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Ellipse Fit (Direct method)

by Nikolai Chernov

14 Jan 2009 (Updated 16 Jan 2009)

Code covered by the BSD License

Fits an ellipse to a set of points on a plane; returns the coefficients of the ellipse's equation

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Description

This is a fast and non-iterative ellipse fit. Usage:

A = EllipseDirectFit(XY)

Input: XY(n,2) is the array of coordinates of n points
x(i)=XY(i,1), y(i)=XY(i,2)

Output: A = [a b c d e f]' is the vector of coefficients
of the equation of the best fitting ellipse:

ax^2 + bxy + cy^2 + dx + ey + f = 0,

To convert this vector to the geometric parameters (semi-axes, center, etc.) use standard formulas, e.g., (19) - (24) in Wolfram Mathworld:
http://mathworld.wolfram.com/Ellipse.html

This ellipse fit was proposed in article
A. W. Fitzgibbon, M. Pilu, R. B. Fisher
"Direct Least Squares Fitting of Ellipses"
IEEE Trans. PAMI, Vol. 21, pages 476-480 (1999)

The authors called it "Direct Ellipse Fit".

My code is based on a numerically stable version
of this fit published by R. Halir and J. Flusser. I only
added data centering to further improve performance.

Note: this fit returns ellipses only! You will get an ellipse
even if points can be better approximated by a hyperbola.

This fit is somewhat biased toward smaller ellipses.

Acknowledgements

The author wishes to acknowledge the following in the creation of this submission:
Ellipse Fit

MATLAB release

MATLAB 6.0 (R12)

Tags for This File
Everyone's Tags	image processing, least squares, optimization, potw, statistics
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Comments and Ratings (4)
16 Jan 2009	OH	??? Input argument "XY" is undefined. Error in ==> EllipseDirectFit at 17 centroid = mean(XY); % the centroid of the data set ??? Input argument "XY" is undefined. Error in ==> EllipseDirectFit at 17 centroid = mean(XY); % the centroid of the data set
16 Jan 2009	Nikolai Chernov	from Author: XY is the Nx2 input array of coordinates of N points, x(i)=XY(i,1), y(i)=XY(i,2). To use the program, type A = EllipseDirectFit(XY)
30 Apr 2009	Francisco	The parameters in your equation (i.e.: a, b, c, d, e and f) do not fit with equation 15 in Wolfram's Mathworld. This needs further explanation.
04 Sep 2009	Laurens	The Author (Fitzgibbon) posted some new code of his implementation at his site: http://research.microsoft.com/en-us/um/people/awf/ellipse/ might be of interest

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Updates
16 Jan 2009	Added the description of input and output.

Tag Activity for this File
Tag	Applied By	Date/Time
image processing	Nikolai Chernov	15 Jan 2009 14:34:20
statistics	Nikolai Chernov	15 Jan 2009 14:34:20
least squares	Nikolai Chernov	15 Jan 2009 14:34:20
optimization	Nikolai Chernov	15 Jan 2009 14:34:20
potw	Shari Freedman	20 Apr 2009 12:44:06

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