Abstract
We survey a number of related methods, which have been published by the author and collaborators, in the field of subdivision schemes for curves and surfaces. The theory presented in these works relies mainly on the notion of polynomial reproduction, i.e. the ability of a scheme to reproduce all polynomials up to a certain degree as limit functions. We demonstrate that the study of polynomial reproduction is central to smoothness analysis and to approximation. In particular, we show how to exploit polynomial reproduction in the context of piecewise-uniform stationary subdivision. The applications include boundary treatments for subdivision surfaces, interpolation of curves by surfaces, subdivision stencils around extraordinary vertices (construction of C 2 schemes), as well as schemes that involve different kinds of grids (triangular / quadrilateral).
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References
Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision. Memoirs of AMS, vol. 453. American Mathematical Society, Providence (1991)
Levin, A.: Polynomial generation and quasi-interpolation in stationary non-uniform subdivision. Computer Aided Geometric Design 20(1), 41–60 (2003)
Levin, A.: Combined Subdivision Schemes. PhD thesis, Tel-Aviv university (2000)
Levin, A., Levin, D.: Analysis of quasi-uniform subdivision. Applied and Computational Harmonic Analysis 15(1), 18–32 (2003)
Schaefer, S., Warren, J.: On c 2 triangle/quad subdivision. ACM Transactions on Graphics (2004) (accepted)
Stam, J., Loop, C.: Quad/triangle subdivision. Computer Graphics Forum 22(1), 79–85 (2003)
Hakenberg, J.: Smooth subdivision for mixed volumetric meshes. Master’s thesis, Rice University, Department of Computer Science (2004)
Zulti, A., Levin, A., Levin, D., Teicher, M.: c 2 subdivision over triangulations with one extraordinary point. Computer Aided Geometric Design (2004) (to appear)
Reif, U.: A unified approach to subdivision algorithms near extraordinary points. Computer Aided Geometric Design 12, 153–174 (1995)
Warren, J., Weimer, H.: Subdivision Methods for Geometric Design. Morgan Kaufmann, San Francisco (2002)
Zorin, D.: Smoothness of stationary subdivision on irregular meshes (1998) (preprint )
Levin, A.: Interpolating nets of curves by smooth subdivision surfaces. In: Proceedings of SIGGRAPH 1999, Computer Graphics Proceedings. Annual Conference Series, pp. 57–64 (1999)
Levin, A.: Filling n-sided holes using combined subdivision schemes. In: Laurent, P.J., Sablonniere, P., Schumaker, L.L. (eds.) Curve And Surface Design, pp. 221–228. Vanderbilt University Press, Nashville (1999)
Dyn, N.: Subdivision schemes in computer aided geometric design. In: Light, W.A. (ed.) Advances in Numerical Analysis II, Subdivision algorithms and radial functions, pp. 36–104. Oxford University Press, Oxford (1992)
Dyn, N., Greogory, J.A., Levin, D.: Piecewise uniform subdivision schemes. In: Dahlen, M., Lyche, T., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces, pp. 111–120. Vanderbilt University Press, Nashville (1995)
Ron, A.: Wavelets and their associated operators. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory IX, pp. 283–317. Vanderbilt University Press, Nashville (1998)
de Boor, C.: Quasiinterpolants and approximation power of multivariate splines. In: Gasca, M., Michelli, C.A. (eds.) Computation of curves and surfaces, pp. 313–345. Kluwer Academic Publishers, Dordrecht (1990)
de Boor, C., Ron, A.: The exponentials in the span of the multiinteger translates of a compactly supported function: quasiinterpolation and approximation order. Journal of London Mathematical Society 45(2), 519–535 (1992)
Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., Stuetzle, W.: Piecewise smooth surface reconstruction. Computer graphics 28(3), 295–302 (1994)
Schweitzer, J.: Analysis and Applications of Subdivision Surfaces. PhD thesis, University of Washington, Seattle (1996)
Biermann, H., Levin, A., Zorin, D.: Piecewise smooth subdivision surfaces with normal control. In: Proceedings of SIGGRAPH 2000, Computer Graphics Proceedings. Annual Conference Series, pp. 113–120 (2000)
Levin, A.: Combined subdivision schemes for the design of surfaces satisfying boundary conditions. Computer Aided Geometric Design 16(5), 345–354 (1999)
Daubechies, I., Guskov, I., Sweldens, W.: Regularity of irregular subdivision. Constructive Approximation 15, 381–426 (1999)
de Boor, C.: Cutting corners always works. Computer Aided Geometric Design 4, 125–131 (1987)
Dyn, N., Greogory, J.A., Levin, D.: A four-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design 4, 257–268 (1987)
Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation. Constructive Approximation 5, 49–68 (1989)
Catmull, E., Clark, J.: Recursively generated b-spline surfaces on arbitrary topological meshes. Computer Aided Design 10, 350–355 (1978)
Loop, C.: Smooth spline surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)
Prautzsch, H., Umlauf, G.: A g 2 subdivision algorithm. In: Farin, G., Bieri, H., Brunnet, G., DeRose, T. (eds.) Geometric Modeling, Computing Supplements, pp. 217–224. Springer, New York (1998)
Prautzsch, H., Reif, U.: Degree estimates of c k piecewise polynomial subdivision surfaces. Advances in Computational Mathematics 10, 209–217 (1999)
Prautzsch, H.: Freeform splines. Computer Aided Geometric Design 14, 201–206 (1997)
Reif, U.: Turbs: Topologically unrestricted b-splines. Constructive Approximation 4, 55–77 (1998)
Zulti, A., Levin, A., Levin, D., Taicher, M.: An electronic appendix to the paper: c2 subdivision over triangulations with one extraordinary point (2004), http://www.math.tau.ac.il/~levin/adi/zulti-levin_c2_sbd.htm
Dyn, N., Greogory, J.A., Levin, D.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9, 160–169 (1990)
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Levin, A. (2005). The Importance of Polynomial Reproduction in Piecewise-Uniform Subdivision. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_17
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DOI: https://doi.org/10.1007/11537908_17
Publisher Name: Springer, Berlin, Heidelberg
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