## Minimum Volume Enclosing Ellipsoid

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Computes the minimum-volume covering ellipoid that encloses N points in a D-dimensional space.

Updated Tue, 20 Jan 2009 17:28:01 +0000

[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);

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)

### Cite As

Nima Moshtagh (2022). Minimum Volume Enclosing Ellipsoid (https://www.mathworks.com/matlabcentral/fileexchange/9542-minimum-volume-enclosing-ellipsoid), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R13
Compatible with any release
##### Platform Compatibility
Windows macOS Linux