condition_Points

% condition points, determine conditioning matrix
 see PCV (6.137), but sigma*sqrt(dim) instead of max

 Usage:
   [xc,M] = condition_Points(x)
  
    x  - Nxd matrix of Cartesian coordinates
 
   xc - Nxd matrix of Cartesian coordinates of conditioned points
   M  - d+1 x d+1 matrix of conditioning

 Wolfgang Förstner 2/2013
 wfoerstn@uni-bonn.de 

 See also uncondition_Points, condition_Homography, uncondition_Homography

0001 %% condition points, determine conditioning matrix
0002 % see PCV (6.137), but sigma*sqrt(dim) instead of max
0003 %
0004 % Usage:
0005 %   [xc,M] = condition_Points(x)
0006 %
0007 %    x  - Nxd matrix of Cartesian coordinates
0008 %
0009 %   xc - Nxd matrix of Cartesian coordinates of conditioned points
0010 %   M  - d+1 x d+1 matrix of conditioning
0011 %
0012 % Wolfgang Förstner 2/2013
0013 % wfoerstn@uni-bonn.de
0014 %
0015 % See also uncondition_Points, condition_Homography, uncondition_Homography
0016 
0017 function [xc,M] = condition_Points(x)
0018 
0019 % number and dimension
0020 [N,d] = size(x);
0021 
0022 % mean and scale
0023 xm = mean(x);
0024 C  = cov(x);
0025 sigma = sqrt(d)*sqrt(trace(C)/d);
0026 
0027 % conditioning matrix
0028 M = [eye(d)/sigma  -xm/sigma;...
0029      zeros(1,d)    1];
0030  
0031 % condition points
0032 xch =  (M * [x ones(N,1)]')';
0033 
0034 % Cartesian coordinates
0035 xc = xch(:,1:d);
0036 
0037