function [S,mo] = precond2(m)
%_PRECOND2 Normalize the coordinates of a 2D point set. 
%
%   [S,mo] = PRECOND(m) scale and translate the 2D point coordinates so that
%   the barycenter is in origin and the average radius  is sqrt(2)
%
%   See also ....

% Algorithm ref.: Hartley
%



[rm2,cm2]=size(m);
if (rm2 ~= 2)
    error('Le coordinate immagine devono essere cartesiane!!');
end

u1=m(1,:)';
v1=m(2,:)';

n_points = size(u1,1);

if n_points > 1
%calcolo centroide
avgx1 = (sum(u1)/n_points);
avgy1 = (sum(v1)/n_points);
%standardizzo punti
u1 = u1 - avgx1;
v1 = v1 - avgy1;
%scalo tutto con distanza sqrt(2)
tscale1 = sum(sqrt(u1.^2 + v1.^2))/n_points/sqrt(2);

u1 = u1/tscale1;
v1 = v1/tscale1;

S = [1/tscale1 0 -avgx1/tscale1; 
     0 1/tscale1 -avgy1/tscale1; 
     0       0          1];
mo = [u1';v1'];

else
  % it does not make sense to normalize 1 single point.
  
  mo = m;
  S = eye(3);
end



% end of scaling


% scale the point coordinates so that centroid is in origin 
% and average radius  is sqr(2)
% $$$ xc = (sum(M(1,:))/n_points);
% $$$ yc = (sum(M(2,:))/n_points);
% $$$ uc = (sum(m(1,:))/n_points);
% $$$ vc = (sum(m(2,:))/n_points);
% $$$ M(1,:) = M(1,:) - xc;
% $$$ M(2,:) = M(2,:) - yc;
% $$$ m(1,:) = m(1,:) - uc;
% $$$ m(2,:) = m(2,:) - vc;
% $$$ rM = sum(sqrt(M(1,:).^2 + M(2,:).^2))/n_points/sqrt(2);
% $$$ rm = sum(sqrt(m(1,:).^2 + m(2,:).^2))/n_points/sqrt(2);
% $$$ M(1,:) = M(1,:)/rM;
% $$$ M(2,:) = M(2,:)/rM;
% $$$ m(1,:) = m(1,:)/rm;
% $$$ m(2,:) = m(2,:)/rm;
% $$$ TM = [1 0 -xc; 
% $$$       0 1 -yc; 
% $$$       0 0  rM];
% $$$ Tm = [1 0 -uc; 
% $$$       0 1 -vc; 
% $$$       0 0  rm];
% $$$   
% $$$ % end of scaling