This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Curve Parametrization by Moments
January 2009 (vol. 31 no. 1)
pp. 15-26
Irina Popovici, United States Naval Academy, Annapolis
William Douglas Withers, United States Naval Academy, Annapolis
We present a method for deriving a parametric description of a conic section (quadratic curve) in an image from the moments of the image with respect to several specially-constructed kernel functions. In contrast to Hough-transform-type methods, the moment approach requires no large accumulator array. Judicious implementation allows the parameters to be determined using five multiplication operations and six addition operations per pixel. The use of moments renders the calculation robust in the presence of high-frequency noise or texture and resistant to small-scale irregularities in the edge. Our method is generalizable to more complex classes of curves with more parameters as well as to surfaces in higher dimensions.

[1] A.S. Aguado, M.E. Montiel, and M.S. Nixon, “Ellipse Detection via Gradient Direction in the Hough Transform,” Proc. Fifth Int'l Conf. Image Processing and Its Applications, pp. 375-378, July 1995.
[2] A.S. Aguado and M.S. Nixon, “A New Hough Transform Mapping for Ellipse Detection,” Research J., Dept. of Electronics and Computer Science, Univ. of Southampton, 1995.
[3] T.J. Atherton and D.J. Kerbyson, “Using Phase to Represent Radius in the Coherent Circle Hough Transform,” Proc. IEE Colloquium on the Hough Transform, pp. 5/1-5/4, May 1999.
[4] T.J. Atherton and D.J. Kerbyson, “Size-Invariant Circle Detection,” Image and Vision Computing, vol. 17, pp. 795-803, 1999.
[5] D.H. Ballard, “Generalizing the Hough Transform to Detect Arbitrary Shapes,” Pattern Recognition, vol. 13, no. 2, pp. 111-122, 1981.
[6] R. Chan and W.-C. Siu, “Fast Detection of Ellipses Using Chord Bisectors,” Proc. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing (ICASSP '90), vol. 4, pp. 2201-2204, Apr. 1990.
[7] E.R. Davies, “Finding Ellipses Using the Generalised Hough Transform,” Pattern Recognition Letters, vol. 9, pp. 87-96, 1989.
[8] D.L. Donoho, “Wedgelets: Nearly Minimax Estimations of Edges,” The Annals of Statistics, vol. 27, no. 3, pp. 859-897, 1999.
[9] R.O. Duda and P.E. Hart, “Use of the Hough Transform to Detect Lines and Curves in Pictures,” Comm. ACM, vol. 15, pp. 11-15, 1972.
[10] S. Ghosal and R. Mehrotra, “Orthogonal Moment Operators for Subpixel Edge Detection,” Pattern Recognition, vol. 26, no. 2, pp.295-306, 1993.
[11] S. Ghosal and R. Mehrotra, “A Moment-Based Unified Approach to Image Feature Detection,” IEEE Trans. Image Processing, vol. 6, no. 6, pp. 781-793, June 1997.
[12] A. Goneid, S. El-Gindi, and A. Sewisy, “A Method for the Hough Transform Detection of Circles and Ellipses Using a 1-Dimensional Array,” Proc. IEEE Int'l Conf. Systems, Man, and Cybernetics (SMC '97), vol. 4, pp. 3154-3157, Oct. 1997.
[13] G. Golub, B. Gustafsson, P. Milanfar, M. Putinar, and J. Varah, “Shape Reconstruction from Moments: Theory, Algorithms, and Applications,” Proc. SPIE, vol. 4116, pp. 406-416, Nov. 2000.
[14] B. Gustafsson, C. He, P. Milanfar, and M. Putinar, “Reconstructing Planar Domains from Their Moments,” Inverse Problems, vol. 16, no. 4, pp. 1053-1070, Aug. 2000.
[15] J. Heikkilä, “Moment and Curvature Preserving Technique for Accurate Ellipse Boundary Detection,” Proc. 14th Int'l Conf. Pattern Recognition (ICPR '98), vol. 1, pp. 734-737, Aug. 1998.
[16] J. Heikkilä and O. Silvén, “A Four-Step Camera Calibration Procedure with Implicit Image Correction,” Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition (CVPR '97), pp. 1106-1112, June 1997.
[17] C.J. Hinton and J.F. Boyce, “Circle, Ellipse, and Parabola Parametrisation Using Integral Transforms,” Proc. IEE Colloquium on Hough Transforms, pp. 6/1-6/4, May 1993.
[18] P.V.C. Hough, Method and Means for Recognizing Complex Patterns, US Patent 3,069,654, Washington, D.C.: Patent and Trademark Office, Dec. 1962.
[19] M.Z. Mat Jafri and F. Deravi, “Efficient Algorithm for Detection of Parabolic Curves,” Proc. SPIE, vol. 2356, pp. 53-61, Jan. 1995.
[20] K. Kanatani, “Statistical Bias of Conic Fitting and Renormalization,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 3, pp. 320-326, Mar. 1994.
[21] M.G. Krein and A.A. Nudelman, “The Markov Moment Problem and Extremal Problems,” Translations of Mathematical Monographs, vol. 50, Am. Math. Soc., 1977.
[22] Y. Lei and K.C. Wong, “Ellipse Detection Based on Symmetry,” Pattern Recognition Letters, vol. 20, pp. 41-47, 1999.
[23] A. Lisowksa, “Geometrical Wavelets and Their Generalizations in Digital Image Coding and Processing,” PhD dissertation, Faculty of Computer Science and Materials Science, Univ. of Silesia, 2005.
[24] E.P. Lyvers, O.R. Mitchell, M.L. Akey, and A.P. Reeves, “Subpixel Measurements Using a Moment-Based Edge Operator,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 12, pp.1293-1309, Dec. 1989.
[25] R.A. McLaughlin, “Randomized Hough Transform: Better Ellipse Detection,” Proc. IEEE Region 10 Conf. Digital Signal Processing Applications (TENCON '96), vol. 1, pp. 409-414, Nov. 1996.
[26] I. Popovici and W.D. Withers, “Custom-Built Moments for Edge Location,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 4, pp. 637-642, Apr. 2006.
[27] I. Popovici and W.D. Withers, “Locating Thin Lines and Roof Edges by Custom-Built Moments,” Proc. IEEE Int'l Conf. Image Processing (ICIP '06), pp. 753-756, 2006.
[28] I. Popovici and W.D. Withers, “Locating Edges and Removing Ringing Artifacts in JPEG Images by Frequency-Domain Analysis,” IEEE Trans. Image Processing, vol. 16, no. 5, pp. 1470-1474, May 2007.
[29] M. Putinar, “Linear Analysis of Quadrature Domains,” Arkiv för Matematik, vol. 33, no. 2, pp. 357-376, Oct. 1995.
[30] A.P. Reeves, M.L. Akey, and O.R. Mitchell, “A Moment Based Two-Dimensional Edge Operator,” Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition (CVPR '83), pp. 312-317, 1983.
[31] J.K. Romberg, M.B. Wakin, and R.G. Baraniuk, “Multiscale Geometric Image Processing,” Proc. SPIE Visual Comm. Image Processing, vol. 5150, pp. 1265-1272, 2003.
[32] M. Sakai, “Quadrature Domains,” Springer-Verlag Lecture Notes in Math, vol. 934, 1982.
[33] R. Safaee-Rad, I. Tchoukanov, B. Benhabib, and K.C. Smith, “Accurate Parameter Estimation of Quadratic Curves from Grey-Level Images,” CVGIP: Image Understanding, vol. 54, no. 2, pp. 259-274, Sept. 1991.
[34] CRC Handbook of Tables for Mathematics, R.C. Weast and S.M. Selby, eds., fourth ed. CRC Press, 1975.
[35] Y.-H. Xie and Q. Ji, “A New Efficient Ellipse Detection Method,” Proc. 16th Int'l Conf. Pattern Recognition (ICPR '02), vol. 2, pp. 957-960, 2002.
[36] L. Xu, E. Oja, and P. Kultanen, “A New Curve Detection Method: Randomized Hough Transform (RHT),” Pattern Recognition Letters, vol. 11, no. 5, pp. 331-338, May 1990.
[37] J.H. Yoo and I.K. Sethi, “Ellipse Detection Method from the Polar and Pole Definition of Conics,” Pattern Recognition, vol. 26, no. 2, pp. 307-315, 1993.

Index Terms:
Edge and feature detection, Moments
Citation:
Irina Popovici, William Douglas Withers, "Curve Parametrization by Moments," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 1, pp. 15-26, Jan. 2009, doi:10.1109/TPAMI.2008.54
Usage of this product signifies your acceptance of the Terms of Use.