This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Orthogonal Distance Fitting of Implicit Curves and Surfaces
May 2002 (vol. 24 no. 5)
pp. 620-638

Dimensional model fitting finds its applications in various fields of science and engineering and is a relevant subject in computer/machine vision and coordinate metrology. In this paper, we present two new fitting algorithms, distance-based and coordinate-based algorithm, for implicit surfaces and plane curves, which minimize the square sum of the orthogonal error distances between the model feature and the given data points. Each of the two algorithms has its own advantages and is to be purposefully applied to a specific fitting task, considering the implementation and memory space cost, and possibilities of observation weighting. By the new algorithms, the model feature parameters are grouped and simultaneously estimated in terms of form, position, and rotation parameters. The form parameters determine the shape of the model feature and the position/rotation parameters describe the rigid body motion of the model feature. The proposed algorithms are applicable to any kind of implicit surface and plane curve. In this paper, we also describe algorithm implementation and show various examples of orthogonal distance fit.

[1] ABW GmbH,http://www.caip.rutgers.edu/riul/research/ theses.htmlhttp://www.abw-3d.dehome_e.html , Feb. 2002.
[2] S.J. Ahn, “Calibration of the Stripe Projecting 3D-Measurement System,” Proc. 13th Korea Automatic Control Conf. (KACC '98), pp. 1857-1862, 1998 (in Korean).
[3] S.J. Ahn and W. Rauh, “Geometric Least Squares Fitting of Circle and Ellipse,” Int'l J. Pattern Recognition and Artificial Intelligence, vol. 13, no. 7, pp. 987-996, 1999.
[4] S.J. Ahn, W. Rauh, and H.-J. Warnecke, “Least-Squares Orthogonal Distances Fitting of Circle, Sphere, Ellipse, Hyperbola, and Parabola,” Pattern Recognition, vol. 34, no. 12, pp. 2283-2303, 2001.
[5] S.J. Ahn, E. Westkämper, and W. Rauh, “Orthogonal Distance Fitting of Parametric Curves and Surfaces,” Proc. Int'l Symp. Algorithms for Approximation IV: Huddersfield 2001, I. Anderson and J. Levesley, eds., 2002.
[6] S.J. Ahn, W. Rauh, and S.I. Kim, “Circular Coded Target for Automation of Optical 3D-Measurement and Camera Calibration,” Int'l J. Pattern Recognition and Artificial Intelligence, vol. 15, no. 6, pp. 905-919, 2001.
[7] M.D. Altschuler, B.R. Altschuler, and J. Taboada, “Measuring Surfaces Space-Coded by a Laser-Projected Dot Matrix,” Proc. SPIE Conf. Imaging Applications for Automated Industrial Inspection and Assembly, vol. 182, pp. 187-191, 1979.
[8] D.H. Ballard, “Generalizing the Hough Transform to Detect Arbitrary Shapes,” Pattern Recognition, vol. 13, no. 2, pp. 111-122, 1981.
[9] A.H. Barr, “Superquadrics and Angle-Preserving Transformations,” IEEE Computer Graphics and Applications, vol. 1, no. 1, pp. 11-23, 1981.
[10] M. Bennamoun and B. Boashash, “A Structural Description Based Vision System for Automatic Object Recognition,” IEEE Trans. Systems, Man, and Cybernetics, Part B, vol. 27, no. 6, pp. 893–906, 1997.
[11] P.T. Boggs, R.H. Byrd, and R.B. Schnabel, “A Stable and Efficient Algorithm for Nonlinear Orthogonal Distance Regression,” SIAM J. Scientific and Statistical Computing, vol. 8, no. 6, pp. 1052-1078, 1987.
[12] F.L. Bookstein, “Fitting Conic Sections to Scattered Data,” Computer Graphics and Image Processing, vol. 9, no. 1, pp. 56-71, 1979.
[13] B.P. Butler, A.B. Forbes, and P.M. Harris, “Algorithms for Geometric Tolerance Assessment,” Report no. DITC 228/94, Nat'l Physical Laboratory, Teddington, U.K., 1994.
[14] X. Cao, N. Shrikhande, and G. Hu, “Approximate Orthogonal Distance Regression Method for Fitting Quadric Surfaces to Range Data,” Pattern Recognition Letters, vol. 15, no. 8, pp. 781-796, 1994.
[15] B.B. Chaudhuri and G.P. Samanta, “Elliptic Fit of Objects in Two and Three Dimensions by Moment of Inertia Optimization,” Pattern Recognition Letters, vol. 12, no. 1, pp. 1-7, 1991.
[16] DIN 32880-1, Coordinate Metrology; Geometrical Fundamental Principles, Terms and Definitions. Berlin: Beuth Verlag, German Standard, 1986.
[17] N.R. Draper and H. Smith, Applied Regression Analysis, third ed. New York: John Wiley and Sons, 1998.
[18] R. Drieschner, B. Bittner, R. Elligsen, and F. Wäldele, “Testing Coordinate Measuring Machine Algorithms: Phase II,” BCR Report no. EUR 13417 EN, Commission of the European Communities, Luxemburg, 1991.
[19] R.O. Duda and P.E. Hart, "Use of the Hough transforms to detect lines and curves in pictures," Comm. ACM, vol. 15, no. 1, pp. 11-15, 1972
[20] R. Fletcher, Practical Methods of Optimization. John Wiley and Sons, second ed., 1987.
[21] W. Gander, G.H. Golub, and R. Strebel, “Least-Squares Fitting of Circles and Ellipses,” BIT, vol. 34, no. 4, pp. 558-578, 1994.
[22] M. Gardiner, “The Superellipse: A Curve that Lies between the Ellipse and the Rectangle,” Scientific Am., vol. 213, no. 3, pp. 222-234, 1965.
[23] C.F. Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections (Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum), first published in 1809, translation by C.H. Davis. New York: Dover, 1963.
[24] R.N. Goldman, “Two Approaches to a Computer Model for Quadric Surfaces,” IEEE Computer Graphics and Applications, vol. 3, no. 9, pp. 21-24, 1983.
[25] G.H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numerische Mathematik, vol. 14, no. 5, pp. 403-420, 1970.
[26] F. Gray, “Pulse Code Communication,” US Patent 2,632,058, Mar. 17, 1953.
[27] H.-P. Helfrich and D. Zwick, “A Trust Region Method for Implicit Orthogonal Distance Regression,” Numerical Algorithms, vol. 5, pp. 535-545, 1993.
[28] P.V.C. Hough, “Method and Means for Recognizing Complex Patterns,” US Patent 3,069,654, Dec. 18, 1962.
[29] G. Hu and N. Shrikhande, “Estimation of Surface Parameters Using Orthogonal Distance Criterion,” Proc. Fifth Int'l Conf. Image Processing and Its Applications, pp. 345-349, 1995.
[30] ISO 10360-6:2001, Geometrical Product Specifications (GPS)—Acceptance and Reverification Tests for Coordinate Measuring Machines (CMM)—Part 6: Estimation of Errors in Computing Gaussian Associated Features, Int'l Standard, ISO, Dec. 2001.
[31] D. Marshall, G. Lukacs, and R. Martin, “Robust Segmentation of Primitives from Range Data in the Presence of Geometric Degeneracy,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 3, pp. 304-314, Mar. 2001.
[32] K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” The Philosophical Magazine, Series 6, vol. 2, no. 11, pp. 559-572, 1901.
[33] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C.Cambridge, England: Cambridge Univ. Press, 1988.
[34] P.L. Rosin, “Fitting Superellipses,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 7, pp. 726-732, July 2000.
[35] R. Safaee-Rad, K.C. Smith, B. Benhabib, and I. Tchoukanov, “Application of Moment and Fourier Descriptors to the Accurate Estimation of Elliptical Shape Parameters,” Pattern Recognition Letters, vol. 13, no. 7, pp. 497-508, 1992.
[36] P.D. Sampson, “Fitting Conic Sections to‘Very Scattered’Data: An Iterative Refinement of the Bookstein Algorithm,” Computer Graphics and Image Processing, vol. 18, no. 1, pp. 97-108, 1982.
[37] 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.
[38] D. Sourlier, “Three Dimensional Feature Independent Bestfit in Coordinate Metrology,” PhD Thesis no. 11319, ETH Zurich, Switzerland, 1995.
[39] S. Sullivan, L. Sandford, and J. Ponce, "Using Geometric Distance Fits for 3D Object Modeling and Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 12, pp. 1,183-1,196, 1994.
[40] 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.
[41] D.A. Turner, “The Approximation of Cartesian Coordinate Data by Parametric Orthogonal Distance Regression,” PhD thesis, School of Computing and Math., Univ. of Huddersfield, U.K., 1999.
[42] K. Voss and H. Suesse, “Invariant Fitting of Planar Objects by Primitives,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, pp. 80-83, 1997.
[43] K. Voss and H. Süße, “A New One-Parametric Fitting Method for Planar Objects,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 7, pp. 646–651, 1999.
[44] F.M. Wahl, “A Coded-Light Approach for 3-Dimensional (3D) Vision,” Research Report no. 1452, IBM Zurich Research Laboratory, Switzerland, Sept. 1984.

Index Terms:
implicit curve, implicit surface, curve fitting, surface fitting, orthogonal distance fitting, geometric distance, orthogonal contacting, nonlinear least squares, parameter estimation, Gauss-Newton method, parameter constraint, parametric model recovery, object segmentation, object classification, object reconstruction
Citation:
S.J. Ahn, W. Rauh, H.S. Cho, H.J. Warnecke, "Orthogonal Distance Fitting of Implicit Curves and Surfaces," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 620-638, May 2002, doi:10.1109/34.1000237
Usage of this product signifies your acceptance of the Terms of Use.