Abstract
The focus of our paper is on the fitting of general curves and surfaces to 3D data. In the past researchers have used approximate distance functions rather than the Euclidean distance because of computational efficiency. We now feel that machine speeds are sufficient to ask whether it is worth considering Euclidean fitting again. Experiments with the real Euclidean distance show the limitations of suggested approximations like the Algebraic distance or Taubin’s approximation. In this paper we present our results improving the known fitting methods by an (iterative) estimation of the real Euclidean distance. The performance of our method is compared with several methods proposed in the literature and we show that the Euclidean fitting guarantees a better accuracy with an acceptable computational cost.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Albano. Representation of digitized contours in terms of conic arcs and straight-line segments. CGIP, 3:23, 1974.
F. E. Allan. The general form of the orthogonal polynomial for simple series with proofs of their simple properties. In Proc. Royal Soc. Edinburgh, pp. 310–320, 1935.
P. J. Besl. Analysis and Interpretation of Rang Images, chapter Geometric Signal Processing. Springer, Berlin-Heidelberg-New York, 1990.
P. J. Besl and R. C. Jain. Three-dimensional object recognition. Computing Survey, 17(1):75–145, März 1985.
F. L. Bookstein. Fitting conic sections to scattered data. CGIP, 9:56–71, 1979.
R. O. Duda and P. E. Hart. The use of Hough transform to detect lines and curves in pictures. Comm. Assoc. Comp. Machine, 15:11–15, 1972.
A. W. Fitzgibbon and R. B. Fisher. A buyer’s guide to conic fitting. In 6th BMVC. IEE, BMVA Press, 1995.
K. Levenberg. A method for the solution of certain nonlinear problems in least squares. Quarterly of Applied Mathematics, 2:164–168, 1944.
D. W. Marquardt. An algorithm for least squares estimation of nonlinear parameters. J. of the Soc. of Industrial and Applied Mathematics, 11:431–441, 1963.
K. A. Paton. Conic sections in chromosome analysis. Pattern Recognition, 2:39, 1970.
W. J. J. Ray. Introduction to Robust and Quasi-Robust Statistical Methods. Springer, Berlin-Heidelberg-New York, 1983.
G. Taubin. Estimation of planar curves, surfaces and non-planar space curves defined by implicit equations, with applications to edge and range image segmentation. IEEE Trans. on PAMI, 13(11):1115–1138, 1991.
G. Taubin. An improved algorithm for algebraic curve and surface fitting. In 4th Int. Conf. on Computer Vision, pages 658–665, 1993.
Z. Zhang. Parameter estimation techniques: a tutorial with application to conic fitting. Image and Vision Computing, 15:59–76, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Faber, P., Fisher, R. (2001). Euclidean Fitting Revisited. In: Arcelli, C., Cordella, L.P., di Baja, G.S. (eds) Visual Form 2001. IWVF 2001. Lecture Notes in Computer Science, vol 2059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45129-3_14
Download citation
DOI: https://doi.org/10.1007/3-540-45129-3_14
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42120-7
Online ISBN: 978-3-540-45129-7
eBook Packages: Springer Book Archive