fitAffineTransform2d

 Compute the affine transform that best register two point sets.

   TRANSFO = fitAffineTransform2d(REF, SRC)
   Returns the affine transform matrix that minimizes the distance between
   the reference point set REF and the point set SRC after transformation.
   Both REF and SRC must by N-by-2 arrays with the same number of rows,
   and the points must be in correspondence.
   The function minimizes the sum of the squared distances:
   CRIT = sum(distancePoints(REF, transformPoint(PTS, TRANSFO)).^2);

   Example
     % computes the transform the register two ellipses
     % create the reference poitn set
     elli = [50 50 40 20 30];
     poly = resamplePolygonByLength(ellipseToPolygon(elli, 200), 5);
     figure; axis equal; axis([0 100 0 100]); hold on;
     drawPoint(poly, 'kx')
     % create the point set to fit on the reference
     trans0 = createRotation([20 60], -pi/8);
     poly2 = transformPoint(poly, trans0);
     poly2 = poly2 + randn(size(poly)) * 2;
     drawPoint(poly2, 'b+');
     % compute the transform that project poly2 onto poly.
     transfo = fitAffineTransform2d(poly, poly2);
     poly2t = transformPoint(poly2, transfo);
     drawPoint(poly2t, 'mo')
     legend('Reference', 'Initial', 'Transformed');

   See also
     transforms2d, transformPoint, transformVector,
     fitPolynomialTransform2d, registerICP, fitAffineTransform3d

0001 function trans = fitAffineTransform2d(ref, src)
0002 % Compute the affine transform that best register two point sets.
0003 %
0004 %   TRANSFO = fitAffineTransform2d(REF, SRC)
0005 %   Returns the affine transform matrix that minimizes the distance between
0006 %   the reference point set REF and the point set SRC after transformation.
0007 %   Both REF and SRC must by N-by-2 arrays with the same number of rows,
0008 %   and the points must be in correspondence.
0009 %   The function minimizes the sum of the squared distances:
0010 %   CRIT = sum(distancePoints(REF, transformPoint(PTS, TRANSFO)).^2);
0011 %
0012 %   Example
0013 %     % computes the transform the register two ellipses
0014 %     % create the reference poitn set
0015 %     elli = [50 50 40 20 30];
0016 %     poly = resamplePolygonByLength(ellipseToPolygon(elli, 200), 5);
0017 %     figure; axis equal; axis([0 100 0 100]); hold on;
0018 %     drawPoint(poly, 'kx')
0019 %     % create the point set to fit on the reference
0020 %     trans0 = createRotation([20 60], -pi/8);
0021 %     poly2 = transformPoint(poly, trans0);
0022 %     poly2 = poly2 + randn(size(poly)) * 2;
0023 %     drawPoint(poly2, 'b+');
0024 %     % compute the transform that project poly2 onto poly.
0025 %     transfo = fitAffineTransform2d(poly, poly2);
0026 %     poly2t = transformPoint(poly2, transfo);
0027 %     drawPoint(poly2t, 'mo')
0028 %     legend('Reference', 'Initial', 'Transformed');
0029 %
0030 %   See also
0031 %     transforms2d, transformPoint, transformVector,
0032 %     fitPolynomialTransform2d, registerICP, fitAffineTransform3d
0033 %
0034 
0035 % ------
0036 % Author: David Legland
0037 % e-mail: david.legland@inrae.fr
0038 % Created: 2009-07-31,    using Matlab 7.7.0.471 (R2008b)
0039 % Copyright 2009 INRAE - Cepia Software Platform.
0040 
0041 % check number of points are equal
0042 N = size(src, 1);
0043 if size(ref, 1) ~= N
0044     error('Requires the same number of points for both arrays');
0045 end
0046 
0047 % main matrix of the problem
0048 A = [...
0049     src(:,1) src(:,2) ones(N,1) zeros(N, 3) ; ...
0050     zeros(N, 3) src(:,1) src(:,2) ones(N,1)  ];
0051 
0052 % conditions initialisations
0053 B = [ref(:,1) ; ref(:,2)];
0054 
0055 % compute coefficients using least square
0056 coefs = A\B;
0057 
0058 % format to a matrix
0059 trans = [coefs(1:3)' ; coefs(4:6)'; 0 0 1];