Journal of Mathematical Imaging and Vision

, Volume 54, Issue 3, pp 287–300 | Cite as

A Fisher–Rao Metric for Curves Using the Information in Edges

  • Stephen J. Maybank


Two curves which are close together in an image are indistinguishable given a measurement, in that there is no compelling reason to associate the measurement with one curve rather than the other. This observation is made quantitative using the parametric version of the Fisher–Rao metric. A probability density function for a measurement conditional on a curve is constructed. The distance between two curves is then defined to be the Fisher–Rao distance between the two conditional pdfs. A tractable approximation to the Fisher–Rao metric is obtained for the case in which the measurements are compound in that they consist of a point \({\mathbf {x}}\) and an angle \(\alpha \) which specifies the direction of an edge at \({\mathbf {x}}\). If the curves are circles or straight lines, then the approximating metric is generalized to take account of inlying and outlying measurements. An estimate is made of the number of measurements required for the accurate location of a circle in the presence of outliers. A Bayesian algorithm for circle detection is defined. The prior density for the algorithm is obtained from the Fisher–Rao metric. The algorithm is tested on images from the CASIA iris interval database.


Bayesian curve detection CASIA iris database Circle detection Hough transform Riemannian metric Step edges 


  1. 1.
    Aguado, A.S., Montiel, M.E., Nixon, M.S.: On using directional information for parameter space decomposition in ellipse detection. Pattern Recognit. 29, 369–381 (1996)CrossRefGoogle Scholar
  2. 2.
    Amari, S.-I.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balasubramanian, V.: Statistical inference, Occam’s razor and statistical mechanics on the space of probability distributions. Neural Comput. 9, 349–368 (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ballard, D.H.: Generalizing the Hough transform to detect arbitrary shapes. Pattern Recognit. 13, 111–122 (1981)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bonci, A., Leo, T., Longhi, S.: A Bayesian approach to the Hough transform for line detection. IEEE Trans. Syst. Man Cybern. Part A 35, 945–955 (2005)CrossRefGoogle Scholar
  6. 6.
    CASIA Iris Image Database: (2010)
  7. 7.
    Ceolin, S., Hancock, E.R.: Distinguishing facial expression using the Fisher-Rao metric. In: Proceedings of IEEE Conference on Image Processing (ICIP), pp. 1437–1440 (2010)Google Scholar
  8. 8.
    Ceolin, S.R., Hancock, E.R.: Computing gender difference using Fisher-Rao metric from facial surface normals. In: Proceedings of Conference on Graphics, Patterns and Images (SIBGRAPI), pp. 336–343 (2012)Google Scholar
  9. 9.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)CrossRefzbMATHGoogle Scholar
  10. 10.
    Forsyth, D.A., Ponce, J.: Computer Vision: A Modern Approach, 2nd edn. Prentice Hall, Upper Saddle River (2011)Google Scholar
  11. 11.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Pearson Education, Singapore (2008)Google Scholar
  12. 12.
    Jaynes, E.T.: Probability Theory: The Logic of Science. CUP, New York (2003)CrossRefGoogle Scholar
  13. 13.
    Jeffreys, H.: Theory of Probability. Oxford Classics Series. OUP, Oxford (1998)zbMATHGoogle Scholar
  14. 14.
    Ji, Q., Haralick, R.M.: An optimal Bayesian Hough transform for line detection. In: Proceedings of the 1999 International Conference on Image Processing, vol. 2, pp. 691–695 (1999)Google Scholar
  15. 15.
    Kanatani, K.-I.: Statistical Computation for Geometrical Optimization. Elsevier, New York (1996)Google Scholar
  16. 16.
    Kimme, C., Ballard, D., Sklansky, J.: Finding circles by an array of accumulators. Commun. Assoc. Comput. Mach. 18, 120–122 (1975)zbMATHGoogle Scholar
  17. 17.
    Leavers, V.F.: Which Hough transform? Comput. Vis. Graph. Image Process. 58, 250–264 (1993)CrossRefGoogle Scholar
  18. 18.
    Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, New York (2000)zbMATHGoogle Scholar
  19. 19.
    Maybank, S.J.: Detection of image structures using the Fisher information and the Rao metric. IEEE Trans. Pattern Anal. Mach. Intell. 26, 1579–1589 (2004)CrossRefGoogle Scholar
  20. 20.
    Maybank, S.J.: Application of the Fisher-Rao metric to ellipse detection. Int. J. Comput. Vis. 72, 287–307 (2007)CrossRefGoogle Scholar
  21. 21.
    Maybank, S.J., Ieng, S., Benosman, R.: A Fisher-Rao metric for paracatadioptric images of lines. Int. J. Comput. Vis. 99, 147–165 (2012). doi: 10.1007/s11263-012-0523-x CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Michor, P.W., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23, 74–113 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Mio, W., Srivastava, S., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73, 307–324 (2006)CrossRefGoogle Scholar
  24. 24.
    Olson, C.F.: Constrained Hough transform for curve detection. Comput. Vis. Image Underst. 73, 329–345 (1999)CrossRefzbMATHGoogle Scholar
  25. 25.
  26. 26.
    Peter, A., Rangarajan, A.: Information geometry for landmark shape analysis: unifying shape representation and deformation. IEEE Trans. Pattern Anal. Mach. Intell. 31, 337–350 (2009)CrossRefGoogle Scholar
  27. 27.
    Rao, C.: Information and the accuracy obtainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Srivastava, A., Jermyn, I., Joshi, S.: Riemannian analysis of probability density functions with applications in vision. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR2007, pp. 1–8 (2007)Google Scholar
  29. 29.
    Srivastava, A., Klassen, E., Joshi, S.H., Jermyn, I.: Shape analysis of elastic curves in Euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33, 1415–1428 (2011)CrossRefGoogle Scholar
  30. 30.
    Sundaramoorthi, G., Mennucci, A.C.G., Soatto, S., Yezzi, A.: A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM J. Imaging Sci. 4, 109–145 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Szeliski, R.: Computer Vision: Algorithms and Applications. Springer, London (2011)CrossRefGoogle Scholar
  32. 32.
    Tatu, A., Lauze, F., Sommer, S., Nielsen, M.: On restricting planar curve evolution to finite dimensional implicit subspaces with non-euclidean metric. J. Math. Imaging Vis. 38, 226–240 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Toronto, N., Morse, B.S., Ventura, D., Seppi, K.: (2007) The Hough transform’s explicit Bayesian foundation. In: Proceedings of the 14th International Conference on Image Processing, IV, pp. 377–380 (2007)Google Scholar
  34. 34.
    Werman, M., Keren, D.: A Bayesian method for fitting parametric and nonparametric models to noisy data. IEEE Trans. Pattern Anal. Mach. Intell. 23, 528–534 (2001)CrossRefGoogle Scholar
  35. 35.
    Woodford, O.J., Pham, M.-T., Maki, A., Porbet, F., Stenger, B.: Demisting the Hough transform for 3D shape recognition and registration. Int. J. Comput. Vis. 106, 332–341 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computer Science and Information SystemsBirkbeck CollegeLondonUK

Personalised recommendations