Abstract
We propose a new method to approximate curves that interpolate a given set of time-labeled data on Riemannian symmetric spaces. First, we present our new formulation on the Euclidean space as a result of minimizing the mean square acceleration. This motivates its generalization on some Riemannian symmetric manifolds. Indeed, we generalize the proposed solution to the the special orthogonal group, the manifold of symmetric positive definite matrices, and Riemannian n-manifolds with constant negative curvature. By means of this generalization, we are able to prove that the approximates enjoy a number of nice properties: The solution exists and is optimal in many common situations. Several examples are provided together with some applications and graphical representations.
Similar content being viewed by others
References
Fang, Y., Hsieh, C., Kim, M., Chang, J., Woo, T.: Real time motion fairing with unit quaternions. Comput. Aided Des. 30, 191–198 (1998)
Barr, A., Currin, B., Gabriel, S., Hughes, J.: Smooth interpolation of orientations with angular velocity constraints using quaternions. ACM Siggraph Comput. Graph. 26, 313–320 (1992)
Zefran, M., Kumar, V., Croke, C.: On the generation of smooth three-dimensional rigid body motions. IEEE Trans. Robot. Autom. 14, 576–589 (1998)
Westin, C.F., Maier, E., Mamata, H., Nabavi, A., Jolesz, F., Kikinis, R.: Process and visualization for diffusion tensor MRI. Med. Image Anal. 6, 93–108 (2002)
Lazar, M., Weinstein, D., Tsuruda, J., Hasan, K., Arfanakis, K., Meyerand, M., Badie, B., Rowley, H., Haughton, V.: White matter tractography using diffusion tensor deflection. Hum. Brain Mapp. 18(4), 306–321 (2003)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. J. Magn. Reson. Med. 56(2), 411–421 (2006)
Sambridge, M., Braun, J., Mcqueen, H.: Geophysical parametrization and interpolation of irregular data using natural neighbours. Geophys. J. Int. 122(3), 837–857 (1995)
Moakher, M., Zerai, M.: The Riemannian geometry of the space of positive definite matrices and its application to the regularization of positive-definite matrix-valued data. J. Math. Imaging Vis. 40, 171–187 (2011)
Schwartzman, A., Walter, F., Jonathan, E.: Inference for eigenvalues and eigenvectors of gaussian symmetric matrices. Ann. Stat. 36, 2886–2919 (2008)
Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inf. 6(4), 465–473 (1998)
Samir, C., Absil, P.A., Srivastava, A., Klassen, E.: A gradient-descent method for curve fitting on Riemannian manifolds. Found. Comput. Math. 12(1), 49–73 (2012)
Crouch, P., Silva Leite, F., Croke, C.: Geometry and the dynamic interpolation problem. In: Proceedings American Control Conference, vol. 14, pp. 1131–1136 (1991)
Crouch, P., Silva Leite, F., Croke, C.: The dynamic interpolation problem: on Riemannian manifolds, lie groups, and symmetric spaces. J. Dyn. Control Syst. 1(2), 177–202 (1995)
Camarinha, M., Silva Leite, F., Crouch, P.: Splines of class \(c^k\) on non-Euclidean spaces. IMA J. Math. Control Inf. 12(4), 399–410 (1995)
Crouch, P., Kun, G., Silva Leite, F.: The de Casteljau algorithm on lie groups and spheres. J. Dyn. Control Syst. 5(3), 397–429 (1999)
Altafini, C.: The de Casteljau algorithm on \(SE(3)\). Nonlinear Control in the Year 2000 5(3), 23–34 (2000)
Popiel, T., Noakes, L.: Bézier curves and c2 interpolation in Riemannian manifolds. J. Approx. Theory 148(2), 111–127 (2007)
Róthb, A.: Control point based exact description of trigonometric/hyperbolic curves. J. Comput. Appl. Math. 290, 74–91 (2015)
Juhásza, I., Róthb, A.: Adjusting the energies of curves defined by control points. Comput. Aided Des. 107, 77–88 (2019)
Mainar, E., Peña, J.M.: Optimal bases for a class of mixed spaces and their associated spline spaces. Comput. Math. Appl. 59(4), 1509–1523 (2010)
Shoemake, K.: Animating rotation with quaternion curves. ACM SIGGRAPH 85(19), 245–254 (1985)
Hart, J., Francis, G., Kaufman, L.: Visualizing quaternion rotation. ACM Trans. Graph. 13(3), 256–276 (1994)
Ge, Q., Ravani, B.: Geometric construction of Bézier motions. ASME J. Mech. Des. 116, 749–755 (1994)
Nielson, G., Heiland, R.: Animated rotations using quaternions and splines on a 4d sphere. Program. Comput. Softw 18(4), 145–154 (1992)
Rodrigues, R.C., Silva Leite, F., Jakubiak, J.: A new geometric algorithm to generate smooth interpolating curves on Riemannian manifolds. LMS J. Comput. Math. 8, 251–266 (2011)
Park, F., Ravani, B.: Bézier curves on Riemannian manifolds and Lie groups with kinematics applications. J. Mech. Des. 117(1), 36–40 (1995)
Scull, S.: Bezier curves on groups. Computers Math. Appl. 18, 691–700 (1989)
Samir, C., Adouani, I.: \(c^{1}\) interpolating Bézier path on Riemannian manifolds, with applications to 3d shape space. Appl. Math. Comput. 348, 371–384 (2019)
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric spaces. Academic Press, New York (1978)
Adams, J.F.: Lectures on Lie Groups. Midway Reprints Series. The University of Chicago Press, Chicago (1983)
Park, F., Ravani, B.: Smooth invariant interpolation of rotations. ACM Trans. Graph (TOG) 16, 277–295 (1997)
Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications II. Springer, New York (1988)
Siegel, C.L.: Symplectic Geometry. Academic Press, New York (1964)
Skovgaard, L.: A Riemannian geometry of the multivariate normal model. Scand. J. Stat. 11, 211–233 (1984)
Fletcher, P., Joshi, S.: Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors, pp. 87–98. Springer, Berlin (2004)
Ratcliffe, John G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. Springer, Berlin (2005)
John, M.: Lee: Riemannian Manifolds: An Introduction to Curvature. Graduate texts in mathematics, Springer Science (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alexandru Kristály.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: List of Detailed Algorithms
Appendix: List of Detailed Algorithms
Rights and permissions
About this article
Cite this article
Adouani, I., Samir, C. A Constructive Approximation of Interpolating Bézier Curves on Riemannian Symmetric Spaces. J Optim Theory Appl 187, 158–180 (2020). https://doi.org/10.1007/s10957-020-01751-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-020-01751-5
Keywords
- Optimization
- Riemannian manifolds
- Mean square acceleration
- Curve fitting
- Bézier curve
- Special orthogonal group
- Symmetric positive-definite matrices
- Affine-invariant metric
- Hyperbolic manifolds