Piecewise Polynomial Monotonic Interpolation of 2D Gridded Data

  • Léo Allemand-Giorgis
  • Georges-Pierre Bonneau
  • Stefanie Hahmann
  • Fabien Vivodtzev
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


A method for interpolating monotone increasing 2D scalar data with a monotone piecewise cubic C1-continuous surface is presented. Monotonicity is a sufficient condition for a function to be free of critical points inside its domain. The standard axial monotonicity for tensor-product surfaces is however too restrictive. We therefore introduce a more relaxed monotonicity constraint. We derive sufficient conditions on the partial derivatives of the interpolating function to ensure its monotonicity. We then develop two algorithms to effectively construct a monotone C1 surface composed of cubic triangular Bézier surfaces interpolating a monotone gridded data set. Our method enables to interpolate given topological data such as minima, maxima and saddle points at the corners of a rectangular domain without adding spurious extrema inside the function domain. Numerical examples are given to illustrate the performance of the algorithm.


Local Extremum Monotonicity Constraint Regular Vertex Grid Vertex Hermite Data 
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.


  1. 1.
    Beatson, R.K.T., Ziegler, Z.: Monotonicity preserving surface interpolation. SIAM J. Numer. Anal. 22(2), 401–411 (1985)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Carlson, R.E., Fritsch, F.N.: An algorithm for monotone piecewise bicubic interpolation. SIAM J. Numer. Anal. 26(1), 230–238 (1989)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Carnicer, J.M., Floater, M.S., Peña, J.M.: Linear convexity conditions for rectangular and triangular Bernstein-Bézier surfaces. Comput. Aided Geom. Des. 15(1), 27–38 (1997)CrossRefMATHGoogle Scholar
  4. 4.
    Cavaretta Jr., A.S., Sharma, A.: Variation diminishing properties and convexity for the tensor product bernstein operator. In: Yadav, B.S., Singh, D. (eds.) Functional Analysis and Operator Theory. Lecture Notes in Mathematics, vol. 1511, pp. 18–32. Springer, Berlin/Heidelberg (1992)CrossRefGoogle Scholar
  5. 5.
    Delgado, J., Peòa, J.M.: Are rational Bézier surfaces monotonicity preserving? Comput. Aided Geom. Des. 24(5), 303–306 (2007)CrossRefMATHGoogle Scholar
  6. 6.
    Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse–Smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geom. 30(1), 87–107 (2003)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Farin, G.E.: Triangular Bernstein-Bézier patches. Comput. Aided Geom. Des. 3(2), 83–127 (1986)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Farin, G.E.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Code, 4th edn. Academic, Orlando (1996)Google Scholar
  9. 9.
    Floater, M.S., Peña, J.M.: Tensor-product monotonicity preservation. Adv. Comput. Math. 9(3–4), 353–362 (1998)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Floater, M.S., Peña, J.M.: Monotonicity preservation on triangles. Math. Comput. 69(232), 1505–1519 (2000)CrossRefMATHGoogle Scholar
  11. 11.
    Floater, M., Beccari, C., Cashman, T., Romani, L.: A smoothness criterion for monotonicity-preserving subdivision. Adv. Comput. Math. 39(1), 193–204 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Han, L., Schumaker, L.L.: Fitting monotone surfaces to scattered data using c1 piecewise cubics. SIAM J. Numer. Anal. 34(2), 569–585 (1997)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Jüttler, B.: Surface fitting using convex tensor-product splines. J. Comput. Appl. Math. 84(1), 23–44 (1997)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kuijt, F., van Damme, R.: Monotonicity preserving interpolatory subdivision schemes. J. Comput. Appl. Math. 101(1–2), 203–229 (1999)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mainar, E., Peña, J.M.: Monotonicity preserving representations of non-polynomial surfaces. J. Comput. Appl. Math. 233(9), 2161–2169 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    McAllister, D.F., Passow, E., Roulier, J.A.: Algorithms for computing shape preserving spline interpolations to data. Math. Comput. 31(139), 717–725 (1977)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Smale, S.: On gradient dynamical systems. Ann. Math. 74(1), 199–206 (1961)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Strom, K.: On convolutions of b-splines. J. Comput. Appl. Math. 55(1), 1–29 (1994)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Willemans, K., Dierckx, P.: Smoothing scattered data with a monotone powell-sabin spline surface. Numer. Algorithms 12(1), 215–231 (1996)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Léo Allemand-Giorgis
    • 1
  • Georges-Pierre Bonneau
    • 1
  • Stefanie Hahmann
    • 1
  • Fabien Vivodtzev
    • 2
  1. 1.Inria - Laboratoire Jean KuntzmannUniversity of GrenobleGrenobleFrance
  2. 2.CEA: French Atomic Energy Commission and Alternative EnergiesLe BarpFrance

Personalised recommendations