sugr_condition_Points

% condition points, determine conditioning matrix

 [xc,M] = sugr_condition_Points(x)

 x  = struct, unconditioned points, non-homogeneous coordinates
      x.e = coordiantes, Cartesian
      x.Cee = covariance matrices 

 xc = struxt: conditioned SURG-points
      x.h   = homogeneous coordiantes
      x.Crr = covariance matrices of reduced coordiantes
      x.e   = nonhomogeneous coordiante
      x.Cee = covariance matrices of x.e
 M  = matrix of conditioning 

 see PCV (6.137), but sigma*sqrt(dim) instead of max

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

0001 %% condition points, determine conditioning matrix
0002 %
0003 % [xc,M] = sugr_condition_Points(x)
0004 %
0005 % x  = struct, unconditioned points, non-homogeneous coordinates
0006 %      x.e = coordiantes, Cartesian
0007 %      x.Cee = covariance matrices
0008 %
0009 % xc = struxt: conditioned SURG-points
0010 %      x.h   = homogeneous coordiantes
0011 %      x.Crr = covariance matrices of reduced coordiantes
0012 %      x.e   = nonhomogeneous coordiante
0013 %      x.Cee = covariance matrices of x.e
0014 % M  = matrix of conditioning
0015 %
0016 % see PCV (6.137), but sigma*sqrt(dim) instead of max
0017 %
0018 % Wolfgang Förstner 2/2013
0019 % wfoerstn@uni-bonn.de
0020 
0021 function [xc,M] = sugr_condition_Points(x)
0022 
0023 % number and dimension
0024 [N,d] = size(x.e);
0025 xc    = x;
0026 
0027 % mean and scale
0028 xm = mean(x.e); 
0029 C  = cov(x.e);
0030 sigma = sqrt(d) * sqrt(trace(C)/d);
0031 
0032 % conditioning-matrix
0033 M = [eye(d)       -xm';...
0034      zeros(1,d) sigma];
0035 %M= eye(d+1);
0036 
0037 % condition point coordiantes
0038 for n=1:N
0039     xhn   = [x.e(n,:) , 1]';      % homogeneous coord.
0040     Chhn  = [squeeze(x.Cee(n,:,:)) zeros(d,1); zeros(1,d+1)];
0041     xcn   = M * xhn;               % homogeneous condit. coord.
0042     Chhcn = M * Chhn * M';         % ... CovM
0043     if d ==2
0044         xen = sugr_Point_2D(xcn,Chhcn);
0045     else
0046         xen = sugr_Point_3D(xcn,Chhcn);
0047     end
0048     xc.h(n,:)     = xen.h';        % spherically normalized cond. coord.
0049     xc.Crr(n,:,:) = xen.Crr;       % ... reduced CovM
0050     xc.e(n,:)     = xcn(1:d)/xcn(d+1);
0051     xc.Cee(n,:,:) = x.Cee(n,:,:)/sigma^2;
0052 end
0053 
0054 %  cond_x  = cond(cov(x.e))
0055 %  cond_xc = cond(cov(xc.h(:,1:d)./(xc.h(:,d+1)*ones(1,d))))
0056  
0057