[A , c] = MinVolEllipse(P, tolerance)
Finds the minimum volume enclosing ellipsoid (MVEE) of a set of data points stored in matrix P. The following optimization problem is solved:
minimize log(det(A))
s.t. (P_i - c)'*A*(P_i - c)<= 1
in variables A and c, where P_i is the i-th column of the matrix P.
The solver is based on Khachiyan Algorithm, and the final solution is different from the optimal value by the pre-specified amount of 'tolerance'.
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Outputs:
c : D-dimensional vector containing the center of the ellipsoid.
A : This matrix contains all the information regarding the shape of the ellipsoid. To get the radii and orientation of the ellipsoid take the Singular Value Decomposition ( svd function in matlab) of the output matrix A:
[U Q V] = svd(A);
the radii are given by:
r1 = 1/sqrt(Q(1,1));
r2 = 1/sqrt(Q(2,2));
...
rD = 1/sqrt(Q(D,D));
and matrix V is the rotation matrix that gives you the orientation of the ellipsoid.
For plotting in 2D or 3D, use MinVolEllipse_plot.m (see the link bellow) |