for interpolation on SE(3)

Skinning with dual quaternions

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