Investigation into Building an Invariant Surface Model from Sparse Data

  • Robert L. Stevenson
  • Edward J. Delp
Conference paper
Part of the NATO ASI Series book series (volume 83)


This paper addresses the problem of forming surface depth estimates from sparse information, which are invariant to three-dimensional rotations and translations of the surface in the chosen coordinate system. We begin this investigation by examining a simplified version of this problem, that of forming invariant curve estimates from sparse data, to help gain insight into the more complex problem in higher dimensions. Two new algorithms are proposed and studied in detail, and several examples are presented to demonstrate their effectiveness. The extension of these algorithms to surfaces in threedimensional space is also briefly discussed.


Surface Reconstruction Location Constraint Invariant Curve Reconstruction Problem Nonconvex Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Poggio, T., T. V. Torre, and C. Koch: Computational vision and regularization theory. Nature 317, 314–319 (1985)CrossRefGoogle Scholar
  2. 2.
    Tikhonov, A. N. and V. Y. Arsenin: Solutions of Ill-Posed Problems. Washington, D. C.:V. H. Winston & Sons 1977MATHGoogle Scholar
  3. 3.
    Guggenheimer, H. W.: Differential Geometry. New York: Dover Publications, Inc. 1977MATHGoogle Scholar
  4. 4.
    Besl, P. J. and R. C. Jain: Invariant surface characteristics for 3D object recognition in range data. Computer Vision, Graphics and Image Processing 33, 33–80 (1886).CrossRefGoogle Scholar
  5. 5.
    Bernoulli, D.: The 26th letter to Euler. In: Correspondence Mathematique et Physique (P. H. Fuss, ed.). St. Petersburg: Academia imperiale des sciences 1843Google Scholar
  6. 6.
    Malcolm, M. A.: On the computation of nonlinear spline functions. SIAM Journal of Numerical Analysis 14, 254–282 (1977)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Blake, A. and A. Zisserman: Visual reconstruction. Cambridge, Massachusetts: MIT Press 1987Google Scholar
  8. 8.
    Grimson, W. E. L.: From images to surfaces: a computational study of the human early visual system. Cambridge, Massachusetts: MIT Press 1981Google Scholar
  9. 9.
    Terzopoulos, D.: Multilevel reconstruction of visual surfaces: variational principles and finite-element representations. In: Multiresolution Image Processing and Analysis (A. Rosenfeld, ed.). New York, NY: Springer-Verlag 1984Google Scholar
  10. 10.
    Terzopoulos, D.: Regularization of inverse visual problems involving discontinuities. IEEE Transactions on Pattem Analysis and Machine Intelligence PAMI-8, 413–424 (1986)CrossRefGoogle Scholar
  11. 11.
    Terzopoulos, D.: The computation of visible-surface representations. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-10, 417–438 (1988)CrossRefGoogle Scholar
  12. 12.
    Boult, T. E. and J. R. Kender: Visual surface reconstruction using sparse depth data. Proceedings of the Computer Vision and Pattern Recognition Conference 1986, Miami, FL, pp. 68–76.Google Scholar
  13. 13.
    Lancaster, P. and K. Salkauskas: Curve and Surface Fitting. London: Academic Press 1986MATHGoogle Scholar
  14. 14.
    Reinsch, C. H.: Smoothing by Spline Functions. Numerische Mathematik 10, 177–183 (1967)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Salkauskas, K.: C1 splines for interpolation of rapidly varying data. Rocky Mountain Journal of Mathematics 14, 239–250 (1984)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Blake, A. and A. Zisserman: Invariant surface reconstruction using weak continuity constraints. Proceedings of the Computer Vision and Pattern Recognition Conference 1986, Miami, FL, pp. 62–67.Google Scholar
  17. 17.
    Terzopoulos, D.: Multilevel computational processes for visual surface reconstruction. Computer Vision, Graphics, and Image Processing 24, 52–96 (1983)CrossRefGoogle Scholar
  18. 18.
    Lipschutz, M. M.: Differential Geometry. New York, NY: McGraw-Hill 1969MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Robert L. Stevenson
    • 1
  • Edward J. Delp
    • 1
  1. 1.Computer Vision and Image Processing Laboratory School of Electrical EngineeringPurdue UniversityWest LafayetteUSA

Personalised recommendations