Smooth Image Surface Approximation by Piecewise Cubic Polynomials

  • Oliver Matias van Kaick
  • Helio Pedrini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4756)


The construction of surfaces from dense data points is an important problem encountered in several applications, such as computer vision, reverse engineering, computer graphics, terrain modeling, and robotics. Moreover, the particular problem of approximating digital images from a set of selected points allows to employ methods that are directed specifically to this task, which take advantage of the fact that all points belong to a common 2D domain. This paper describes a method for approximating images by fitting smooth surfaces to scattered points, where the smooth surfaces are constructed using piecewise cubic approximation. An incremental triangulation algorithm is used to iteratively refine a mesh until a specified error tolerance is achieved. The resulting surface is represented by a network of piecewise cubic triangular patches possessing C 1 continuity. The proposed method is compared against other surface approximation approaches and applied to several data sets in order to demonstrate its performance.


Surface modeling image surface smooth interpolation 


  1. 1.
    van Kaick, O.M., da Silva, M.V.G., Schwartz, W.R., Pedrini, H.: Fitting Smooth Surfaces to Scattered 3D Data Using Piecewise Quadratic Approximation. In: IEEE International Conference on Image Processing, Rochester, New York, USA, pp. 493–496. IEEE Computer Society Press, Los Alamitos (2002)CrossRefGoogle Scholar
  2. 2.
    Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10(6), 350–355 (1978)CrossRefGoogle Scholar
  3. 3.
    Dyn, N., Levin, D., Gregory, J.A.: A Butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9(2), 160–169 (1990)zbMATHCrossRefGoogle Scholar
  4. 4.
    Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)Google Scholar
  5. 5.
    Stam, J.: On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. Computer Aided Geometric Design 18, 383–396 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Forsey, D.R., Bartels, R.H.: Surface fitting with hierarchical splines. ACM Transactions on Graphics 14(2), 134–161 (1995)CrossRefGoogle Scholar
  7. 7.
    Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design - A Practical Guide. Academic Press, London (1992)Google Scholar
  8. 8.
    Hahmann, S., Bonneau, G.P.: Polynomial surfaces interpolating arbitrary triangulations. IEEE Transactions on Visualization and Computer Graphics 9(1), 99–109 (2003)CrossRefGoogle Scholar
  9. 9.
    Chaikin, G.M.: An algorithm for high-speed curve generation. Computer Graphics and Image Processing 3(4), 346–349 (1974)Google Scholar
  10. 10.
    Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10(6), 356–360 (1978)CrossRefGoogle Scholar
  11. 11.
    Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. In: SIGGRAPH 1996 Conference Proceedings, New Orleans, Louisiana, USA, pp. 189–192 (August 1996)Google Scholar
  12. 12.
    Eck, M., Hoppe, H.: Automatic reconstruction of B-spline surfaces of arbitray topological type. In: ACM SIGGRAPH 1996, pp. 325–334. ACM Press, New York (1996)CrossRefGoogle Scholar
  13. 13.
    Zheng, J.J., Zhang, J.J., Zhou, H.J., Shen, L.G.: Smooth spline surface generation over meshes of irregular topology. The Visual Computer 21(8–10), 858–864 (2005)CrossRefGoogle Scholar
  14. 14.
    Krishnamurthy, V., Levoy, M.: Fitting smooth surfaces to dense polygonal meshes. In: Computer Graphics Proceedings, Annual Conference Series, pp. 313–324 (1996)Google Scholar
  15. 15.
    Loop, C.: Smooth splines surfaces over irregular meshes. In: Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, pp. 303–310. ACM Press, New York (1994)CrossRefGoogle Scholar
  16. 16.
    Peters, J.: C1 surface splines. SIAM Journal of Numerical Analysis 32(2), 645–666 (1995)zbMATHCrossRefGoogle Scholar
  17. 17.
    Clough, R., Tocher, J.: Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of Conference on Matrix Methods in Structural Analysis (1965)Google Scholar
  18. 18.
    Quak, E., Schumaker, L.L.: Cubic spline fitting using data dependent triangulations. Computer-Aided Geometric Design 7(1–4), 293–301 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Oliver Matias van Kaick
    • 1
  • Helio Pedrini
    • 2
  1. 1.School of Computing Science, Simon Fraser University, Burnaby, BC, V5A 1S6Canada
  2. 2.Department of Computer Science, Federal University of Paraná, Curitiba, PR, 81531-990Brazil

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