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Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

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Abstract

Convergence and normal continuity analysis of a bivariate nonstationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, nonstationary subdivision scheme near an extraordinary vertex/face.

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References

  1. Badoual, A., Novara, P., Romani, L., Schmitter, D., Unser, M.: A non-stationary subdivision scheme for the construction of deformable models with sphere-like topology. Graph. Models 94, 38–51 (2017)

    Article  MathSciNet  Google Scholar 

  2. Catmull, E., Clark, J.: Recursively generated B-splines surfaces on arbitrary topological meshes. Comput. Aided Des. 10(6), 350–355 (1978)

    Article  Google Scholar 

  3. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, vol. 453. Memoirs of American Mathematical Society, Providence (1991)

    MATH  Google Scholar 

  4. Charina, M., Conti, C.: Polynomial reproduction of multivariate scalar subdivision schemes. J. Comput. Appl. Math. 240, 51–61 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charina, M., Conti, C., Romani, L.: Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix. Numer. Math. 127(2), 223–254 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Charina, M., Conti, C., Guglielmi, N., Protasov, V.: Regularity of non-stationary subdivision: a matrix approach. Numer. Math. 135(3), 639–678 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Q., Prautzsch, H.: Subdivision by WAVES—weighted averaging schemes. In: Proceedings of DWCAA12, Dolomites Research Notes on Approximation, vol. 6, pp. 9–19 (2013)

  8. Conti, C., Cotronei, M., Sauer, T.: Factorization of Hermite subdivision operators preserving exponentials and polynomials. Adv. Comput. Math. 42, 1055–1079 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conti, C., Cotronei, M., Sauer, T.: Convergence of level-dependent Hermite subdivision schemes. Appl. Numer. Math. 116, 119–128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conti, C., Dyn, N., Manni, C., Mazure, M.-L.: Convergence of univariate non-stationary subdivision schemes via asymptotic similarity. Comput. Aided Geom. Des. 37, 1–8 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Conti, C., Romani, L., Unser, M.: Ellipse-preserving Hermite interpolation and subdivision. J. Math. Anal. Appl. 426, 211–227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Conti, C., Romani, L., Yoon, J.: Approximation order and approximate sum rules in subdivision. J. Approx. Theory 207, 380–401 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cotronei, M., Sissouno, N.: A note on Hermite multiwavelets with polynomial and exponential vanishing moments. Appl. Numer. Math. 120, 21–34 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Delgado-Gonzalo, R., Thevenaz, P., Seelamantula, C.S., Unser, M.: Snakes with an ellipse-reproducing property. IEEE Trans. Image Process. 21, 1258–1271 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Delgado-Gonzalo, R., Thevenaz, P., Unser, M.: Exponential splines and minimal-support bases for curve representation. Comput. Aided Geom. Des. 29, 109–128 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Delgado-Gonzalo, R., Unser, M.: Spline-based framework for interactive segmentation in biomedical imaging. IRBM Ingenierie et Recherche Biomedicale/BioMed. Eng. Res. 34, 235–243 (2013)

    Google Scholar 

  17. Doo, D., Sabin, M.: Behavior of recursive division surfaces near extraordinary points. Comput. Aided Des. 10(6), 356–360 (1978)

    Article  Google Scholar 

  18. Dyn, N., Kounchev, O., Levin, D., Render, H.: Regularity of generalized Daubechies wavelets reproducing exponential polynomials with real-valued parameters. Appl. Comput. Harmonic Anal. 37, 288–306 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dyn, N., Levin, D.: Analysis of asymptotically equivalent binary subdivision schemes. J. Math. Anal. Appl. 193, 594–621 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dyn, N., Levin, D.: Subdivision schemes in geometric modeling. Acta Numer. 11, 73–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dyn, N., Levin, D., Liu, D.: Interpolatory convexity-preserving subdivision schemes for curves and surfaces. Comput. Aided Des. 24(4), 211–216 (1992)

    Article  MATH  Google Scholar 

  22. Fang, M., Ma, W., Wang, G.: A generalized surface subdivision scheme of arbitrary order with a tension parameter. Comput. Aided Des. 49, 8–17 (2014)

    Article  MathSciNet  Google Scholar 

  23. Gotsman, C., Gumhold, S., Kobbelt, L.: Simplification and compression of 3D meshes. Iske, A., Quak, E., Floater, M.S. (eds.) Tutorials on Multiresolution in Geometric Modelling, Part of the Series Mathematics and Visualization, pp. 319–361 (2002)

  24. Han, B.: Non homogeneous wavelet systems in high dimensions. Appl. Comput. Harmonic Anal. 32, 169–196 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jena, M.K., Shunmugaraj, P., Das, P.C.: A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes. Comput. Aided Geom. Des. 20, 61–77 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jeong, B., Yoon, J.: Construction of Hermite subdivision schemes reproducing polynomials. J. Math. Anal. Appl. 451(1), 565–582 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jia, R.Q., Lei, J.J.: Approximation by piecewise exponentials. SIAM J. Math. Anal. 22, 1776–1789 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lee, Y.-J., Yoon, J.: Non-stationary subdivision schemes for surface interpolation based on exponential polynomials. Appl. Numer. Math. 60, 130–141 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lu, Y., Wang, G., Yang, X.: Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom. Des. 19(6), 335–343 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Micchelli, C.A., Prautzsch, H.: Refinement and subdivision for spaces of integer translates of compactly supported functions. In: Griffith, C.F., Watson, G.A. (eds.) Numerical Analysis, pp. 192–222. Academic Press, London (1987)

    Google Scholar 

  32. Peters, J., Reif, U.: Subdivision Surfaces. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  33. Reif, U.: A unified approach to subdivision algorithms near extraordinary vertices. Comput. Aided Geom. Des. 12, 153–174 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Romani, L.: A circle-preserving \(C^2\) Hermite interpolatory subdivision scheme with tension control. Comput. Aided Geom. Des. 27(1), 36–47 (2010)

    Article  MATH  Google Scholar 

  35. Romani, L., Mederos, V.H., Sarlabous, J.E.: Exact evaluation of a class of nonstationary approximating subdivision algorithms and related applications. IMA J. Numer. Anal. 36(1), 380–399 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Stam, J.: On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. Comput. Aided Geom. Des. 18(5), 383–396 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Uhlmann, V., Delgado-Gonzalo, R., Conti, C., Romani, L., Unser, M.: Exponential Hermite splines for the analysis of biomedical images. In: ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing—Proceedings 6853874, pp. 1631–1634 (2014)

  38. Umlauf, G.: Analyzing the characteristic map of triangular subdivision schemes. Constr. Approx. 16, 145–155 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vonesch, C., Blu, T., Unser, M.: Generalized Daubechies wavelet families. IEEE Trans. Signal Process. 55, 4415–4429 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Warren, J., Weimer, H.: Subdivision Methods for Geometric Design. Morgan-Kaufmann, New York (2002)

    Google Scholar 

  41. Zorin, D.: A method for analysis of \(C^1\)-continuity of subdivision surfaces. SIAM J. Numer. Anal. 35(5), 1677–1708 (2000)

    Article  MATH  Google Scholar 

  42. Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. In: Proceedings of 23rd International Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’96), pp. 189–192 (1996)

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Acknowledgements

This research has been accomplished within RITA (Research ITalian network on Approximation). The authors are members of the INdAM Research group GNCS, which has partially supported this work. The authors wish to thank the reviewers for their constructive comments that allowed them to improve the presentation of the results and to clarify all the details of their proofs.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Università di Firenze, Viale Morgagni 40/44, 50134, Firenze, Italy

    Costanza Conti

  2. Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Marco Donatelli & Paola Novara

  3. Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Lucia Romani

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  1. Costanza Conti
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  2. Marco Donatelli
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  3. Lucia Romani
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  4. Paola Novara
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Correspondence to Lucia Romani.

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Communicated by Wolfgang Dahmen.

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Conti, C., Donatelli, M., Romani, L. et al. Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces. Constr Approx 50, 457–496 (2019). https://doi.org/10.1007/s00365-019-09477-y

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  • DOI: https://doi.org/10.1007/s00365-019-09477-y

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