The parameter space for the ellipses in a two dimensional image is a five dimensional manifold, where each point of the manifold corresponds to an ellipse in the image. The parameter space becomes a Riemannian manifold under a Fisher-Rao metric, which is derived from a Gaussian model for the blurring of ellipses in the image. Two points in the parameter space are close together under the Fisher-Rao metric if the corresponding ellipses are close together in the image. The Fisher-Rao metric is accurately approximated by a simpler metric under the assumption that the blurring is small compared with the sizes of the ellipses under consideration. It is shown that the parameter space for the ellipses in the image has a finite volume under the approximation to the Fisher-Rao metric. As a consequence the parameter space can be replaced, for the purpose of ellipse detection, by a finite set of points sampled from it. An efficient algorithm for sampling the parameter space is described. The algorithm uses the fact that the approximating metric is flat, and therefore locally Euclidean, on each three dimensional family of ellipses with a fixed orientation and a fixed eccentricity. Once the sample points have been obtained, ellipses are detected in a given image by checking each sample point in turn to see if the corresponding ellipse is supported by the nearby image pixel values. The resulting algorithm for ellipse detection is implemented. A multiresolution version of the algorithm is also implemented. The experimental results suggest that ellipses can be reliably detected in a given low resolution image and that the number of false detections can be reduced using the multiresolution algorithm.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Abramowitz, M. and Stegun, I.A. (eds.) 1965. Handbook of Mathematical Functions with formulas, graphs, and mathematical tables. Dover.
Agrawal, M. and Davis, L.S. 2003. Camera calibration using spheres: a semi-definite programming approach. In Proc. 9th International Conference on Computer Vision, Nice, vol. 2, pp. 782–789.
Aguado, A.S., Montiel, M.E. and Nixon, M.S. 1995. Ellipse extraction via gradient direction in the Hough transform. In Proc. 5th Int. Conf. on Image Processing and its Applications, Edinburgh, UK, pp. 375–378, IEE.
Amari, S.-I. 1985. Differential-Geometrical Methods in Statistics. Lecture Notes in Computer Science, 28. Springer.
Arias-Castro, E., Donoho, D. and Huo, X. 2005. Near-optimal detection of geometric objects by fast multiscale methods. IEEE Transactions on Information Theory, 51(7): 2402–2425.
Chavel, I. 1996. Riemannian Geometry: a modern introduction. Cambridge Tracts in Mathematics, vol. 108, CUP.
Conway, J.H. and Sloane, N.J.A. 1999. Sphere Packings, Lattices and Groups. Grundlehren der mathematischen Wissenschaften, vol. 290. Springer.
Cover, T.M. and Thomas, J.A. 1991. Elements of Information Theory. John Wiley and Sons.
Desolneux, A., Moisan, L. and Morel, J.-M. 2000. Meaningful alignments. International Journal of Computer Vision, 40(1): 7–23.
Desolneux, A., Moisan, L. and Morel, J.-M. 2003. Maximal meaningful events and applications to image analysis. The Annals of Statistics, 31(6):1822–1851.
Fitzgibbon, A., Pilu, M., and Fisher, R.B. 1999. Direct least squares fitting of ellipses. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(5): 476–480.
Gallot, S., Hulin, D. and LaFontaine, J. 1990. Riemannian Geometry. 2nd edition, Universitext, Springer.
Hartley, R. and Zisserman, A. 2003. Multiple View Geometry in Computer Vision. CUP.
van Hateren, J.H. and van der Schaaf, A. 1998. Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings of the Royal Society of London, Series B, 265:359–366.
Ji, Q. and Haralick, R.M. 1999. A statistically efficient method for ellipse detection. IEEE International Conference on Image Processing, ICIP99, vol. 2, pp. 730–734.
Kanatani, K. 1994. Statistical bias of conic fitting and renormalisation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(3): 320–326.
Kanatani, K. 1996. Statistical Computation for Geometric Optimization. Elsevier.
Kanatani, K. and Ohta, N. 2003. Automatic detection of circular objects by ellipse growing. International Journal of Image and Graphics, 4(1): 35–50.
Kwolek, B. 2004. Stereo-vision based head tracking using color and ellipse fitting in a particle filter. In Pajdla, T. and Matas, J. (eds.) 2004. Computer Vision - ECCV 2004, vol. 4, pp. 192–204. Lecture Notes in Computer Science, vol. 3024, Springer.
Lazebnik, S., Schmid, C. and Ponce, J. 2004. Semi-local affine parts for object recognition. In Hoppe, A., Barman, S. and Ellis, T. (eds.) British Machine Vision Conference 2004, vol. 2, pp. 959–968, BMVA.
Leavers, V.F. 1992. Shape Detection in Computer Vision Using the Hough Transform. Springer Verlag.
Leedan, Y. and Meer, P. Estimation with bilinear constraints in computer vision. Sixth International Conference on Computer Vision, ICCV’98, Bombay, India, pp. 733–738.
Lutton, E. and Martinez, P. 1994. A genetic algorithm for the detection of 2D geometric primitives in images. Proc. 12th International Conference on Pattern Recognition, vol. 1, pp. 526–529.
Maio, D. and Maltoni, D. 1998. Fast face location in complex backgrounds. In Wechsler, H., Phillips, P.J., Bruce, V., Soulié, F. and Huang, T.S. (eds.) Face Recognition from Theory to Applications, NATO LSI Series, pp. 568–577, Springer-Verlag.
Maybank, S.J. 2003. Fisher information and model selection for projective transformations of the line. Proceedings of the Royal Society of London, Series A, 459: 1–21.
Maybank, S.J. 2004. Detection of image structures using Fisher information and the Rao metric. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(12): 1579–1589.
Maybank, S.J. 2005. The Fisher-Rao metric for projective transformations of the line. International Journal of Computer Vision, 63(3): 191–206.
Maybank, S.J. 2006. Application of the Fisher-Rao metric to structure detection. Journal of Mathematical Imaging and Vision, 25(1).
Rosin, P.L. 1996. Assessing error of fit functions for ellipses. Graphical Models and Image Processing, 58(5): 494–502.
Rucklidge, W.J. 1997. Efficiently locating objects using the Hausdorff distance. International Journal of Computer Vision, 24(3): 251–270.
Scaggiante, A., Frezza, R. and Zampato, M. 1999. Identifying and tracking ellipses: a technique based on elliptical deformable templates. In Proc. 10th International Conference on Image Analysis and Processing, pp. 582–587.
Taubin, G. 1991. Estimation of planar curves, surfaces, and non planar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(11):1115–1138.
Torr, P.H.S. and Fitzgibbon, A.W. 2004. Invariant fitting of two view geometry. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(5): 648–650.
Wang, J.G., Sung, E. and Venkateswarlu, R. 2003. Eye gaze estimation from a single image of one eye. In Proc. 9th International Conference on Computer Vision, Nice, vol. 1, pp. 136–143.
Wolfram, S. The Mathematica Book. 5th Edition, Wolfram Media, 2003.
Xie, Y. and Ji, Q. 2002. A new efficient ellipse detection method. Proc. International Conference on Pattern Recognition, vol. 2, pp. 957–960.
Xu, L. and Oja, E. 1993. Randomized Hough transform (RHT): basic mechanisms, algorithms and computational complexities. Computer Vision, Graphics, and Image Processing: Image Understanding, 57(2): 131–154.
Yao, J., Kharma, N. and Grogono, P. 2004. Fast, robust GA-based ellipse detection. In Proc. 17th International Conference on Pattern Recognition, ICPR 2004, vol. 2, pp. 859–862.
Younes, L. 1998. Computable elastic distances between shapes. SIAM Journal on Applied Mathematics, 58(2): 565–586.
Zhang, Z. 1997. Parameter estimation techniques: a tutorial with application to conic fitting. Image and Vision Computing Journal, 15(1): 59–76.
About this article
Cite this article
Maybank, S.J. Application of the Fisher-Rao Metric to Ellipse Detection. Int J Comput Vision 72, 287–307 (2007). https://doi.org/10.1007/s11263-006-9033-z
- ellipse detection
- Fisher-Rao metric
- flat metric
- Hough transform
- Kullback-Leibler distance
- Riemannian manifold
- volume of a Riemannian manifold
- Voronoi’s principal lattice