A Novel Curvature Estimator for Digital Curves and Images

  • Oliver Fleischmann
  • Lennart Wietzke
  • Gerald Sommer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6376)


We propose a novel curvature estimation algorithm which is capable of estimating the curvature of digital curves and two-dimensional curved image structures. The algorithm is based on the conformal projection of the curve or image signal to the two-sphere. Due to the geometric structure of the embedded signal the curvature may be estimated in terms of first order partial derivatives in ℝ3. This structure allows us to obtain the curvature by just convolving the projected signal with the appropriate kernels. We show that the method performs an implicit plane fitting by convolving the projected signals with the derivative kernels. Since the algorithm is based on convolutions its implementation is straightforward for digital curves as well as images. We compare the proposed method with differential geometric curvature estimators. It turns out that the novel estimator is as accurate as the standard differential geometric methods in synthetic as well as real and noise perturbed environments.


Curvature Estimator Ground Truth Image Projected Signal Inverse Radon Conformal Method 
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.


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  1. 1.
    Hermann, S., Klette, R.: Global Curvature Estimation for Corner Detection. Technical report, The University of Auckland, New Zealand (2005)Google Scholar
  2. 2.
    Williams, D.J., Shah, M.: A Fast Algorithm for Active Contours and Curvature Estimation. CVGIP: Image Underst. 55(1), 14–26 (1992)zbMATHCrossRefGoogle Scholar
  3. 3.
    Morse, B., Schwartzwald, D.: Isophote-based Interpolation. In: International Conference on Image Processing, vol. 3, p. 227 (1998)Google Scholar
  4. 4.
    Lachaud, J.O., Vialard, A., de Vieilleville, F.: Fast, Accurate and Convergent Tangent Estimation on Digital Contours. Image Vision Comput. 25(10), 1572–1587 (2007)CrossRefGoogle Scholar
  5. 5.
    Coeurjolly, D., Miguet, S., Tougne, L.: Discrete Curvature Based on Osculating Circle Estimation. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 303–312. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Hermann, S., Klette, R.: Multigrid Analysis of Curvature Estimators. In: Proc. Image Vision Computing, New Zealand, pp. 108–112 (2003)Google Scholar
  7. 7.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  8. 8.
    Wietzke, L., Fleischmann, O., Sommer, G.: 2D Image Analysis by Generalized Hilbert Transforms in Conformal Space. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part II. LNCS, vol. 5303, pp. 638–649. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  10. 10.
    Needham, T.: Visual Complex Analysis. Oxford University Press, USA (1999)Google Scholar
  11. 11.
    Zayed, A.: Handbook of Function and Generalized Function Transformations. CRC, Boca Raton (1996)zbMATHGoogle Scholar
  12. 12.
    Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier Science Inc., New York (1996)zbMATHGoogle Scholar
  13. 13.
    Lindeberg, T.: Scale-space Theory in Computer Vision. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  14. 14.
    Felsberg, M., Sommer, G.: The Monogenic Scale-space: A Unifying Approach to Phase-based Image Processing in Scale-space. Journal of Mathematical Imaging and vision 21(1), 5–26 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gander, W., Golub, G.H., Strebel, R.: Least-Squares Fitting of Circles and Ellipses. BIT (4), 558–578 (1994)Google Scholar
  16. 16.
    Coope, I.D.: Circle Fitting by Linear and Nonlinear Least Squares. Journal of Optimization Theory and Applications 76(2), 381–388 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Romeny, B.M.: Geometry-Driven Diffusion in Computer Vision. Kluwer Academic Publishers, Norwell (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Oliver Fleischmann
    • 1
  • Lennart Wietzke
    • 2
  • Gerald Sommer
    • 1
  1. 1.Cognitive Systems Group, Department of Computer ScienceKiel UniversityGermany
  2. 2.Raytrix GmbHKielGermany

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