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Box Splines and Applications

  • M. Dæhlen
  • T. Lyche
Part of the Computer Graphics — Systems and Applications book series (COMPUTER GRAPH.)

Abstract

We give an elementary introduction to box spline methods for the representation of surfaces. First, we derive basic properties of box splines starting with the univariate cardinal case. Proofs of most of the results are included. We proceed with a detailed presentation of refinement and evaluation methods for box splines. We discuss shape preserving properties, the construction of non-rectangular box spline surfaces, applications of box splines to surface modelling and problems related to an imbedding of box spline surfaces within a tensor product surface.

Keywords

Tensor Product Control Polygon Cardinal Spline Multivariate Spline Cardinal Interpolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arge, E. and M. Dæhlen, Grid point interpolation on finite regions by box splines, preprint.Google Scholar
  2. 2.
    Boehm, W., Subdividing multivariate splines, Comput. Aided Design 15 (1983), 345–352.CrossRefGoogle Scholar
  3. 3.
    Boehm, W., Prautzsch, H., and P. Amer, On triangular splines, Constr. Approx. 2 (1987), 157–167.MathSciNetCrossRefGoogle Scholar
  4. 4.
    de Boor, C., Splines as linear combinations of B-splines. A survey, in Approximation Theory II, G. G. Lorentz, C. K. Chui, and L. L. Schumaker (eds.), Academic Press, New York, 1976, 1–47.Google Scholar
  5. 5.
    de Boor, C., A Practical Guide to Splines, Springer-Verlag, New York, (1978).MATHGoogle Scholar
  6. 6..
    de Boor, C., Multivariate approximation, in The State of the Art in Numerical analysis, A. Iserles and M.J.D. Powell (eds.), Claredon Press, Oxford 1987, 87–109.Google Scholar
  7. 7.
    de Boor, C. and R. DeVore, Approximations by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), 775–785.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    de Boor, C.and K. Höllig, B-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99–115.MATHCrossRefGoogle Scholar
  9. 9.
    de Boor, C. and K. Höllig, Bivariate box splines and pp functions on a three-direction mesh, J. Comput. Appl. Math. 9 (1983), 13–28.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Cavaretta, A. S. and C. A. Micchelli, Subdivision algorithms, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker (eds.), Academic Press, N. Y., 1989, 115–153.Google Scholar
  11. 11.
    Chaikin, G. M., An algorithm for high speed curve generation, Computer Graphics and Image Processing 3 (1974), 346–349.CrossRefGoogle Scholar
  12. 12.
    Cheney, E. W., Multivariate approximation theory: Selected topics, CBMS-NSF Reg. Conf. series in applied mathematics 51, SIAM, Philadelphia, 1986.Google Scholar
  13. 13.
    Chui, C. K., Multivariate Splines, CBMS-NSF, Reg. Conf. series in Appl. Math., SIAM, Philadelphia, (1988).Google Scholar
  14. 14.
    Chui, C. K., Diamond, H. and L. Raphael, Interpolation by multivariate splines, Math. Comp. 51, (1988), 203–218.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Chui, C. K., Diamond, H. and L. Raphael, Shape-preserving quasi-interpolation and interpolation bybox spline surfaces, J. Comput. Appl. Math. 25, (1989), 169–198.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Cohen, E., Lyche, T., and R. Riesenfeld, Discrete box-splines and refinement algorithms, Comput. Aided Geom. Design 1 (1984), 131–148.MATHGoogle Scholar
  17. 17.
    Cohen, E., Lyche, T., and R. Riesenfeld, Cones and recurrence relations for simplex splines, Constr. Approx. 3 (1987), 131–141.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Cohen, E., Lyche, T., and L. L. Schumaker, Degree raising for splines, J. Approx. Theory 46 (1986), 170–181.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Dæhlen, M., An example of bivariate interpolation with translates of C 0-quadratic box splines on a three direction mesh, Comput. Aided Geom. Design 4 (1987), 251255.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Dæhlen, M., On the evaluation of box splines, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker (eds.), Academic Press, N. Y., 1989, 167–179.Google Scholar
  21. 21.
    Dæhlen, M., Box splines and applications of polynomial splines, dissertation, University of Oslo, Research Report 128, Institute for informatics, ISBN 82 7368 032 0, (1989).Google Scholar
  22. 22.
    Dæhlen, M., Knotline removal on box spline surfaces, preprint.Google Scholar
  23. 23.
    Dæhlen, M., Modelling with box spline surfaces, in Curve and Surface Design, H. Hagen (ed.), SIAM Conference Series, to appear.Google Scholar
  24. 24.
    Dæhlen, M. and T. Lyche, Bivariate interpolation with quadratic box splines, Math. Comp. 51 (1988), 219–230.MathSciNetMATHGoogle Scholar
  25. 25.
    Dæhlen, M. and V. Skytt, Modelling non-rectangular surfaces using box splines, in Mathematics of Surfaces III, D. Handscomb (ed.), (1989), 285–300.Google Scholar
  26. 26.
    Dahmen, W., Multivariate B-splines — Recurrence relations and linear combinations of truncated powers, in Multivariate Approximation Theory, W. Schempp and K. Zeller (eds.), Birkhäuser, Basel, (1979), 64–82.Google Scholar
  27. 27.
    Dahmen, W. and C. A. Micchelli, Recent progress in multivariate splines, in Approximation Theory IV, C. K. Chui, L. L. Schumaker, and J. Ward, (eds.), Academic Press, New York, (1983), 27–121.Google Scholar
  28. 28.
    Dahmen, W. and C. A. Micchelli, Subdivision algorithms for generation of of box-spline surfaces, Comput. Aided Geom. Design 1 (1984), 115–129.MATHGoogle Scholar
  29. 29.
    Dahmen, W., and C. A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), 243–262.MathSciNetMATHGoogle Scholar
  30. 30.
    Dahmen, W., Dyn, N., and D. Levin, On the convergence rates of subdivision algorithms for box spline surfaces, Constr. Approx. 1 (1985), 305–322.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Goodman, T.N.T., Polyhedral splines, in Computation of Curves and Surfaces, W. Dahmen, M. Gasca and C.A. Micchelli (eds.), Kluwer, Dordrecht, 1990, 347–382.Google Scholar
  32. 32.
    Höllig, K., Box-splines, in Approximation Theory V, C. K. Chui, L. L. Schumaker, and J. Ward, (eds.), Academic Press, (1986), 71–95.Google Scholar
  33. 33.
    Höllig, K., Box-splines surfaces, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker (eds.), Academic Press, N. Y., 1989, 385–402.Google Scholar
  34. 34.
    Jetter, K., A short survey on cardinal interpolation by box splines, in Topics in Multivariate Approximation,C. K. Chui, L. L. Schumaker, and F. Utreras, (eds.), Academic Press, Boston, (1987), 125–139Google Scholar
  35. 35.
    Jia, Rong-Qing, Linear independence of translates of box splines, J. Approx. Theory 40 (1984), 158–160.MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Lane, J. M. and R. F. Riesenfeld, A theoretical development for the computer generation of piecewise polynomial surfaces, IEEE Trans. on Pattern Analysis and Machine Intelligence 2 (1980), 34–46.CrossRefGoogle Scholar
  37. 37.
    Lane, J. M. and R. F. Riesenfeld, A geometric proof of the variation diminishing property of B-spline approximation, J. Approx. Theory 37 (1983), 1–4.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Lyche, T., Discrete B-splines and conversion problems, in Computation of Curves and Surfaces, W. Dahmen, M. Gasca and C.A. Micchelli (eds.), Kluwer, Dordrecht, 1990, 117–134.Google Scholar
  39. 39.
    Mueller, T. I., Geometric Modelling with multivariate B-splines, dissertation, Dept. of Comp. Science, Univ. of Utah, (1986).Google Scholar
  40. 40.
    Micchelli, C. A., Algebraic aspects of interpolation, in Approximation Theory, AMS Symposium in Applied Mathematics 36, Providence, R. I., 1986, 81–102.Google Scholar
  41. 41.
    Prautzsch, H., Unterteilungsalgorithmen für multivariate splines, Ein geometrische zugang, dissertation, Technische Universität Braunschweig, (1984).MATHGoogle Scholar
  42. 42.
    Prautzsch, H., Degree elevation of B-spline curves, Comput. Aided Geom. Design 1 (1984), 193–198.MATHGoogle Scholar
  43. 43.
    Ron, A., Exponential box splines, Constr. Approx. 4, (1988), 357–378.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Sabin, M. A., The use of piecewise forms for the numerical representation of shapes, dissertation, Hungarian National Academy of Sciences, 1977.Google Scholar
  45. 45.
    Sablonnière, P., A catalog of B-splines of degree ≤ 10 on a three direction mesh, Report ANO-132, Université de Lille, France, 1984.Google Scholar
  46. 46.
    Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946), 45–99, 112–141.MathSciNetGoogle Scholar
  47. 47.
    Schoenberg, I. J., letter to Philip J. Davis dated May 31, 1965.Google Scholar
  48. 48.
    Schoenberg, I. J., Cardinal Spline Interpolation, Reg. Conf. series in Applied Mathematics 12 SIAM, Philadelphia, 1973.Google Scholar
  49. 49.
    Schumaker, L. L., Spline functions: Basic Theory, Whiley & Sons, New York, (1981).MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • M. Dæhlen
    • 2
  • T. Lyche
    • 1
  1. 1.Institutt for informatikkOslo 3Norway
  2. 2.Center for Industrial ResearchOslo 3Norway

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