ellipse fitting with matlab using fittype

Hello,

I want to use fitoptions and fittype for a staistical nonlinear least squares fitting approach. The problem is, I cant find the correct equation for my model (ellipse). I tried multiple like: to fittype ('sqrt(b^2*(1-(x^2/a^2)))' ). In this example, the square root has to be positive. I tried to tightening the upper and lower bounderies, but it doesnt work.

Can someone make an example of the following form:

fo = fitoptions('Method','NonLinearLeastSquares')

ft = fittype('ellipse equation')

[myfit, gof] = fit(x_data, y_data, ft, fo)

Answer 1

[Matt J]

You could use fit() if you convert your data to polar cooridnates. In polar coordinates, an ellipse is given by an explicit function,

https://en.wikipedia.org/wiki/Ellipse#Polar_form_relative_to_center

[Johannes Tischer]

After this I converted it back to cartesian coordinates and plotted the data with my original data.

The pictures shows the original data (blue) and the ellipse fit (red).

Thank you Matt !

[Matt J]

The problem with this method, of course, is that it assumes that the ellipse is centered at the origin. In general, that may not be true or known to be true.

Answer 2

[Alan Stevens]

Have a look here: https://uk.mathworks.com/help/curvefit/fittype.html?s_tid=srchtitle

It shows you how to construct your own function correctly.

[Johannes Tischer]

Thank you for your fast response.

There are only basic problems. I need a function like f(x,y)= a*x + b*x*y + c* y + d , as model function. How can I transform this function correctly ?

I tried fittype( @(a, b, c ,d, x, y) 'a*x + b*x*y + c* y + d' ), but the syntax isnt correctly, right ?

[Alan Stevens]

You could try using a linear fit! Because you know the values of your x’s and y’s you could define, say, Xi = xi^2 and Yi = yi^2. Then your ellipse equation becomes uX + vY = 1, where u and v are the unknown constants to be found, from which the semi-major axes, a and b, of your ellipse can be determined. No need for awkward square roots!

Just how successful the end result would be I can’t say, but it might be worth a try.

[Johannes Tischer]

This is all I got... Do you have any suggestions ? It's not even close to an ellipse...

Thank you for all you help !!!!

I tried a^2 and a in my model. But it doesnt matter...

[Alan Stevens]

No, it doesn't look good! Did you subtract the mean of the x values from all the x-values and the mean of the y values from all the y-values first, to ensure you are centred on zero?

Are you able to upload the basic data that you are trying to fit? I've done some tests with arbitrarily generated ellipse data and the "linear" manipulation works out ok (I didn't use Matlab's fittype - just did a simple matrix least squares fit in core Matlab).

[Johannes Tischer]

This is the basic data. I want to use fittype because it has so many different options to fit and compare.

[Alan Stevens]

Hmm. I'm struggling to see your basic data as elliptical. Nevertheless, the code below tries to fit them using an ellipse (I've deleted a few obvious non-elliptical outliers below; however, you could easily retain them if you wish):

load('EllipseData.mat')
% The next three lines remove outliers that are obviously not part of an
% ellipse!  Delete the three lines if you wish to retain the outliers.
ix = find(data_x<-600); data_x(ix) = [];data_y(ix) = [];
iylo = find(data_y<-220); data_x(iylo) = [];data_y(iylo) = [];
iyhi = find(data_y>220); data_x(iyhi) = [];data_y(iyhi) = [];

% Set the origin at the means of the x and y data
x = data_x - mean(data_x);
y = data_y - mean(data_y);

% "Linearise" and do a straightforward least-squares fit
N = length(x);
X = x.^2; Y = y.^2;
M = [N  -sum(X); sum(X) -sum(X.^2)];
V = [sum(Y); sum(X.*Y)];
AB = M\V;  % AB = [A; B]

% Extract constants
A = AB(1);
B = AB(2);

% Create ellipse semimajor axes from "linear" constants
a = sqrt(A/B);
b = sqrt(A);

% Separate upper and lower halves of "elliptical" values.
xhi = x(x>=0); xlo = x(x<0);
yhi = b*sqrt(1 - (xhi/a).^2);
ylo = -b*sqrt(1 - (xlo/a).^2);

% Put them back together and shift the origin back
xf = [xhi xlo] + mean(data_x);
yf = [yhi ylo] + mean(data_y);

% Plot and compare Data and fit.
plot(data_x,data_y,'o',x,yf,'*'), grid
xlabel('x'),ylabel('y')
legend('Data','"Elliptical" fit')

This is the result:

Answer 3

[Matt J]

You cannot use fit() for implicit function fitting, but you could use dedicated ellipse fitting methods from the File Exchange, e.g.,

https://www.mathworks.com/matlabcentral/fileexchange/22683-ellipse-fit-taubin-method

[Johannes Tischer]

so theres no way to use this method for ellipse fitting ?

Thank you both !!! I really appreciate your support.

Answer 4

[Image Analyst]

For what it's worth, attached is a paper that describes the method.

Least Squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola.pdf