function [lre] = LRE(I, phi)
% 
% Written by:
% -- 
% John L. Weatherwax                2005-08-04
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
% Epage 
% 
%-----

if( nargin>1 ) % compute the LRE for just one displacement and angular direction

  Q         = Q_RL(I,phi);
  [N_g,N_r] = size(Q);
  J2        = repmat( 1:N_r, [N_g,1] ).^2;
  tmp       = Q .* J2;
  num       = sum(tmp(:));
  den       = sum(Q(:));
  
  lre       = num/den;
  
else % compute the LRE for all angles and average 
  
  all_phis = [ 0, 45, 90, 135 ];
  all_lres = zeros(1,length(all_phis));
  for ii=1:length(all_phis)
    
    a = LRE(I,all_phis(ii)); % recursively call the three argument case  
    all_lres(ii) = a;
    
  end
  
  lre = mean(all_lres);
  
end