how to approximate set of point to a given function

Hello,

I am have a set of points (about 100 points) that are supposed to represent a rotated ellipse, given by this formula:

(a^2*sin(tr)^2+b^2*cos(tr)^2)*(x-x0)^2+2*(b^2-a^2)*sin(tr)*cos(tr)*(x-x0)*(y-y0)+(a^2+cos(tr)^2+b^2*sin(tr))*(y-y0)^2=a^2*b^2;

where x0,y0 are the coordinates of the center of the ellipse, a and b are the semi axes of the ellipse and tr is the rotation angle.

How would I go about finding the a,b,x0,y0 and tr so the points would be close as possible to the analytical formula.

I tried to use a multi variable minimization routine that minmize the fifference between the data points and the curve, but it seems to complicated and somewhat prone to errors.

I was wondering if there were a simple way to do that in MATLAB.

Thank you

[Jan]

Please elaborate this: "it seems to complicated and somewhat prone to errors." We do not see, what you have exactly tried and this estimation is too vague to be clear.

[Robert Jones]

what I was thinking to do is set an optimization scheme in which I assume the variables are [a b xo, y0 tr] and the Goal function something like

Goal=0
For i=1:100
Goal=sqrt(sum(YDi-YF)i^2)+Goal
end

where YD is the Y coordinate from the data point set and YF is the Y-coordinate from the formula

The optimization algorithms I used (GA and Newton) did not converge

[Jan]

"The optimization algorithms I used (GA and Newton) did not converge" - Then I assume, they contain a programming error or your initial estimation was too far apart. If you post your code, the readers can check this.

Answer 1

[Bora Eryilmaz]

This is an optimization problem that can be solved using fminsearch and a least-squares cost function.

% "Unknown" model parameters
r = 3.5;
x0 = 2.8;
y0 = -1.6;

% Set of data points.
th = 0:0.05:2*pi;
x = r * cos(th) + x0;
y = r * sin(th) + y0;
plot(x, y, 'b', x0, y0, 'r+')
grid
axis equal

% Estimate model parameteres [r, x0, y0]
params0 = [0, 0, 0]; % Initial estimates.
params = fminsearch(@(p) costFcn(p,x,y), params0) % Passes data x and y to the function.

params = 1×3

3.5000 2.8000 -1.6000

function cost = costFcn(params, x, y)
% Grid of points. Should generate xh and yh below of the same size as x and y data.
th = 0:0.05:2*pi;

r = params(1);
x0 = params(2);
y0 = params(3);

xh = r * cos(th) + x0;
yh = r * sin(th) + y0;

cost = sum((xh-x).^2 + (yh-y).^2); % Least-squares cost
end

[Robert Jones]

thanks will check

Answer 2

[Jan]

Search in the net for "Matlab fit ellipse":

• https://www.mathworks.com/matlabcentral/fileexchange/3215-fit_ellipse
• https://www.mathworks.com/matlabcentral/fileexchange/22684-ellipse-fit-direct-method
• https://www.mathworks.com/matlabcentral/answers/98522-how-do-i-fit-an-ellipse-to-my-data-in-matlab
• https://www.mathworks.com/matlabcentral/fileexchange/22683-ellipse-fit-taubin-method
• https://www.mathworks.com/matlabcentral/fileexchange/15125-fitellipse-m

[Robert Jones]

Thanks but Matalb fit ellipse just givres a polynomial approximation.

The goal here is not just getting the points but the ellipse properties

[Jan]

"Thanks but Matalb fit ellipse just givres a polynomial approximation." - I've posted links to 6 different approaches (the last one contains 2 methods, a linear and a non-linear fit).

"The goal here is not just getting the points but the ellipse properties" - The shown methods determine the parameters of a ellipse, which fits a given set of points.