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Fitting quadratic curves and surfaces

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Fitting quadratic curves and surfaces

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03 Feb 2014 (Updated )

Fit ellipses, ellipsoids and other quadratic curves and surfaces to noisy data.

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Description

Generating points along an ellipse or ellipsoid, plotting ellipses and ellipsoids in various parametric representations, and fitting ellipses, ellipsoids or other quadratic curves and surfaces to noisy data occur frequently in fields such as computer vision, pattern recognition and system identification.
This toolbox provides a fairly comprehensive toolset of estimating quadratic curves and surfaces in an errors-in-variables context, with and without constraints. In addition to classical fitting methods such as least squares (with and without curve or surface normals), Taubin's method, direct ellipse fit by Fitzgibbon et al. [1] and direct ellipsoid fit by Qingde Li and John G. Griffiths [4], the toolbox features an estimation algorithm by the author [2,3], based on and extending the work of István Vajk and Jenő Hetthéssy [5]. The proposed quadratic curve and surface fitting algorithm combines direct fitting with a noise cancellation step, producing consistent estimates close to maximum likelihood but without iterations.
REFERENCES

[1] Andrew W. Fitzgibbon, Maurizio Pilu and Robert B. Fisher, "Direct Least Squares Fitting of Ellipses", IEEE Trans. PAMI 21, 1999, pp476-480.
[2] Levente Hunyadi, "Estimation methods in the errors-in-variables context", PhD dissertation, Budapest University of Technology and Economics, 2013.
[3] Levente Hunyadi and István Vajk, "Constrained quadratic errors-in-variables fitting", The Visual Computer, 12 pages, in print, available on-line from October 2013.
[4] Qingde Li and John G. Griffiths, "Least Squares Ellipsoid Specific Fitting", Proceedings of the Geometric Modeling and Processing, 2004.
[5] István Vajk and Jenő Hetthéssy, "Identification of nonlinear errors-in-variables models", Automatica 39, 2003, pp2099-2107.

CONTACT INFORMATION

Levente Hunyadi
http://hunyadi.info.hu/

Please use my private e-mail address to submit bug reports, which will be addressed upon short notice; reviews, however, are not monitored. Any feedback is most welcome.

Acknowledgements

Quadratic Curves And Quadric Surfaces In Implicit Form and Symbolic Polynomials inspired this file.

Required Products Optimization Toolbox
Symbolic Math Toolbox
MATLAB release MATLAB 7.14 (R2012a)
Other requirements This submission depends "Quadratic curves and quadric surfaces in implicit form" and "Symbolic polynomials" by the same author, both of which are included. Optimization Toolbox and Symbolic Math Toolbox are optional and required only by a few features.
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Comments and Ratings (6)
18 Sep 2014 Martti K

In file quad3dfit_taubin.m, line 39 is written as

dX = dx.^2 + dy.^2 + dz.^2;

but it should be

dX = [dx; dy; dz];

15 Sep 2014 Martti K  
08 Aug 2014 Torsten Schönfeld  
14 Mar 2014 Levente Hunyadi

optimset is a standard MATLAB function, see: http://www.mathworks.com/help/matlab/ref/optimset.html

13 Mar 2014 Carey Smith

Although the zip contains 97 files (many examples, I believe), it does not contain the needed file imconic.m.
This (by the same author) can also be downloaded, imconic.zip. It has 24 files, but does not have optimset.m.
So I couldn't run ellipsefit.
(A search for optimset returns many files, but none by the same author, so I gave up.)

11 Mar 2014 Jean-Yves Tinevez  
Updates
25 Feb 2014

Improved numerical robustness for standard least-squares estimation of ellipsoid parameters.

14 Mar 2014

Added automatic dependency check for Optimization Toolbox and suggestion which function to use when the toolbox is not installed.

12 May 2014

Included all external dependencies into a single package.

22 Sep 2014

Fixed an issue with 3D Taubin fit (contributed by Martti K).

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