This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Nonparametric Segmentation of Curves into Various Representations
December 1995 (vol. 17 no. 12)
pp. 1140-1153

Abstract—This paper describes and demonstrates the operation and performance of an algorithm for segmenting connected points into a combination of representations such as lines, circular, elliptical and superelliptical arcs, and polynomials. The algorithm has a number of interesting properties including being scale invariant, nonparametric, general purpose, and efficient.

[1] G.J. Agin,“Fitting ellipses and general second-order curves,” Technical Report CMU-RI-TR-81-5, Robotics Inst., Carnegie Mellon Univ., Pittsburgh, 1981.
[2] A. Albano,“Representation of digitised contours in terms of conic arcs andstraight line segments,” Computer Graphics and Image Processing, vol. 5, pp. 23-33, 1974.
[3] N. Ansari and K.-W. Huang,“Non-parametric dominant point detection,” Pattern Recognition, vol. 24, pp. 849-862, 1991.
[4] H. Aoyama and M. Kawagoe,“A piecewise linear approximation methodpreserving visual feature points of original figures,” CVGIP: Graphical Models and Image Processing, vol. 53, no. 5, pp. 435-446, 1991.
[5] H. Asada and M. Brady, “The Curvature Primal Sketch,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 2-14, 1986.
[6] R. Bellman,“On the approximation of curves by line segments using dynamicprogramming,” Comm. ACM, vol. 4, no. 6, p. 284, 1961.
[7] H. Bendtsen and I. Wright,“Fitting ellipses to noisy spatial data,” technical report, Dept. of Mathematics and Statistics, Curtin Univ., 1992.
[8] F.L. Bookstein,“Fitting conic sections to scattered data,” Computer Vision, Graphics, and Image Processing, vol. 9, pp. 56-71, 1979.
[9] D. Cooper and N. Yalabik,“On the cost of approximating and recognizing anoise-perturbed straight line or quadratic arc segment in the plane,” Technical Report NASA NSG5036/1, Div. Eng., Brown Univ., Providence, R.I., 1975.
[10] D. Cooper and N. Yalabik,“On the computational cost of approximating andrecognizing noise-perturbed straight lines and quadratic arcs in theplane,” IEEE Trans. Computers, vol. 25, no. 10, pp. 1,020-1,032, 1976.
[11] J.G. Dunham,“Optimum uniform piecewise linear approximation of planar curves,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 1, pp. 67-75, 1986.
[12] T. Ellis, “Ellipse Detection and Matching with Uncertainty,” Image and Vision Computing, vol. 10, no. 2, pp. 271-276, 1992.
[13] M.A. Fischler and R.C. Bolles, “Perceptual Organization and Curve Partitioning,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 100-105, 1986.
[14] M.A. Fischler and H.C. Wolf,“Locating perceptually salient points on planar curves,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 2, pp. 113-129, 1994.
[15] D. Forsyth,J.L. Mundy,A. Zisserman,, and C.M. Brown,“Projectively invariant representations using implicit algebraic curves,” First European Conf. Computer Vision, pp. 427-436, Springer-Verlag, 1990.
[16] W. Gander,G.H. Golub,, and R. Strebel,“Fitting of circles and ellipses least squares solution,” Technical Report TR-217, Institut für Wissenschaftliches Rechen, ETH, Zurich, 1994.
[17] A.D. Gross and T.E. Boult,“Error of fit for recovering parametric solids,” Second Int’l Conf. Computer Vision, pp. 690-694,Tampa, Fla., 1988.
[18] A. Gueziec and N. Ayache,“Smoothing and matching of 3D space curves,” Second European Conf. Computer Vision, pp. 620-629,Santa Margherita Ligure, Italy, Springer-Verlag, 1992.
[19] A. Gupta and R. Bajcsy,“Surface and volumetric segmentation of rangeimages using biquadrics and superquadrics,” Int’l Conf. Pattern Recognition, vol. 1, pp. 158-162, The Hague, 1992.
[20] R.M. Haralick and L.G. Shapiro, Computer and Robot Vision. New York: Addison-Wesley, 1993.
[21] D. Hoffman and W. Richards,“Parts of recognition,” Cognition, vol. 18, pp. 65-96, 1984.
[22] C.L. Huang,“Elliptical feature extraction via an improved Hough transform,” Pattern Recognition Letters, vol. 10, pp. 93-100, 1989.
[23] 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.
[24] N. Kehtarnavaz and R.J.P. de Figueiredo,“A 3D contour segmentation scheme based on curvature and torsion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, pp. 707-713, 1988.
[25] D. Keren,D. Cooper,, and J. Subrahmonia,“Describing complicated objects by implicit polynomials,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 1, pp. 38-53, 1994.
[26] A. Leonardis and R. Bajcsy,”, Finding parametric curves in an image,” Second European Conf. Computer Vision, pp. 653-657,Santa Margherita Ligure, Italy, Springer-Verlag, 1992.
[27] Y-Z. Liao,“A two-stage method of fitting conic arcs and straight linesegments to digitised contours,” Conf. Pattern Recognition and Image Processing, pp. 224-229, 1981.
[28] D.G. Lowe, “Three-Dimensional Object Recognition from Single Two-Dimensional Images,” Artificial Intelligence, vol. 31, pp. 355-395, 1987.
[29] D.G. Lowe,“Organization of smooth image curves at multiple scales,” Int’l J. Computer Vision, vol. 3, pp. 119-130, 1989.
[30] G. Medioni and Y. Yasumoto,“Corner detection and curve representationusing cubic B-splines,” Computer Vision, Graphics, and Image Processing, pp. 267-278, 1987.
[31] F. Mokhatarian,“Multi-scale description of scale curves and three-dimensional objects,” Proc. Computer Vision and Pattern Recognition, pp. 298-303, 1988.
[32] Y. Nakagawa and A. Rosenfeld,“A note on polygonal and ellipticalapproximation of mechanical parts,” Pattern Recognition, vol. 11, pp. 133-142, 1979.
[33] A.M. Paterson,G. Dowling,, and D.A. Chamberlain,“Identifying key features in a building using a single uncalibrated camera,” Proc. 11th Int’l. Symp.Automation and Robotics in Construction, pp. 657-664,Brighton, UK, 1994.
[34] T. Pavlidis,“Curve fitting with conic splines,” ACM Trans. Graphics, vol. 2, no. 1, pp. 1-31, 1983.
[35] A. Pentland,“Perceptual organization and the representation of natural form,” Artificial Intelligence, vol. 28, pp. 293-331, 1986.
[36] A.P. Pentland,“Automatic extraction of deformable part models,” Int’l J. Computer Vision, vol. 4, pp. 107-126, 1990.
[37] S. Pollard and J. Porrill,“Robust recovery of 3D ellipse data,” British Machine Vision Conf., pp. 39-48,Leeds, UK, Springer-Verlag, 1992.
[38] J. Porill, “Fitting Ellipses and Predicting Confidence Envelopes Using a Bias Corrected Kalman Filter,” Image and Vision Computing, vol. 8, no. 1, pp. 37-41, 1990.
[39] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C.Cambridge, England: Cambridge Univ. Press, 1988.
[40] A.P. Pridmore,J. Porrill,, and J.E.W. Mayhew,“Segmentation and description of binocularly viewed contours,” Image and Vision Computing, vol. 5, pp. 132-138, 1987.
[41] A. Rattarangsi and R.T. Chin,“Scale-based detection of corners of planar curves,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 4, pp. 430-449, 1992.
[42] J. Rissanen, Stochastic Complexity in Statistical Inquiry. World Scientific Series in Computer Science, vol. 15, 1989.
[43] A. Rosenfeld and E. Johnson,“Angle detection on digital curves,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 2, no. 2, pp. 875-878, 1973.
[44] P.L. Rosin,“Ellipse fitting by accumulating five-point fits,” Pattern Recognition Letters, vol. 14, pp. 661-669, 1993.
[45] P.L. Rosin, “A Note on the Least Square Fitting of Ellipses,” Pattern Recognition Letters, vol. 14, no. 10, pp. 799-808, 1993.
[46] P.L. Rosin,“Non-parametric multi-scale curve smoothing,” Int’l J. Pattern Recognition and Artificial Intellegence, vol. 8, no. 6, 1994.
[47] P.L. Rosin and G.A.W. West, “Segmentation of Edges into Lines and Arcs,” Image Vision Computing, vol. 7, pp. 109-114, 1989.
[48] P.L. Rosin and G.A.W. West,“Segmenting curves into elliptic arcs and straight lines,” Third Int’l Conf. Computer Vision, pp. 75-78,Osaka, Japan, 1990.
[49] P.L. Rosin and G.A.W. West,“Curve segmentation and representation by superellipses,” Australia and New Zealand Conf. Intelligent InformationSystems, pp. 530-534,Perth, Australia, Dec. 1993.
[50] G. Roth and M.D. Levine,“Extracting geometric primitives,” CVGIP: Image Understanding, vol. 58, no. 1, pp. 1-22, 1993.
[51] P. Rousseeuw and A. Leory, Robust Regression and Outlier Detection. Wiley Series in Probability and Statistics, 1987.
[52] R. Safee-Rad,I. Tchoukanov,B. Benhabib,, and K.C. Smith,“Accurate parameter estimation of quadratic curves from grey levelimages,” CVGIP: Image Understanding, vol. 54, pp. 259-274, 1991.
[53] P.D. Sampson,“Fitting conic sections to very scattered data: An iterativerefinement to the Bookstein algorithm,” Computer Vision, Graphics, andImage Processing, vol. 18, pp. 97-108, 1982.
[54] D. Sarkar,“A simple algorithm for detection of significant vertices forpolygonal approximation of chain-coded curves,” Pattern Recognition Letters, vol. 14, pp. 959-964, 1993.
[55] Y. Shirai,“Recognition of real-world objects using edge cues,” Computer Vision Systems, A.R Hanson and E.M Riseman, eds., pp. 353-362.New York: Academic Press, 1978.
[56] F. Solina and R. Bajcsy,“Recovery of parametric models from range images: The case for superquadrics with global deformations,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, pp. 131-147, Feb. 1990.
[57] G. Taubin,“Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1137, Nov. 1991.
[58] C.H. Teh and R.T. Chin, “On the Detection of Dominant Points on Digital Curves,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 859-872, 1989.
[59] S.M. Thomas and Y.T. Chan,“A simple approach for the estimation of circular arc center and its radius,” Computer Vision, Graphics, and Image Processing, vol. 45, 1989.
[60] S. Tsuji and F. Matsumoto,“Detection of ellipses by a modified Hough transform,” IEEE Trans. Computers, vol. 27, pp. 777-781, 1979.
[61] R. Wang,A.R. Hanson,, and E.M. Riseman,“Fast extraction of ellipses,” Ninth Int’l. Conf. Pattern Recognition, pp. 508-510, 1988.
[62] G.A.W. West and P.L. Rosin, “Techniques for Segment Image Curves into Meaningful Descriptions,” Pattern Recognition, vol. 24, pp. 643-652, 1991.
[63] J. Wu and T. Caelli,“Model-based 3D object localisation and recognitionfrom a single intensity image,” Conf. Vision Interfaces, pp. 21-67,Edmonton, Alta., Canada, 1988.
[64] D.M. Wuescher and K.L. Boyer,“Robust contour decomposition using a constant curvature criterion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 1, pp. 41-51, 1991.
[65] S. Yi, R.M. Haralick, and L.G. Shapiro, “Error Propagation in Machine Vision,” Machine Vision and Applications, vol. 7, pp. 93-114, 1994.
[66] N. Yokoya,M. Kaneta,, and K. Yamamoto,“Recovery of superquadric primitives from a range image using simulated annealing,” Int’l Conf. Pattern Recognition, vol. 1, pp. 168-172, The Hague, 1992.

Index Terms:
2D, 3D, curve representation, segmentation, circular, elliptical, superelliptical, parabolic arcs, least square elllipse fitting, multistage, significance, nonparametric.
Citation:
Paul L. Rosin, Geoff A.W. West, "Nonparametric Segmentation of Curves into Various Representations," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 12, pp. 1140-1153, Dec. 1995, doi:10.1109/34.476507
Usage of this product signifies your acceptance of the Terms of Use.