Adaptive Constructive Polynomial Fitting

  • Francis Deboeverie
  • Kristof Teelen
  • Peter Veelaert
  • Wilfried Philips
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6474)


To extract geometric primitives from edges, we use an incremental linear-time fitting algorithm, which is based on constructive polynomial fitting. In this work, we propose to determine the polynomial order by observing the regularity and the increase of the fitting cost. When using a fixed polynomial order under- or even overfitting could occur. Second, due to a fixed treshold on the fitting cost, arbitrary endpoints are detected for the segments, which are unsuitable as feature points. We propose to allow a variable segment thickness by detecting discontinuities and irregularities in the fitting cost. Our method is evaluated on the MPEG-7 core experiment CE-Shape-1 database part B [1]. In the experimental results, the edges are approximated closely by the polynomials of variable order. Furthermore, the polynomial segments have robust endpoints, which are suitable as feature points. When comparing adaptive constructive polynomial fitting (ACPF) to non-adaptive constructive polynomial fitting (NACPF), the average Hausdorff distance per segment decreases by 8.85% and the object recognition rate increases by 10.24%, while preserving simplicity and computational efficiency.


Feature Point Segmentation Result Canny Edge Detector Elemental Subset Average Euclidean Distance 
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.
    Jeannin, S., Bober, M.: Description of core experiments for mpeg-7 motion/shape, Technical Report ISO/IEC JTC 1/SC 29/WG 11 MPEG99/N2690 (1999)Google Scholar
  2. 2.
    Dorst, L., Smeulders, A.W.M.: Length estimators for digitized contours. Computer Vision, Graphics, and Image Processing 40(3), 311–333 (1987)CrossRefGoogle Scholar
  3. 3.
    Debled-Rennesson, I., Reveills, J.-P.: A Linear Algorithm for Segmentation of Digital Curves. Int. J. of Pattern Recognition and Artificial Intelligence 9(4), 635–662 (1995)CrossRefGoogle Scholar
  4. 4.
    Buzer, L.: An Incremental Linear Time Algorithm for Digital Line and Plane Recognition Using a Linear Incremental Feasibility Problem. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 372–381. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Piegl, L.: On NURBS: A Survey. IEEE Computer Graphics and Applications 11(1), 55–71 (1991)CrossRefGoogle Scholar
  6. 6.
    Goshtasby, A.: Design and Recovery of 2D and 3D Shapes Using Rational Gaussian Curves and Surfaces. International Journal of Computer Vision 10(3), 233–256 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Turk, G., O’Brien, J.F.: Variational Implicit Surfaces. Technical Report GIT-GVU-99-15, Graphics, Visualization, and Usability Center, Georgia Technical Univ. (1999)Google Scholar
  8. 8.
    Taubin, G.: Estimation of Planar Curves, Surfaces and Nonplanar Space Curves Defined by Implicit Equations, with Applications to Edge and Range Image Segmentation. IEEE Trans. PAMI 13(11), 1115–1138 (1991)CrossRefGoogle Scholar
  9. 9.
    Blane, M.M., Lei, Z., Civi, H., Cooper, D.B.: The 3L Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data. IEEE Trans. PAMI 22(3), 298–313 (2000)CrossRefGoogle Scholar
  10. 10.
    Sahin, T., Unel, M.: Stable Algebraic Surfaces for 3D Object Representation. Journal of Mathematical Imaging and Vision 32(2), 127–137 (2008)CrossRefGoogle Scholar
  11. 11.
    Keren, D., Gotsman, C.: Fitting Curves and Surfaces with Constrained Implicit Polynomials. IEEE Trans. PAMI 21(1), 31–41 (1999)CrossRefGoogle Scholar
  12. 12.
    Keren, D.: Topologically Faithful Fitting of Simple Closed Curves. IEEE Trans. PAMI 26(1), 118–123 (2004)CrossRefGoogle Scholar
  13. 13.
    Tasdizen, T., Tarel, J., Cooper, D.: Improving the Stability of Algebraic Curves for Applications. IEEE Trans. Image Processing 9(3), 405–416 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Helzer, A., Barzohar, M., Malah, D.: Stable Fitting of 2D Curves and 3d Surfaces by Implicit Polynomials. IEEE Trans. PAMI 26(10), 1283–1294 (2004)CrossRefGoogle Scholar
  15. 15.
    Veelaert, P., Teelen, K.: Fast polynomial segmentation of digitized curves. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 482–493. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Deboeverie, F., Veelaert, P., Teelen, K., Philips, W.: Face Recognition Using Parabola Edge Map. In: Blanc-Talon, J., Bourennane, S., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2008. LNCS, vol. 5259, pp. 994–1005. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Deboeverie, F., Teelen, K., Veelaert, P., Philips, W.: Vehicle tracking using geometric features. In: Blanc-Talon, J., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2009. LNCS, vol. 5807, pp. 506–515. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Canny, J.F.: A computational approach to edge detection. IEEE Trans. PAMI, 679–698 (1986)Google Scholar
  19. 19.
  20. 20.
    Yang, X., Koknar-Tezel, S., Latecki, L.J.: Locally Constrained Diffusion Process on Locally Densified Distance Spaces with Applications to Shape Retrieval. In: CVPR, pp. 357–364 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Francis Deboeverie
    • 1
    • 2
  • Kristof Teelen
    • 1
    • 2
  • Peter Veelaert
    • 1
    • 2
  • Wilfried Philips
    • 1
    • 2
  1. 1.Image Processing and Interpretation/IBBTGhent UniversityGhentBelgium
  2. 2.Engineering SciencesUniversity College GhentGhentBelgium

Personalised recommendations