how to approximate set of point to a given function

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Robert Jones
Robert Jones 2022-12-14
评论： Jan 2022-12-14
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
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Robert Jones
Robert Jones 2022-12-14
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
Jan 2022-12-14
"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.

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回答（2 个）

Bora Eryilmaz
Bora Eryilmaz 2022-12-14
编辑：Bora Eryilmaz 2022-12-14
Ran in:
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
Jan
Jan 2022-12-14
  2 个评论
Robert Jones
Robert Jones 2022-12-14
Thanks but Matalb fit ellipse just givres a polynomial approximation.
The goal here is not just getting the points but the ellipse properties
Jan
Jan 2022-12-14
"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.

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