×

zbMATH — the first resource for mathematics

Non-stationary subdivision schemes for surface interpolation based on exponential polynomials. (English) Zbl 1195.65015
A subdivision scheme for generating curves and surfaces from a finite set of control points is proposed. The main fact is that the subdivision scheme is non-stationary: the mask used to compute the new points changes from level to level. The definition of the mask at each level goes as follows: Given some finite set of exponential polynomials (functions of the type \(x^\alpha e^{\beta x}\)) the mask is the one fitting a kind of butterfly-shaped stencil for the set of exponential polynomials. Thus, the computation of the mask at each level is equivalent to solve a linear system. Examples of how the algorithm works for parametric surfaces as torus and spheres are shown.
A careful analysis of the convergence and of the smoothness of the subdivision scheme is done proving that these non-stationary schemes have the same smoothness and approximation order as the classical butterfly interpolatory scheme.

MSC:
65D17 Computer-aided design (modeling of curves and surfaces)
65D10 Numerical smoothing, curve fitting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beccari, C.; Casciola, G.; Romani, L., A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics, Comput. aided geom. design, 24, 1-9, (2007) · Zbl 1171.65325
[2] Beccari, C.; Casciola, G.; Romani, L., An interpolating 4-point \(C^2\) ternary non-stationary subdivision scheme with tension control, Comput. aided geom. design, 24, 210-219, (2007) · Zbl 1171.65326
[3] Cavaretta, A.; Dahmen, W.; Micchelli, C.A., Stationary subdivision, Mem. amer. math. soc., 93, 1-186, (1991) · Zbl 0741.41009
[4] Chaikin, G., An algorithm for high speed curve generation, Computer graphics and image processing, 3, 346-349, (1974)
[5] Choi, Y.-J.; Lee, Y.-J.; Yoon, J.; Lee, B.-G.; Kim, Y.-J., A new class of non-stationary interpolatory subdivision schemes based on exponential polynomials, (), 563-570, (SpringerLink, PDF-file) · Zbl 1160.68616
[6] Cohen, E.; Lyche, T.; Riesenfeld, R., Discrete B-spline and subdivision techniques in computer-aided geometric design and computer graphics, Computer graphics and image processing, 14, 87-111, (1980)
[7] Doo, D.; Sabin, M., Behaviour of recursively division surfaces near extraordinary points, Comput.-aided design, 10, 356-360, (1978)
[8] Deslauriers, G.; Dubuc, S., Symmetric iterative interpolation, Constr. approx., 5, 49-68, (1989) · Zbl 0659.65004
[9] Dyn, N., Subdivision schemes in computer-aided geometric design, (), 36-104 · Zbl 0760.65012
[10] Dyn, N.; Gregory, J.A.; Levin, D., A four-point interpolatory subdivision scheme for curve design, Comput. aided geom. design, 4, 257-268, (1987) · Zbl 0638.65009
[11] Dyn, N.; Gregory, J.A.; Levin, D., A butterfly subdivision scheme for surface interpolation with tension control, ACM trans. graph., 9, 160-169, (1990) · Zbl 0726.68076
[12] Dyn, N.; Levin, D., Analysis of asymptotically equivalent binary subdivision schemes, J. of math. anal. appl., 193, 594-621, (1995) · Zbl 0836.65012
[13] Dyn, N.; Levin, D.; Luzzatto, A., Exponential reproducing subdivision scheme, Found. comp. math., 3, 187-206, (2003) · Zbl 1095.41001
[14] Dyn, N.; Levin, D.; Yoon, J., Analysis of univariate non-stationary subdivision schemes with application to Gaussian-based interpolatory schemes, SIAM J. math. anal., 39, 470-488, (2007) · Zbl 1132.41302
[15] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), John Hopkins University Press Baltimore · Zbl 0865.65009
[16] Jena, M.J.; Shunmugaraj, P.; Das, P.J., A subdivision algorithm for trigonometric spline curves, Comput. aided geom. design, 19, 71-88, (2002) · Zbl 0984.68165
[17] Jena, M.J.; Shunmugaraj, P.; Das, P.J., A non-stationary subdivision scheme for generalizing trigonometric surfaces to arbitrary meshes, Comput. aided geom. design, 20, 61-77, (2003) · Zbl 1069.65557
[18] Jena, M.K.; Shunmugaraj, P.; Das, P.C., A non-stationary subdivision scheme for curve interpolation, Anziam J., 44, E, 216-235, (2003) · Zbl 1078.65528
[19] Morin, G.; Warren, J.; Weimer, H., A subdivision scheme for surfaces of revolution, Comput. aided geom. design, 18, 483-502, (2001) · Zbl 0970.68177
[20] Romani, L., From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms, J. of comp. and appl. math., 224, 383-396, (2009) · Zbl 1159.65019
[21] Warren, J.; Weimer, H., Subdivision methods for geometric design, (2002), Academic Press New York
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.