Abstract
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.
Similar content being viewed by others
References
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)
Amari, S.-I.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Springer, New York (1985)
Balasubramanian, V.: Statistical inference, Occam’s razor and statistical mechanics on the space of probability distributions. Neural Comput. 9, 349–368 (1997)
Ballard, D.H.: Generalizing the Hough transform to detect arbitrary shapes. Pattern Recognit. 13, 111–122 (1981)
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)
CASIA Iris Image Database: http://biometrics.idealtest.org (2010)
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)
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)
Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)
Forsyth, D.A., Ponce, J.: Computer Vision: A Modern Approach, 2nd edn. Prentice Hall, Upper Saddle River (2011)
Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Pearson Education, Singapore (2008)
Jaynes, E.T.: Probability Theory: The Logic of Science. CUP, New York (2003)
Jeffreys, H.: Theory of Probability. Oxford Classics Series. OUP, Oxford (1998)
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)
Kanatani, K.-I.: Statistical Computation for Geometrical Optimization. Elsevier, New York (1996)
Kimme, C., Ballard, D., Sklansky, J.: Finding circles by an array of accumulators. Commun. Assoc. Comput. Mach. 18, 120–122 (1975)
Leavers, V.F.: Which Hough transform? Comput. Vis. Graph. Image Process. 58, 250–264 (1993)
Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, New York (2000)
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)
Maybank, S.J.: Application of the Fisher-Rao metric to ellipse detection. Int. J. Comput. Vis. 72, 287–307 (2007)
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
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)
Mio, W., Srivastava, S., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73, 307–324 (2006)
Olson, C.F.: Constrained Hough transform for curve detection. Comput. Vis. Image Underst. 73, 329–345 (1999)
OpenCV: http://www.docs.opencv.org/modules/imgproc/doc/feature_detection.html#houghcircles (2014). Accessed 20 Feb 2014
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)
Rao, C.: Information and the accuracy obtainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)
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)
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)
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)
Szeliski, R.: Computer Vision: Algorithms and Applications. Springer, London (2011)
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)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maybank, S.J. A Fisher–Rao Metric for Curves Using the Information in Edges. J Math Imaging Vis 54, 287–300 (2016). https://doi.org/10.1007/s10851-015-0603-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-015-0603-y