Extending Neural Networks for B-Spline Surface Reconstruction

  • G. Echevarría
  • A. Iglesias
  • A. Gálvez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2330)


Recently, a new extension of the standard neural networks, the so-called functional networks, has been described [5]. This approach has been successfully applied to the reconstruction of a surface from a given set of 3D data points assumed to lie on unknown Bézier [17] and B-spline tensor-product surfaces [18]. In both cases the sets of data were fitted using Bézier surfaces. However, in general, the Bézier scheme is no longer used for practical applications. In this paper, the use of B-spline surfaces (by far, the most common family of surfaces in surface modeling and industry) for the surface reconstruction problem is proposed instead. The performance of this method is discussed by means of several illustrative examples. A careful analysis of the errors makes it possible to determine the number of B-spline surface fitting control points that best fit the data points. This analysis also includes the use of two sets of data (the training and the testing data) to check for overfitting, which does not occur here.


  1. 1.
    Anand, V.: Computer Graphics and Geometric Modeling for Engineers. John Wiley and Sons, New York (1993)Google Scholar
  2. 2.
    Barhak, J., Fischer, A.: Parameterization and reconstruction from 3D scattered points based on neural network and PDE techniques. IEEE Trans. on Visualization and Computer Graphics 7(1) (2001) 1–16CrossRefGoogle Scholar
  3. 3.
    Bolle, R.M., Vemuri, B.C.: On three-dimensional surface reconstruction methods. IEEE Trans. on Pattern Analysis and Machine Intelligence 13(1) (1991) 1–13CrossRefGoogle Scholar
  4. 4.
    Brinkley, J.F.: Knowledge-driven ultrasonic three-dimensional organ modeling. IEEE Trans. on Pattern Analysis and Machine Intelligence 7(4) (1985) 431–441CrossRefGoogle Scholar
  5. 5.
    Castillo, E.: Functional Networks. Neural Processing Letters 7 (1998) 151–159CrossRefMathSciNetGoogle Scholar
  6. 6.
    Farin, G.E.: Curves and Surfaces for Computer-Aided Geometric Design (Fifth Edition). Morgan Kaufmann, San Francisco (2001)Google Scholar
  7. 7.
    Foley, T.A.: Interpolation to scattered data on a spherical domain. In: Cox, M., Mason, J. (eds.), Algorithms for Approximation II, Chapman and Hall, London (1990) 303–310Google Scholar
  8. 8.
    Freeman, J.A.: Simulating Neural Networks with Mathematica. Addison Wesley, Reading, MA, (1994)Google Scholar
  9. 9.
    Fuchs, H., Kedem, Z.M., Uselton, S.P.: Optimal surface reconstruction form planar contours. Communications of the ACM, 20(10) (1977) 693–702zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gu, P., Yan, X.: Neural network approach to the reconstruction of free-form surfaces for reverse engineering. CAD, 27(1) (1995) 59–64Google Scholar
  11. 11.
    Hastie, T., Stuetzle, W.: Principal curves. JASA, 84 (1989) 502–516zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hertz, J., Krogh, A., Palmer, R.G.: Introduction to the Theory of Neural Computation. Addison Wesley, Reading, MA (1991)Google Scholar
  13. 13.
    Hoffmann, M., Varady, L.: Free-form surfaces for scattered data by neural networks. Journal for Geometry and Graphics, 2 (1998) 1–6zbMATHMathSciNetGoogle Scholar
  14. 14.
    Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. Proc. of SIGGRAPH’92, Computer Graphics, 26(2) (1992) 71–78CrossRefGoogle Scholar
  15. 15.
    Hoppe, H.: Surface reconstruction from unorganized points. Ph. D. Thesis, Department of Computer Science and Engineering, University of Washington (1994)Google Scholar
  16. 16.
    Kohonen, T.: Self-Organization and Associative Memory (3rd. Edition). Springer-Verlag, Berlin (1989)Google Scholar
  17. 17.
    Iglesias, A., Gálvez, A.: A new Artificial Intelligence paradigm for Computer-Aided Geometric Design. In: Artificial Intelligence and Symbolic Computation. Campbell, J. A., Roanes-Lozano, E. (eds.), Springer-Verlag, Lectures Notes in Artificial Intelligence, Berlin Heidelberg 1930 (2001) 200–213.Google Scholar
  18. 18.
    Iglesias, A., Gálvez, A.: Applying functional networks to fit data points from B-spline surfaces. In: Proceedings of the Computer Graphics International, CGI’2001, Ip, H.H.S., Magnenat-Thalmann, N., Lau, R.W.H., Chua, T.S. (eds.) IEEE Computer Society Press, Los Alamitos, California (2001) 329–332Google Scholar
  19. 19.
    Lim, C., Turkiyyah, G., Ganter, M., Storti, D.: Implicit reconstruction of solids from cloud point sets. Proc. of 1995 ACM Symposium on Solid Modeling, Salt Lake City, Utah, (1995) 393–402Google Scholar
  20. 20.
    Meyers, D., Skinnwer, S., Sloan, K.: Surfaces from contours. ACM Transactions on Graphics, 11(3) (1992) 228–258zbMATHCrossRefGoogle Scholar
  21. 21.
    Meyers, D.: Reconstruction of Surfaces from Planar Sections. Ph. D. Thesis, Department of Computer Science and Engineering, University of Washington (1994)Google Scholar
  22. 22.
    Nilroy, M., Bradley, C., Vickers, G., Weir, D.: G1 continuity of B-spline surface patches in reverse engineering. CAD, 27(6) (1995) 471–478Google Scholar
  23. 23.
    Park, H., Kim, K.: 3-D shape reconstruction from 2-D cross-sections. J. Des. Mng., 5 (1997) 171–185Google Scholar
  24. 24.
    Park, H., Kim, K.: Smooth surface approximation to serial cross-sections. CAD, 28(12) (1997) 995–1005Google Scholar
  25. 25.
    Piegl, L., Tiller, W.: The NURBS Book (Second Edition). Springer Verlag, Berlin Heidelberg (1997)Google Scholar
  26. 26.
    Pratt, V.: Direct least-squares fitting of algebraic surfaces. Proc. of SIGGRAPH’87, Computer Graphics, 21(4) (1987) 145–152CrossRefMathSciNetGoogle Scholar
  27. 27.
    Schmitt, F., Barsky, B.A., Du, W.: An adaptive subdivision method for surface fitting from sampled data. Proc. of SIGGRAPH’86, Computer Graphics, 20(4) (1986) 179–188CrossRefGoogle Scholar
  28. 28.
    Sclaroff, S., Pentland, A.: Generalized implicit functions for computer graphics. Proc. of SIGGRAPH’91, Computer Graphics, 25(4) (1991) 247–250CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • G. Echevarría
    • 1
  • A. Iglesias
    • 1
  • A. Gálvez
    • 1
  1. 1.Department of Applied Mathematics and Computational SciencesUniversity of CantabriaSantanderSpain

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