×

zbMATH — the first resource for mathematics

Higher-order smoothing splines versus least squares problems on Riemannian manifolds. (English) Zbl 1203.65028
Summary: We present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a higher-order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval. We show that the Riemannian mean of the given points is achieved as a limiting process of the above. Also, when the Riemannian manifold is an Euclidean space, our approach generates, in the limit, the unique polynomial curve which is the solution of the classical least squares problem. These results support our belief that the approach presented in this paper is the natural generalization of the classical least squares problem to Riemannian manifolds.

MSC:
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
49K15 Optimality conditions for problems involving ordinary differential equations
53A35 Non-Euclidean differential geometry
47J30 Variational methods involving nonlinear operators
53C22 Geodesics in global differential geometry
41A15 Spline approximation
49N60 Regularity of solutions in optimal control
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Baillieul, Kinematic redundancy and the control of robots with flexible components. IEEE Int. Conf. Robotics Automation, Nice, France (1992).
[2] F. Bullo and M. Zefran, On mechanical systems with nonholonomic constraints and symmetries. Systems Control Lett. 42 (1998), Nos. 1/2, 135–164.
[3] S. R. Buss and J. P. Fillmore, Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 2 (2001), No. 20, 95–126. · Zbl 05457176
[4] M. Camarinha, The geometry of cubic polynomials on Riemannian manifolds. Ph.D. Thesis, Depart. de Matemática, Univ. de Coimbra, Portugal (1996).
[5] M. Camarinha, F. Silva Leite, and P. Crouch, Splines of class C k on non-Euclidean spaces. IMA J. Math. Control Inform. 12 (1995), 399–410. · Zbl 0860.58013
[6] _____, On the geometry of Riemannian cubic polynomials. Differ. Geom. Appl. (2001), No. 15, 107–135. · Zbl 1034.53041
[7] C. Lin Chang and P. J. Luh, Formulation and optimization of cubic polynomial joint trajectories for industrial robots. IEEE Trans. Automat. Control AC28 (1983), 1066–1074. · Zbl 0525.93035
[8] P. Crouch, G. Kun, and F. Silva Leite, The De Casteljau algorithm on Lie groups and spheres. J. Dynam. Control Systems 5 (1999), No. 3, 397–429. · Zbl 0961.53027
[9] P. Crouch and F. Silva Leite, Geometry and the dynamic interpolation problem. Proc. Amer. Control Conf. Boston (1991), 1131–1137. · Zbl 0736.93006
[10] _____, The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces. J. Dynam. Control Systems 1 (1995), No. 2, 177–202. · Zbl 0946.58018
[11] P. De Casteljau, Outillages méthodes calcul. Technical Report, A. Citroen, Paris (1959).
[12] M. P. do Carmo, Riemannian geometry. Mathematics: Theory and Applications, Birkäuser (1992). · Zbl 0752.53001
[13] G. Farin, Curves and surfaces for computer aided geometric design. Academic Press (1990). · Zbl 0702.68004
[14] R. Giamgò, F. Giannoni, and P. Piccione, An analytical theory for Riemannian cubic polynomials. IMA J. Math. Control Inform. 19 (2002), 445–460. · Zbl 1138.58307
[15] K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on S n , SO n , and Grassmann manifolds. J. Dynam. Control Systems 13 (2007), No. 4, 467–502. · Zbl 1140.58005
[16] K. Hüper and J. H. Manton, Numerical methods to compute the Karcher mean of points on the special orthogonal group (to appear).
[17] P. E. Jupp and J. T. Kent, Fitting smooth paths to spherical data. Appl. Statist. 36 (1987), No. 1, 34–46. · Zbl 0613.62086
[18] H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30 (1977), 509–541. · Zbl 0354.57005
[19] A. K. Krakowski, Geometrical methods of inference. Ph.D. Thesis, Department of Mathematics and Statistics, The University of Western Australia (2002).
[20] V. Kumar, M. Zefran, and J. Ostrowski, Motion planning in humans and robots. 8th Int. Symp. of Robotics Research, Hayama, Japan (1997).
[21] P. Lancaster and K. Salkauskas, Curve and surface fitting. Academic Press 1990. · Zbl 0649.65012
[22] L. Machado, Least squares problems on Riemannian manifolds. Ph.D. Thesis, Department of Mathematics, University of Coimbra, Portugal (2006).
[23] L. Machado, F. Silva Leite, and K. Hüper, Riemannian means as solutions of variational problems. LMS J. Comput. Math. (2006), No. 8, 86–103. · Zbl 1111.58010
[24] L. Machado and F. Silva Leite, Fitting smooth paths on Riemannian manifolds. Int. J. Appl. Math. Statist. 4 (2006), No. J06, 25–53. · Zbl 1138.65012
[25] J. W. Milnor, Morse theory. Princeton University Press, Princeton, New Jersey 1963.
[26] M. Moakher, A differential geometric approach to the arithmetic and geometric means of operators in some symmetric spaces. SIAM. J. Matrix Anal. Appl. 26 (2005), No. 3, 735–747. · Zbl 1079.47021
[27] L. Noakes, G. Heinzinger, and B. Paden, Cubic splines on curved spaces. IMA J. Math. Control Inform. 6 (1989), 465–473. · Zbl 0698.58018
[28] F. Park and B. Ravani, Bézier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME J. Mech. Design 117 (1995), 36–40.
[29] T. Popiel and L. Noakes, Higher-order geodesics in Lie groups. Math. Control Signals Systems (2007), No. 19, 235–253. · Zbl 1163.53028
[30] C. H. Reinsch, Smoothing by spline functions. Numer. Math. 10 (1967), 177–183. · Zbl 0161.36203
[31] F. Silva Leite and P. Crouch, Closed forms for the exponential mapping on matrix Lie groups based on Putzer’s method. J. Math. Phys. 40 (1999), 3561–3568. · Zbl 0942.22009
[32] F. Silva Leite and K. Krakowski, Covariant differentiation under rolling maps. Pré-Publica\c{}cões do Departamento de Matemática, Univ. of Coimbra, Portugal (2008), No. 08-22, 1–8.
[33] G. Wahba, Spline models for observational data. SIAM. CBMS-NSF Regional Conf. Ser. Appl. Math. 59 (1990).
[34] M. Zefran, V. Kumar, and C. Croke, On the generation of smooth three-dimensional rigid body motions. IEEE Trans. Robotics Automat. 14 (1995), No. 4, 579–589.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.