A NOVEL APPROACH TO ORTHOGONAL DISTANCE LEAST SQUARES FITTING OF GENERAL CONICS

Sudanthi Wijewickrema, Charles Esson, Andrew Papliński

Abstract

Fitting of conics to a set of points is a well researched area and is used in many fields of science and engineering. Least squares methods are one of the most popular techniques available for conic fitting and among these, orthogonal distance fitting has been acknowledged as the ’best’ least squares method. Although the accuracy of orthogonal distance fitting is unarguably superior, the problem so far has been in finding the orthogonal distance between a point and a general conic. This has lead to the development of conic specific algorithms which take the characteristics of the type of conic as additional constraints, or in the case of a general conic, the use of an unstable closed form solution or a non-linear iterative procedure. Using conic specific constraints produce inaccurate fits if the data does not correspond to the type of conic being fitted and in iterative solutions too, the accuracy is compromised. The method discussed in this paper aims at overcoming all these problems, in introducing a direct calculation of the orthogonal distance, thereby eliminating the need for conic specific information and iterative solutions. We use the orthogonal distances in a fitting algorithm that identifies which type of conic best fits the data. We then show that this algorithm requires less accurate initializations, uses simpler calculations and produces more accurate results.

References

  1. Ahn, S. J. (2004). Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space. Lecture Notes in Computer Science, Vol.3151, Springer.
  2. Ahn, S. J., Rauh, W., and Warnecke, H. J. (2001). Least squares orthogonal distance fitting of circle, sphere, hyperbola and parabola. Pattern Recognition, 34:2283-2303.
  3. Boggs, P. T., Byrd, R. H., and Schnabel, R. B. (1987). A stable and efficient algorithm for nonlinear orthogonal distance regression. SIAM Journal of Scientific and Statistical Computing, 8(6):1052-1078.
  4. Chaudhuri, B. B. and Samanta, G. P. (1991). Elliptic fit of objects in two and three dimensions by moment of inertia optimization. Pattern Recognition Letters, 12(1):1-7.
  5. Faber, P. and Fisher, R. B. (2001). Euclidean fitting revisited. Lecture Notes in Computer Science, 2059:165- 172.
  6. Fitzgibbon, A. W. and Fisher, R. B. (1995). A buyer's guide to conic fitting. In British Machine Vision Conference, pages 513-522, Birmingham , UK.
  7. Gander, W., Golub, G. H., and Strebel, R. (1994). Leastsquares fitting of circles and ellipses. BIT, 34:558- 578.
  8. Gauss, C. F. (1963). Theory of the motion of heavenly bodies moving about the sun in conic sections (theoria motus corporum coelestium in sectionibus conicis solem ambientium). First published in 1809, Translation by C. H. Davis. New York: Dover.
  9. Hartley, R. and Zisserman, A. (2003). Multiple View Geometry in Computer Vision. Cambridge University Press.
  10. Helfrich, H.-P. and Zwick, D. (1993). A trust region method for implicit orthogonal distance regression. Numerical Algorithms, 5:535-545.
  11. Hough, P. V. C. (1962). Methods and means for recognizing complex patterns. US Patent 3 069 654.
  12. Ma, Y., Soatto, S., Kosecka, J., and Sastry, S. S. (2004). An Invitation to 3D Vision. Springer.
  13. Miller, J. R. (1988). Analysis of Quadric Surface based Solid Models. IEEE Computer Graphics and Applications, 8(1):28-42.
  14. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1992). Numerical Recipes in C. Cambridge University Press, 2nd Edition.
  15. Semple, J. G. and Kneebone, G. T. (1956). Algebraic Projective Geometry. Oxford University Press.
  16. Spath, H. (1995). Orthogonal squared distance fitting with parabolas. In G. Alefeld, J. Herzberger (Eds.), Proceedings of the IMACS-GAMM International Symposium on Numerical Methods and Error-Bounds, University of Oldenburg, pages 261-269.
  17. Taubin, G. (1991). Estimation of planar curves, surfaces and nonplanar 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.
  18. Wijewickrema, S. N. R., PapliÁski, A. P., and Esson, C. E. (2006). Orthogonal distance fitting revisited. Technical Report 2006/205, Clayton School of Information Technology, Monash University, Melbourne, Australia.
  19. Young, J. W. (1930). Projective Geometry. The Mathematical Association of America.
Download


Paper Citation


in Harvard Style

Wijewickrema S., Esson C. and Papliński A. (2009). A NOVEL APPROACH TO ORTHOGONAL DISTANCE LEAST SQUARES FITTING OF GENERAL CONICS . In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009) ISBN 978-989-8111-69-2, pages 137-144. DOI: 10.5220/0001771901370144


in Bibtex Style

@conference{visapp09,
author={Sudanthi Wijewickrema and Charles Esson and Andrew Papliński},
title={A NOVEL APPROACH TO ORTHOGONAL DISTANCE LEAST SQUARES FITTING OF GENERAL CONICS},
booktitle={Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)},
year={2009},
pages={137-144},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0001771901370144},
isbn={978-989-8111-69-2},
}


in EndNote Style

TY - CONF
JO - Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)
TI - A NOVEL APPROACH TO ORTHOGONAL DISTANCE LEAST SQUARES FITTING OF GENERAL CONICS
SN - 978-989-8111-69-2
AU - Wijewickrema S.
AU - Esson C.
AU - Papliński A.
PY - 2009
SP - 137
EP - 144
DO - 10.5220/0001771901370144