% NORMALISE2DPTS - normalises 2D homogeneous points
%
% Function translates and normalises a set of 2D homogeneous points
% so that their centroid is at the origin and their mean distance from
% the origin is sqrt(2). This process typically improves the
% conditioning of any equations used to solve homographies, fundamental
% matrices etc.
%
% Usage: [newpts, T] = normalise2dpts(pts)
%
% Argument:
% pts - 3xN array of 2D homogeneous coordinates
%
% Returns:
% newpts - 3xN array of transformed 2D homogeneous coordinates. The
% scaling parameter is normalised to 1 unless the point is at
% infinity.
% T - The 3x3 transformation matrix, newpts = T*pts
%
% If there are some points at infinity the normalisation transform
% is calculated using just the finite points. Being a scaling and
% translating transform this will not affect the points at infinity.
% Peter Kovesi
% School of Computer Science & Software Engineering
% The University of Western Australia
% pk at csse uwa edu au
% http://www.csse.uwa.edu.au/~pk
%
% May 2003 - Original version
% February 2004 - Modified to deal with points at infinity.
% December 2008 - meandist calculation modified to work with Octave 3.0.1
% (thanks to Ron Parr)
function [newpts, T] = normalise2dpts(pts)
if size(pts,1) ~= 3
error('pts must be 3xN');
end
% Find the indices of the points that are not at infinity
finiteind = find(abs(pts(3,:)) > eps);
if length(finiteind) ~= size(pts,2)
warning('Some points are at infinity');
end
% For the finite points ensure homogeneous coords have scale of 1
pts(1,finiteind) = pts(1,finiteind)./pts(3,finiteind);
pts(2,finiteind) = pts(2,finiteind)./pts(3,finiteind);
pts(3,finiteind) = 1;
c = mean(pts(1:2,finiteind)')'; % Centroid of finite points
newp(1,finiteind) = pts(1,finiteind)-c(1); % Shift origin to centroid.
newp(2,finiteind) = pts(2,finiteind)-c(2);
dist = sqrt(newp(1,finiteind).^2 + newp(2,finiteind).^2);
meandist = mean(dist(:)); % Ensure dist is a column vector for Octave 3.0.1
scale = sqrt(2)/meandist;
T = [scale 0 -scale*c(1)
0 scale -scale*c(2)
0 0 1 ];
newpts = T*pts;