Skip to main content

Partition of unity parametrics: a framework for meta-modeling

Abstract

We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate non-trivial effects. This serves as a basis for a system where users model the tools used for modeling, here weight-functions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting non-tensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Barsky, B.: Computer Graphics and Geometric Modeling using Beta-Splines. Springer, Berlin (1988)

    MATH  Google Scholar 

  2. 2.

    de Boor, C.: A Practical Guide to Splines (revised edn.). Springer, Berlin (1978)

    MATH  Google Scholar 

  3. 3.

    Brunn, M., Sousa, M., Samavati, F.: Capturing and re-using artistic styles with multiresolution analysis. Int. J. Image Graph. 7(4), 593–615 (2007)

    Article  Google Scholar 

  4. 4.

    Buhmann, M.D.: Radial basis functions. Acta Numer. 9, 1–38 (2000)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cashman, T.J., Augsdörfer, U.H., Dodgson, N.A., Sabin, M.A.: NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes. In: SIGGRAPH’09, pp. 1–9 (2009)

    Google Scholar 

  6. 6.

    Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Schempp, W., Zeller, K. (eds.) Constructive Theory of Functions of Several Variables, vol. 571, pp. 85–100 (1977)

    Chapter  Google Scholar 

  7. 7.

    Eck, M., Hoppe, H.: Automatic reconstruction of B-spline surfaces of arbitrary topological type. In: Proceedings of SIGGRAPH’96, pp. 325–334 (1996)

    Google Scholar 

  8. 8.

    Floater, M.S.: Parametrization and smooth approximation of surface triangulations. Comput. Aided Geom. Des. 14(3), 231–250 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19–27 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Franke, R., Nielson, G.: Smooth interpolation of large sets of scattered data. Int. J. Numer. Methods Eng. 15(11), 1691–1704 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Grimm, C.M., Hughes, J.F.: Modeling surfaces of arbitrary topology using manifolds. In: Proceedings of SIGGRAPH’95, pp. 359–368 (1995)

    Google Scholar 

  12. 12.

    Hormann, K., Lévy, B., Sheffer, A.: Mesh parameterization: theory and practice. In: SIGGRAPH 2007 Course Notes, pp. 1–87 (2007)

    Google Scholar 

  13. 13.

    Li, Q., Tian, J.: 2d piecewise algebraic splines for implicit modeling. ACM Trans. Graph. 28(2), 1–19 (2009)

    Article  Google Scholar 

  14. 14.

    Meyer, M., Barr, A., Lee, H., Desbrun, M.: Generalized barycentric coordinates on irregular polygons. J. Graph. Tools 7(1), 13–22 (2002)

    MATH  Google Scholar 

  15. 15.

    Nealen, A., Igarashi, T., Sorkine, O., Alexa, M.: Fibermesh: Designing free-form surfaces with 3D curves. In: SIGGRAPH’07 p. 41. (2007)

    Google Scholar 

  16. 16.

    Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., Seidel, H.P.: Multi-level partition of unity implicits. ACM Trans. Graph. 22(3), 463–470 (2003)

    Article  Google Scholar 

  17. 17.

    Olsen, L., Samavati, F., Sousa, M., Jorge, J.: Sketch-based mesh augmentation. In: Proceedings of the Sketch-Base Interfaces and Modeling (SBIM) Workshop (2005)

    Google Scholar 

  18. 18.

    Olsen, L., Samavati, F., Sousa, M., Jorge, J.: Sketch-based modeling: a survey. Comput. Graph. 33(1), 85–103 (2009)

    Article  Google Scholar 

  19. 19.

    Piegl, L., Tiller, W.: The NURBS Book. Springer, Berlin (1997)

    Google Scholar 

  20. 20.

    Piegl, L., Tiller, W.: Filling n-sided regions with NURBS patches. Vis. Comput. 15(2), 77–89 (1999)

    MATH  Article  Google Scholar 

  21. 21.

    Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graph. 22(3), 477–484 (2003)

    Article  Google Scholar 

  22. 22.

    Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23rd ACM National Conference, pp. 517–524 (1968)

    Chapter  Google Scholar 

  23. 23.

    Turk, G., O’ Brien, J.: Variational implicit surfaces. Tech. rep., Georgia Institute of Technology (1999)

  24. 24.

    Wang, Q., Hua, W., Guiqing, L., Bao, H.: Generalized NURBS curves and surfaces. In: Geometric Modeling and Processing, pp. 365–368 (2004)

    Chapter  Google Scholar 

  25. 25.

    Ying, L., Zorin, D.: A simple manifold-based construction of surfaces of arbitrary smoothness. In: SIGGRAPH’04, pp. 271–275 (2004)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Adam Runions.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Runions, A., Samavati, F.F. Partition of unity parametrics: a framework for meta-modeling. Vis Comput 27, 495–505 (2011). https://doi.org/10.1007/s00371-011-0567-x

Download citation

Keywords

  • Meta-modeling
  • Parametric curves and surfaces
  • Sketch-based modeling
  • Geometric modeling