How to fit an elliptic cone to a set of data?

Question

I have a set of 3d data (300 points) that create a surface which looks like two cones or ellipsoids connected to each other. I want a way to find the equation of a best fit ellipsoid or cone to this dataset. The regression method is not important, the easier it is the better. I basically need a way, a code or a matlab function to calculate the constants of the elliptic equation for these data.

Answer 1

You can also try with fminsearch, but to avoid falling on local minima you will need a good starting point given the amount of coefficients (try to eliminate some of them).

Here is an example with a 2D ellipse:

% implicit equation
fxyc = @(x, y, c_) c_(1)*x.^2 + c_(2).*y.^2 + c_(3)*x.*y + c_(4)*x + c_(5).*y - 1; % free term locked to -1

% solution (ellipse)
c_ = [1, 2, 1, 0, 0]; % x^2, y^2, x*y, x, y (free term is locked to -1)

[X,Y] = meshgrid(-2:0.01:2);
figure(1);
fxy = @(x, y) fxyc(x, y, c_);
c = contour(X, Y, fxy(X, Y), [0, 0], 'b');
axis equal;
grid on;
xlabel('x');
ylabel('y');          
title('solution');          

% we sample the solution to have some data to fit
N = 100; % samples
sample = unique(2 + floor((length(c) - 2)*rand(1, N)));
x = c(1, sample).';
y = c(2, sample).';

x = x + 5e-2*rand(size(x)); % add some noise
y = y + 5e-2*rand(size(y));

fc = @(c_) fxyc(x, y, c_); % function in terms of the coefficients
e = @(c) fc(c).' * fc(c); % squared error function

% we start with a circle
c0 = [1, 1, 0, 0, 0];

copt = fminsearch(e, c0)

figure(2);
plot(x, y, 'rx');
hold on
fxy = @(x, y) fxyc(x, y, copt);
contour(X, Y, fxy(X, Y), [0, 0], 'b');
hold off;
axis equal;
grid on;
legend('data', 'fit');
xlabel('x');                    %# Add an x label
ylabel('y');          
title('fitted solution');