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 2019-2024 INRAE - Cepia Software Platform
0030 
0031 % compute centroid of points to estimate center
0032 center = mean(points);
0033 
0034 % compute the covariance matrix
0035 covPts = cov(points);
0036 
0037 % perform a principal component analysis to extract principal axes
0038 [rotMat, S] = svd(covPts);
0039 
0040 % extract length of each semi axis
0041 radii = sqrt(diag(S));
0042 
0043 % sort axes from greater to lower
0044 [radii, ind] = sort(radii, 'descend');
0045 radii = radii';
0046 
0047 % format U to ensure first axis points to positive x direction
0048 rotMat = rotMat(ind, :);
0049 if rotMat(1,1) < 0 && size(points, 2) > 2
0050     rotMat = -rotMat;
0051     % keep matrix determinant positive
0052     rotMat(:,3) = -rotMat(:,3);
0053 end
0054 
0055 % format output
0056 if nargout == 2
0057     varargout = {center, rotMat};
0058 elseif nargout == 3
0059     varargout = {center, rotMat, radii};
0060 end