Advertisement

Bias-variance tradeoff for adaptive surface meshes

  • Richard C. Wilson
  • Edwin R. Hancock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1407)

Abstract

This paper presents a novel statistical methodology for exerting control over adaptive surface meshes. The work builds on a recently reported adaptive mesh which uses split and merge operations to control the distribution of planar or quadric surface patches. Hitherto, we have used the target variance of the patch fit residuals as a control criterion. The novelty of the work reported in this paper is to focus on the variance-bias tradeoff that exists between the size of the fitted patches and their associated parameter variances. In particular, we provide an analysis which shows that there is an optimal patch area which minimises the variance in the fitted patch parameters. This area offers the best compromise between the noise-variance, which decreases with increasing area, and the model-bias, which increases in a polynomial manner with area. The computed optimal areas of the local surface patches are used to exert control over the facets of the adaptive mesh. We use a series of split and merge operations to distribute the faces of the mesh so that each resembles as closely as possible its optimal area. In this way the mesh automatically selects its own model-order by adjusting the number of control-points or nodes. We provide experiments on both real and synthetic data. This experimentation demonstrates that our mesh is capable of efficiently representing high curvature surface detail.

Keywords

Noise Variance Adaptive Mesh Surface Patch Patch Area Bias Term 
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.

References

  1. 1.
    A.J. Bulpitt and N.D. Efford, “An efficient 3d deformable model with a self-optimising mesh”, Image and Vision Computing, 14, pp. 573–580, 1996.CrossRefGoogle Scholar
  2. 2.
    P. J. Besl and R. C. Jain, “Segmentation through variable order surface fitting”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 10:2 pp l67–192 1988CrossRefGoogle Scholar
  3. 3.
    X. Chen and F. Schmill. Surface modeling of range data by constrained triangulation”, Computer-Aided Design, 26, pp. 632–645 1994.zbMATHCrossRefGoogle Scholar
  4. 4.
    I. Cohen, L.D. Cohen and N. Ayache, “Using deformable surfaces to segment 3D images and infer differential structure”, CVGIP, 56, pp. 243–263, 1993.Google Scholar
  5. 5.
    I. Cohen and L.D. Cohen, “A hybrid hyper-quadric model for 2-d and 3-d data fitting”, Computer Vision and Image Understanding, 63, pp. 527–541, 1996.CrossRefGoogle Scholar
  6. 6.
    L. De Floriani, “A pyramidal data structure for triangle-based surface description”, IEEE Computer Graphics and Applications, 9, pp. 67–78, 1987.CrossRefGoogle Scholar
  7. 7.
    L. De Floriani, P. Marzano and E. Puppo, “Multiresolution models for topographic surface description.”, Visual Computer, 12:7, pp. 317–345, 1996.Google Scholar
  8. 8.
    H. Delingette, “Adaptive and deformable models based on simplex meshes”, IEEE Computer Society Workshop on Motion of Non-rigid and Articulated Objects, pp. 152–157, 1994.Google Scholar
  9. 9.
    H. Delingette, M. Hebert and K. Ikeuchi, “Shape representation and image segmentation using deformable surfaces”, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 467–472, 1991.Google Scholar
  10. 10.
    D. Geman, E. Bienenstock and R. Doursat, “Neural networks and the bias variance dilemma”, Neural Computation, 4, pp.1–58, 1992.Google Scholar
  11. 11.
    J. O. Lachaud and A. Montanvert, “Volumic segmentation using hierarchical representation and triangulates surface”, Computer Vision, ECCV'96, Edited by B. Buxton and R. Cipolla, Lecture Notes in Computer Science, Volume 1064, pp. 137–146, 1996.Google Scholar
  12. 12.
    R. Lengagne, P. Fua and O. Monga, “Using crest-lines to guide surface reconstruction from stereo”, Proceedings of th 13th International Conference on Pattern Recognition, Volume A, pp. 9–13, 1996.Google Scholar
  13. 13.
    D. McInerney and D. Terzopoulos, “A finite element model for 3D shape reconstruction and non-rigid motion tracking”, Fourth International Conference on Computer Vision, pp. 518–532, 1993.Google Scholar
  14. 14.
    D. McInerney and D. Terzopoulos, “A dynamic finite-element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image-analysis”, Computerised Medical Imaging and Graphics, 19, pp. 69–83, 1995.CrossRefGoogle Scholar
  15. 15.
    M. Moshfeghi, S. Ranganath and K. Nawyn, “Three dimensional elastic matching of volumes”, IEEE Transactions on Image Processing, 3, pp. 128–138Google Scholar
  16. 16.
    W. Neuenschwander, P. Fua, G. Szekely and O. Kubler, “Deformable Velcro Surfaces”, Fifth International Conference on Computer Vision, pp. 828–833, 1995.Google Scholar
  17. 17.
    P.T. Sander and S.W. Zucker, “Inferring surface structure and differential structure from 3D images”, IEEE PAMI, 12, pp 833–854, 1990.Google Scholar
  18. 18.
    F. J. M. Schmitt, B. A. Barsky and Wen-Hui Du, “An adaptive subdivision method for surface fitting from sampled data”, SIGGRAPH '86, 20, pp. 176–188, 1986Google Scholar
  19. 19.
    W.J. Schroeder, J.A. Zarge and W.E. Lorenson, “Decimation of triangle meshes”, Computer Graphics, 26 pp. 163–169, 1992.Google Scholar
  20. 20.
    A.J. Stoddart, A. Hilton and J. Illingworth, “SLIME: A new deformable surface”, Proceedings British Machine Vision Conference, pp. 285–294, 1994.Google Scholar
  21. 21.
    A. J. Stoddart, J. Illingworth and T. Windeatt, ”Optimal Parameter Selection for Derivative Estimation from Range Images” Image and Vision Computing, 13, pp629–635, 1995.CrossRefGoogle Scholar
  22. 22.
    M. Turner and E.R. Hancock, “Bayesian extraction of differential surface structure”, in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, Volume 970, Edited by V. Havlac and R. Sara, pp. 784–789, 1995.Google Scholar
  23. 23.
    M. Turner and E.R. Hancock, “A Bayesian approach to 3D surface fitting and refinement”, Proceedings of the British Machine Vision Conference, pp. 67–76, 1995.Google Scholar
  24. 24.
    G. Turk, “Re-tiling polygonal surfaces”, Computer Graphics, 26, pp. 55–64, 1992.Google Scholar
  25. 25.
    M. Vasilescu and D. Terzopoulos, “Adaptive meshes and shells”, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 829–832, 1992.Google Scholar
  26. 26.
    R. C. Wilson and E. R. Hancock, ”Refining Surface Curvature with Relaxation Labeling”, Proceedings of ICIAP97, Ed, A. Del Bimbo, Lecture Notes in Computer Science 1310, Springer pp. 150–157 1997.Google Scholar
  27. 27.
    R. C. Wilson and E. R. Hancock, ”A Minimum-Variance Adaptive Surface Mesh”, CVPR'97, pp. 634–639 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

Personalised recommendations