Sampling sets and quadrature formulae on the rotation group

Numerical Functional Analysis and Optimization

Volumn 30, Issue 7-8, 2009, Pages 665-688

  Gräf, Manuel ^a   Potts, Daniel ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Quadrature rule; Rotation group SO(3); Sampling sets; Scattered data; Spherical harmonics

Indexed keywords

FAST ALGORITHMS; MESH NORM; NUMERICAL TESTS; QUADRATURE FORMULA; QUADRATURE RULE; QUADRATURE WEIGHTS; ROTATION GROUP; ROTATION GROUP SO(3); SCATTERED DATA; SPHERICAL HARMONICS; THEORETICAL RESULT;

COMPUTATION THEORY; HARMONIC ANALYSIS; ROTATION; SPHERES;

SET THEORY;

EID: 70149100291     PISSN: 01630563     EISSN: 15322467     Source Type: Journal    
DOI: 10.1080/01630560903163508     Document Type: Article

References (21)

1. M. Edén and M.H. Levitt (1998). Computation of orientational averages in solid-state NMR by gaussian spherical quadrature. J. Magn. Reson. 132: 220-239.
2. M. Gräf, S. Kunis, and D. Potts (2009). On the computation of nonnegative quadrature weights on the sphere. Appl. Comput. Harmon. Anal. 27:124-132.
3. M. Gräf and D. Schmid (2009). Pobabilistic Marcinkiewicz-Zygmund inequalities on the rotation group. Preprint (submitted).
4. R.H. Hardin and N.J.A. Sloane (2002). Spherical designs, a library of putatively optimal spherical t-designs. Available at: http://www.research.att. com/ñjas/sphdesigns.
5. C.F.F. Karney (2006). Nearly optimal coverings of orientation space. Available at: http://charles.karney.info/orientation/.
6. C.F.F. Karney (2007). Quaternions in molecular modeling. J. Mol. Graph. Mod. 25:595-604.
7. J.J. Kuffner (2004). Effective sampling and distance metrics for 3d rigid body path planning. In: IEEE International Conference on Robotics and Automation. Vol.4, ICRA, Orleans, LA, USA, pp. 3993-3998.
8. V.I. Lebedev (1976). Quadratures on the sphere. Ž. Vyčisl. Mat. i Mat. Fiz. 16:293-306, 539.
9. V.I. Lebedev and A.L. Skorokhodov (1992). Quadrature formulas for a sphere of orders 41, 47 and 53. Dokl. Akad. Nauk. 324:519-524.
10. A.D. McLaren (1963). Optimal numerical integration on a sphere. Math. Comput. 17:361-383.
11. H.N. Mhaskar, F.J. Narcowich, and J.D. Ward (2001). Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comput. 70:1113-1130. [Corrigendum on the positivity of the quadrature weights in 2002; 71:453-454.]
12. J.C. Mitchell (2008). Sampling rotation groups by successive orthogonal images. SIAM J. Sci. Comput. 30:525-547.
13. A.S. Popov (2008). Cubature formulas on a sphere that are invariant with respect to the octahedron rotation group. Numer. Anal. Appl. 1:355-361.
14. D. Potts, J. Prestin, and A. Vollrath (2009). A fast algorithm for nonequispaced Fourier transforms on the rotation group. Numer. Algorithms (accepted).
15. M. Reimer (2003). Multivariate Polynomial Approximation. Birkhäuser Verlag, Basel.
16. R. Schaback (1995). Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3:251-264.
17. D. Schmid (2008). Marcinkiewicz-Zygmund inequalities and polynomial approximation from scattered data on SO(3). Numer. Funct. Anal. Optim. 29:855-882.
18. D. Schmid (2008). A trade-off principle in connection with the approximation by positive definite kernels. In: C.K. Chui, M. Neamtu, and L.L. Schumaker, eds. Approximation Theory XII: San Antonio. Nashboro Press, Brentwood, pp. 348-359.
19. D. Schmid (2009). Scattered data approximation on the rotation group and generalizations. Dissertation, Fakultät für Mathematik, Technische Universität München.
20. D. Varshalovich, A. Moskalev, and V. Khersonski (1988). Quantum Theory of Angular Momentum. World Scientific Publishing, Singapore.
21. A. Yershova and S.M. LaValle (2004). Deterministic sampling methods for spheres and SO(3). In: Proceedings. IEEE International Conference on Robotics and Automation. Vol.4. ICRA, New Orleans, LA, USA, pp. 3974-3980.