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## Ellipse Fit (Taubin method)

version 1.0.0.0 (2.02 KB) by
Fits an ellipse to a set of points on a plane; returns coefficients of the ellipse's equation.

Updated 14 Jan 2009

Editor's Note: This file was selected as MATLAB Central Pick of the Week

This is a fast non-iterative ellipse fit, and among fast non-iterative ellipse fits this is the most accurate and robust.

It takes the xy-coordinates of data points, and returns the coefficients of the equation of the ellipse:

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

i.e. it returns the vector A=(a,b,c,d,e,f). To convert this vector to the geometric parameters (semi-axes, center, etc.), use standard formulas, see e.g., (19) - (24) in Wolfram Mathworld: http://mathworld.wolfram.com/Ellipse.html

This fit was proposed by G. Taubin in article "Estimation Of Planar Curves, Surfaces And Nonplanar Space Curves Defined By Implicit Equations, With Applications To Edge And Range Image Segmentation", IEEE Trans. PAMI, Vol. 13, pages 1115-1138, (1991).

Note: this method fits a quadratic curve (conic) to a set of points; if points are better approximated by a hyperbola, this fit will return a hyperbola. To fit ellipses only, use "Direct Ellipse Fit".

### Cite As

Nikolai Chernov (2020). Ellipse Fit (Taubin method) (https://www.mathworks.com/matlabcentral/fileexchange/22683-ellipse-fit-taubin-method), MATLAB Central File Exchange. Retrieved .

### Comments and Ratings (8)

Harry Graber

The function, and the pointer to the page on wolfram.com, are very useful. What I would like to know is if it is a straightforward matter to compute a goodness-of-fit parameter from the data points and conic-section coefficients. Likewise, standard errors or confidence limits on the elements of the returned A vector.

youkang Wang

Ali Shahid

thanks very much

Tom Cunningham

This worked very well, thanks, and thanks also to David Christian Berg for his conversion code.

Would it be possible to also calculate and output the MSE (goodness of fit)?

Mathias Funk

Great file. Very reliable fit to almost every data, even the smallest section of an ellipse.

I've got a small question though:
Is it possible to modify the script to fit to hyperbolas only?

I'd be more than thankful for any pointers.

David Christian Berg

Well, found it a bit impractical to not have the code for conversion standard form here. That's why I would like to share it:

A = EllipseFitByTaubin(points);
a = A(1); b = A(2)/2; c = A(3); d = A(4)/2; f = A(5)/2; g = A(6);

center(1) = (c*d - b*f)/(b^2-a*c);
center(2) = (a*f - b*d)/(b^2-a*c);

sem(1) = sqrt( 2*(a*f^2+c*d^2+g*b^2-2*b*d*f-a*c*g) / ((b^2-a*c)*(sqrt((a-c)^2+4*b^2)-(a+c))));
sem(2) = sqrt( 2*(a*f^2+c*d^2+g*b^2-2*b*d*f-a*c*g) / ((b^2-a*c)*(-sqrt((a-c)^2+4*b^2)-(a+c))));

if b == 0 && a < c
phi = 0;
elseif b == 0 && a > c
phi = 0.5*pi;
elseif b ~= 0 && a < c
phi = 0.5* acot((a-c)/(2*b));
else
phi = 0.5*pi + 0.5* acot((a-c)/(2*b));
end

Hope this makes it a bit faster to implement.

Christian

The information from Ed Shen (in the comments about Ellipse Fit (Direct method)) is crucial for this function too: "The code works, however you must divde A(2), A(4) and A(5) by 2 to be able to use it with the Mathworld equations."
Without these divisions the rusulting elippse does not really fit to the points.

K. Titievsky

##### MATLAB Release Compatibility
Created with R12
Compatible with any release
##### Platform Compatibility
Windows macOS Linux
##### Acknowledgements

Inspired by: Ellipse Fit