how to approximate set of point to a given function

Question

Robert Jones 2022-12-14

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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 2022-12-14

在 MATLAB Online 中打开

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 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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Answer 1

Bora Eryilmaz 2022-12-14

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编辑：Bora Eryilmaz 2022-12-14

在 MATLAB Online 中打开

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

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Robert Jones 2022-12-14

thanks will check

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Answer 2

Jan 2022-12-14

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1878202-how-to-approximate-set-of-point-to-a-given-function#answer_1127857

Search in the net for "Matlab fit ellipse":

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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 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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