Abstract
When faced with an ellipse fitting problem, practitioners frequently resort to algebraic ellipse fitting methods due to their simplicity and efficiency. Currently, practitioners must choose between algebraic methods that guarantee an ellipse fit but exhibit high bias, and geometric methods that are less biased but may no longer guarantee an ellipse solution. We address this limitation by proposing a method that strikes a balance between these two objectives. Specifically, we propose a fast stable algorithm for fitting a guaranteed ellipse to data using the Sampson distance as a data-parameter discrepancy measure. We validate the stability, accuracy, and efficiency of our method on both real and synthetic data. Experimental results show that our algorithm is a fast and accurate approximation of the computationally more expensive orthogonal-distance-based ellipse fitting method. In view of these qualities, our method may be of interest to practitioners who require accurate and guaranteed ellipse estimates.
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Ahn, S.J., Rauh, W., Warnecke, H.: Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola. Pattern Recognition 34(12), 2283–2303 (2001)
Al-Sharadqah, A., Chernov, N.: A doubly optimal ellipse fit. Comput. Statist. Data Anal. 56(9), 2771–2781 (2012)
Bai, X., Sun, C., Zhou, F.: Splitting touching cells based on concave points and ellipse fitting. Pattern Recognition 42(11), 2434–2446 (2009)
Brooks, M.J., Chojnacki, W., Gawley, D., van den Hengel, A.: What value covariance information in estimating vision parameters? In: Proc. Eighth Int. Conf. Computer Vision, vol. 1, pp. 302–308 (2001)
Chernov, N.: On the convergence of fitting algorithms in computer vision. J. Math. Imaging and Vision 27(3), 231–239 (2007)
Chojnacki, W., Brooks, M.J.: Revisiting Hartley’s normalized eight-point algorithm. IEEE Trans. Pattern Anal. Mach. Intell. 25(9), 1172–1177 (2003)
Chojnacki, W., Brooks, M.J., van den Hengel, A., Gawley, D.: On the fitting of surfaces to data with covariances. IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1294–1303 (2000)
Ding, L., Martinez, A.M.: Features versus context: An approach for precise and detailed detection and delineation of faces and facial features. IEEE Trans. Pattern Anal. Mach. Intell. 32(11), 2022–2038 (2010)
Duan, F., Wang, L., Guo, P.: RANSAC Based Ellipse Detection with Application to Catadioptric Camera Calibration. In: Wong, K.W., Mendis, B.S.U., Bouzerdoum, A. (eds.) ICONIP 2010, Part II. LNCS, vol. 6444, pp. 525–532. Springer, Heidelberg (2010)
Fitzgibbon, A., Pilu, M., Fisher, R.B.: Direct least square fitting of ellipses. IEEE Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999)
Gander, W., Golub, G.H., Strebel, R.: Least-squares fitting of circles and ellipses. BIT 34(4), 558–578 (1994)
Halíř, R., Flusser, J.: Numerically stable direct least squares fitting of ellipses. In: Proc. Sixth Int. Conf. in Central Europe on Computer Graphics and Visualization, vol. 1, pp. 125–132 (1998)
Hu, M., Worrall, S., Sadka, A.H., Kondoz, A.A.: A fast and efficient chin detection method for 2D scalable face model design. In: Proc. Int. Conf. Visual Information Engineering, pp. 121–124 (2003)
Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier, Amsterdam (1996)
Kanatani, K., Sugaya, Y.: Compact algorithm for strictly ML ellipse fitting. In: Proc. 19th Int. Conf. Pattern Recognition, pp. 1–4 (2008)
Kanatani, K., Rangarajan, P.: Hyper least squares fitting of circles and ellipses. Comput. Statist. Data Anal. 55(6), 2197–2208 (2011)
Kanatani, K., Sugaya, Y.: Performance evaluation of iterative geometric fitting algorithms. Comput. Statist. Data Anal. 52(2), 1208–1222 (2007)
Kim, I.: Orthogonal distance fitting of ellipses. Commun. Korean Math. Soc. 17(1), 121–142 (2002)
Lee, K., Cham, W., Chen, Q.: Chin contour estimation using modified Canny edge detector. In: Proc. 7th Int. Conf. Control, Automation, Robotics and Vision, vol. 2, pp. 770–775 (2002)
Mulchrone, K.F., Choudhury, K.R.: Fitting an ellipse to an arbitrary shape: implications for strain analysis. J. Structural Geology 26(1), 143–153 (2004)
O’Leary, P., Zsombor-Murray, P.: Direct and specific least-square fitting of hyperbolæ and ellipses. J. Electronic Imaging 13(3), 492–503 (2004)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. Cambridge University Press, Cambridge (1995)
Rosin, P.L.: A note on the least squares fitting of ellipses. Pattern Recognition Lett. 14(10), 799–808 (1993)
Sampson, P.D.: Fitting conic sections to ‘very scattered’ data: An iterative refinement of the Bookstein algorithm. Computer Graphics and Image Processing 18(1), 97–108 (1982)
Sarzi, M., Rix, H., Shields, J.C., Rudnick, G., Ho, L.C., McIntosh, D.H., Filippenko, A.V., Sargent, W.L.W.: Supermassive black holes in bulges. The Astrophysical Journal 550(1), 65–74 (2001)
Sturm, P., Gargallo, P.: Conic Fitting Using the Geometric Distance. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds.) ACCV 2007, Part II. LNCS, vol. 4844, pp. 784–795. Springer, Heidelberg (2007)
Yu, D., Pham, T.D., Zhou, X.: Analysis and recognition of touching cell images based on morphological structures. Computers in Biology and Medicine 39(1), 27–39 (2009)
Yu, J., Kulkarni, S.R., Poor, H.V.: Robust ellipse and spheroid fitting. Pattern Recognition Lett. 33(5), 492–499 (2012)
Zhang, G., Jayas, D.S., White, N.D.: Separation of touching grain kernels in an image by ellipse fitting algorithm. Biosystems Engineering 92(2), 135–142 (2005)
Zhang, Z.: Parameter estimation techniques: a tutorial with application to conic fitting. Image and Vision Computing 15(1), 59–76 (1997)
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Szpak, Z.L., Chojnacki, W., van den Hengel, A. (2012). Guaranteed Ellipse Fitting with the Sampson Distance. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds) Computer Vision – ECCV 2012. ECCV 2012. Lecture Notes in Computer Science, vol 7576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33715-4_7
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DOI: https://doi.org/10.1007/978-3-642-33715-4_7
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