principalAxes

PRINCIPALAXES Principal axes of a set of ND points.

   [CENTER, ROTMAT] = principalAxes(PTS)
   [CENTER, ROTMAT, SCALES] = principalAxes(PTS)
   Computes the principal axes of a set of points given in a N-by-ND array
   and returns the result in two or three outputs:
   CENTER  is the centroid of the points, as a 1-by-ND row vector
   ROTMAT  represents the orientation of the point cloud, as a ND-by-ND
           rotation matrix
   SCALES  is the scaling factor along each dimension, as a 1-by-ND row
           vector.

   Example
     pts = randn(100, 2);
     pts = transformPoint(pts, createScaling(5, 2));
     pts = transformPoint(pts, createRotation(pi/6));
     pts = transformPoint(pts, createTranslation(3, 4));
     [center, rotMat] = principalAxes(pts);

   See also
     equivalentEllipse, equivalentEllipsoid, principalAxesTransform

0001 function varargout = principalAxes(points)
0002 %PRINCIPALAXES Principal axes of a set of ND points.
0003 %
0004 %   [CENTER, ROTMAT] = principalAxes(PTS)
0005 %   [CENTER, ROTMAT, SCALES] = principalAxes(PTS)
0006 %   Computes the principal axes of a set of points given in a N-by-ND array
0007 %   and returns the result in two or three outputs:
0008 %   CENTER  is the centroid of the points, as a 1-by-ND row vector
0009 %   ROTMAT  represents the orientation of the point cloud, as a ND-by-ND
0010 %           rotation matrix
0011 %   SCALES  is the scaling factor along each dimension, as a 1-by-ND row
0012 %           vector.
0013 %
0014 %   Example
0015 %     pts = randn(100, 2);
0016 %     pts = transformPoint(pts, createScaling(5, 2));
0017 %     pts = transformPoint(pts, createRotation(pi/6));
0018 %     pts = transformPoint(pts, createTranslation(3, 4));
0019 %     [center, rotMat] = principalAxes(pts);
0020 %
0021 %   See also
0022 %     equivalentEllipse, equivalentEllipsoid, principalAxesTransform
0023 %
0024  
0025 % ------
0026 % Author: David Legland
0027 % e-mail: david.legland@inrae.fr
0028 % Created: 2019-08-12,    using Matlab 9.6.0.1072779 (R2019a)
0029 % Copyright 2019 INRAE - Cepia Software Platform.
0030 
0031 % number and dimension of points
0032 n = size(points, 1);
0033 nd = size(points, 2);
0034 
0035 % compute centroid
0036 center = mean(points);
0037 
0038 % compute the covariance matrix
0039 covPts = cov(points) / n;
0040 
0041 % perform a principal component analysis to extract principal axes
0042 [rotMat, S] = svd(covPts);
0043 
0044 % extract length of each semi axis
0045 radii = sqrt(diag(S) * n);
0046 
0047 % sort axes from greater to lower
0048 [radii, ind] = sort(radii, 'descend');
0049 radii = radii';
0050 
0051 % format U to ensure first axis points to positive x direction
0052 rotMat = rotMat(ind, :);
0053 if rotMat(1,1) < 0 && nd > 2
0054     rotMat = -rotMat;
0055     % keep matrix determinant positive
0056     rotMat(:,3) = -rotMat(:,3);
0057 end
0058 
0059 % format output
0060 if nargout == 2
0061     varargout = {center, rotMat};
0062 elseif nargout == 3
0063     varargout = {center, rotMat, radii};
0064 end