Smoothness equivalence properties of manifold-valued data subdivision schemes based on the projection approach

SIAM Journal on Numerical Analysis

Volumn 45, Issue 3, 2007, Pages 1200-1225

  Xie, Gang ^a   Yu, Thomas P Y ^a  

^a DREXEL UNIVERSITY   (United States)

Author keywords

Approximation order; Closest point projection; Interpolation; Lie groups; Linear subdivision schemes; Nonlinear subdivision schemes; Riemanian manifold; Singular value decomposition; Smoothness; SO(n); Sphere

Indexed keywords

EMBEDDED SURFACE; EUCLIDEAN SPACES; FUNDAMENTAL PROBLEMS; MULTIRESOLUTION (MR); NONLINEAR SUBDIVISION; PROJECTION OPERATORS; REAL-VALUED DATA; SUBDIVISION METHODS; SUBDIVISION SCHEMES; SYMMETRIC SPACES;

EIGENVALUES AND EIGENFUNCTIONS; INTERPOLATION; MAPS;

MATHEMATICAL OPERATORS;

EID: 38649127310     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060652944     Document Type: Article

References (37)

1. D. ASIMOV, The grand tour: A tool for viewing multidimensional data, SIAM J. Sci. Stat. Comp., 6 (1985), pp. 128 143.
2. C. BELTA AND V. KUMAR, An SVD-based projection method for interpolation on SE(3), IEEE Trans. Robotics Automat., 18 (2002), pp. 334 345.
3. W. M. BOOTHBY, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., Academic Press, New York, 2002.
4. A. S. CAVARETTA. W. DAHMEN, AND C. A. MICCHELLI, Stationary Subdivision, Mem. Amer. Math. Soc. 453, AMS., Providence, RI, 1991.
5. Z. CIESIELSKI, Approximation of splines and its application to Lipschitz classes and to stochastic processes, in Approximation Theory of Functions, Proceedings of the Conference in Kaluga, 1975, Nauka, Moscow, 1977, pp. 397-400.
6. I. DAUBECHIES AND J. C. LACARIAS, Two-scale difference equations 1. Existence and global regularity of solutions, SlAM J. Math. Anal., 22 (1991), pp. 1388 1410.
7. I. DAUBECHIES AND J. C. LACARIAS, Two-scale difference equations II. Local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal., 23 (1992), pp. 1031-1079.
8. I. DAUBECHIES. O. RUNBORG, AND W. SWELDENS, Normal multiresolution approximation of curves, Constructive Approximation, 3 (2004), pp. 399-463.
9. C. DESLAURIERS AND S. DUBUC. Symmetric iterative interpolation processes, Constr. Approx., 5 (1989), pp. 49-68.
10. Z. DITZIAN, Moduli of smoothness using discrete data, J. Approx. Theory, 49 (1987), pp. 115-129.
11. M. P. DO CARMO, Riemannian Geometry (translated by F. Flaherty), Birkhäuser Boston, Cambridge, MA, 1992.
12. S. DUBUC, Interpolation through an iterative scheme, J. Math. Anal. Appl., 114 (1986), pp. 185-204.
13. T. DUCH AMP, personal communication, 2005.
14. N. DYN AND D. LEVIN, Subdivision schemes in geometric modelling, Acta Numer., 11 (2002), pp. 73-144.
15. G. GOLUB AND C. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, London, 1996.
16. B. HAN, T. P.-Y. YU, AND Y. XUE, Non-interpolatory Hermite subdivision schemes, Math. Comput., 74 (2005), pp. 1345-1367.
17. A. ISERLES, H. Z. MUNTHE-KAAS, S. P. NORSETT, AND A. ZANNA, Lie-group methods, Acta Numer., 9 (2000), pp. 215-365.
18. A. ISERLES AND S. P. NØRSETT, On the solution of linear differential equations on Lie groups, Phil. Trans. Royal Soc. A, 357 (1999), pp. 983-1019.
19. V. IVANCEVIC, Symplectic rotational geometry in human biomechanics, SIAM Rev., 46 (2004), pp. 455-474.
20. A. MARTHINSEN, Interpolation in Lie groups, SIAM J. Numer. Anal., 37 (1999), pp. 269-285.
21. J. L. MERRIEN, A family of Hermite interpolants by bisection algorithms, Numer. Algorithms, 2 (1992), pp. 187-200.
22. P. OSWALD, Smoothness of a nonlinear subdivision scheme, in Curves and Surface Fitting: Saint-Malo 2002, A. Cohen, J.-L. Merrien, and L. L. Schumaker, eds., Nashboro Press, Brentwood, TN, 2003, pp. 323-332.
23. P. OSWALD, Smoothness of nonlinear median-interpolation subdivision, Adv. Comput. Math., 20 (2004), pp. 401-423.
24. I. UR RAHMAN, I. DRORI, V. C. STODDEN, D. L. DONOHO, AND P. SCHRÖDER, Multiscale representations for manifold-valued data, Multiscale Model. Simul., 4 (2005), pp. 1201-1232.
25. O. RIOUL, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal., 23 (1992), pp. 1544-1576.
26. S. SCHAEFER, A Factored, Interpolatory Subdivision Scheme for Surfaces of Revolution, Mas-ter's thesis, Department of Computer Science, Rice University, Houston, TX, 2003.
27. C. H. SEQUIN AND J. A. YEN, Fair and robust curve interpolation on the sphere, in Sketches and Application (SIGGRAPH'01), 2001, p. 182.
28. J. WALLNER, Gliding spline motions and applications, Comput. Aided Geom. Design, 21 (2004), pp. 3-21.
29. J. WALLNER, Smoothness analysis of subdivision schemes by proximity, Constr. Approx., 24 (2006), pp. 289-318.
30. 1 analysis of subdivision schemes on manifolds by proximity, Comput. Aided Geom. Design, 22 (2005), pp. 593-622.
31. J. WALLNER AND H. POTTMANN, Intrinsic subdivision with smooth limits for graphics and animation, ACM Trans. Graphics, 25 (2005), pp. 356-374.
32. G. XIE AND T. P.-Y. YU, On a linearization principle for nonlinear p-mean subdivision schemes, in Advances in Constructive Approximation, M. Neamtu and E. B. Saff, eds., Nashboro Press, Brentwood, TN, 2004, pp. 519-533.
33. G. XIE AND T. P.-Y. YU, Smoothness analysis of nonlinear subdivision schemes of homoge-neous and affine invariant type, Construct. Approx., 22 (2005), pp. 219-254.
34. T. P.-Y. YU, On the regularity analysis of interpolatory Hermite subdivision schemes, J. Math. Anal. Appl., 302 (2005), pp. 201-216.
35. T. P.-Y. YU, Approximation order equivalence properties of manifold-valued data subdivision schemes, 2006, in preparation.
36. T. P.-Y. YU, Cutting corners on the sphere, in Wavelets and Splines: Athens 2005, G. Chen and M.-J. Lai, eds., Nashboro Press, Brentwood, TN, 2006, pp. 496-506.
37. T. P.-Y. YU, How data dependent is a nonlinear subdivision scheme? A case study based on convexity preserving subdivision, SIAM J. Numer. Anal., 44 (2006), pp. 936-948.