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// Copyright (c) 2005 INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org); you may redistribute it under
// the terms of the Q Public License version 1.0.
// See the file LICENSE.QPL distributed with CGAL.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: svn+ssh://gankit@scm.gforge.inria.fr/svn/cgal/trunk/Principal_component_analysis/include/CGAL/linear_least_squares_fitting_rectangles_2.h $
// $Id: linear_least_squares_fitting_2.h 37882 2007-04-03 15:15:30Z spion $
//
// Author(s) : Pierre Alliez and Sylvain Pion and Ankit Gupta
#ifndef CGAL_LINEAR_LEAST_SQUARES_FITTING_RECTANGLES_2_H
#define CGAL_LINEAR_LEAST_SQUARES_FITTING_RECTANGLES_2_H
#include <CGAL/basic.h>
#include <CGAL/Object.h>
#include <CGAL/centroid.h>
#include <CGAL/eigen_2.h>
#include <CGAL/eigen.h>
#include <CGAL/Linear_algebraCd.h>
#include <CGAL/PCA_util.h>
#include <iterator>
#include <vector>
#include <cmath>
CGAL_BEGIN_NAMESPACE
namespace internal {
// Fits a line to a 2D rectangle set.
// Returns a fitting quality (1 - lambda_min/lambda_max):
// 1 is best (zero variance orthogonally to the fitting line);
// 0 is worst (isotropic case, returns a line with horizontal
// direction by default)
template < typename InputIterator, typename K >
typename K::FT
linear_least_squares_fitting_2(InputIterator first,
InputIterator beyond,
typename K::Line_2& line, // best fit line
typename K::Point_2& c, // centroid
const typename K::Iso_rectangle_2*,// used for indirection
const K&, // kernel
const CGAL::Dimension_tag<2>& tag)
{
// types
typedef typename K::FT FT;
typedef typename K::Line_2 Line;
typedef typename K::Point_2 Point;
typedef typename K::Vector_2 Vector;
typedef typename K::Iso_rectangle_2 Iso_rectangle;
typedef typename CGAL::Linear_algebraCd<FT> LA;
typedef typename LA::Matrix Matrix;
typedef typename K::Segment_2 Segment_2;
// precondition: at least one element in the container.
CGAL_precondition(first != beyond);
// compute centroid
c = centroid(first,beyond,K(),tag);
// assemble covariance matrix as a semi-definite matrix.
// Matrix numbering:
// 0
// 1 2
//Final combined covariance matrix for all rectangles and their combined mass
FT mass = 0.0;
FT covariance[3] = {0.0,0.0,0.0};
// assemble 2nd order moment about the origin.
FT temp[4] = {1/3.0, 0.25,
0.25, 1/3.0};
Matrix moment = init_matrix<K>(2,temp);
for(InputIterator it = first;
it != beyond;
it++)
{
// Now for each rectangle, construct the 2nd order moment about the origin.
// assemble the transformation matrix.
const Iso_rectangle& t = *it;
// defined for convenience.
// FT example = CGAL::to_double(t[0].x());
FT x0 = t.xmin();
FT y0 = t.ymin();
FT x1 = t.xmax();
FT y2 = t.ymax();
FT delta[4] = {x1-x0, 0.0,
0.0, y2-y0};
Matrix transformation = init_matrix<K>(2,delta);
FT area = (x1-x0)*(y2-y0);
CGAL_assertion(area != 0.0);
// Find the 2nd order moment for the rectangle wrt to the origin by an affine transformation.
// Transform the standard 2nd order moment using the transformation matrix
transformation = area * transformation * moment * LA::transpose(transformation);
// Translate the 2nd order moment to the center of the rectangle.
FT xav0 = (x1-x0)/2.0;
FT yav0 = (y2-y0)/2.0;
// and add to covariance matrix
covariance[0] += transformation[0][0] + area * (x0*xav0*2 + x0*x0);
covariance[1] += transformation[0][1] + area * (x0*yav0 + xav0*y0 + x0*y0);
covariance[2] += transformation[1][1] + area * (y0*yav0*2 + y0*y0);
mass += area;
}
// Translate the 2nd order moment calculated about the origin to
// the center of mass to get the covariance.
covariance[0] += mass * (-1.0 * c.x() * c.x());
covariance[1] += mass * (-1.0 * c.x() * c.y());
covariance[2] += mass * (-1.0 * c.y() * c.y());
// solve for eigenvalues and eigenvectors.
// eigen values are sorted in descending order,
// eigen vectors are sorted in accordance.
std::pair<FT,FT> eigen_values;
std::pair<Vector,Vector> eigen_vectors;
FT eigen_vectors1[4];
FT eigen_values1[2];
eigen_symmetric<FT>(covariance,2, eigen_vectors1, eigen_values1);
eigen_values = std::make_pair(eigen_values1[0],eigen_values1[1]);
eigen_vectors = std::make_pair(Vector(eigen_vectors1[0],eigen_vectors1[1]),Vector(eigen_vectors1[2],eigen_vectors1[3]));
// check unicity and build fitting line accordingly
if(eigen_values.first != eigen_values.second)
{
// regular case
line = Line(c, eigen_vectors.first);
return (FT)1.0 - eigen_values.second / eigen_values.first;
}
else
{
// isotropic case (infinite number of directions)
// by default: assemble a line that goes through
// the centroid and with a default horizontal vector.
line = Line(c, Vector(1.0, 0.0));
return (FT)0.0;
}
} // end linear_least_squares_fitting_2 for rectangle set with 2D tag
template < typename InputIterator, typename K >
typename K::FT
linear_least_squares_fitting_2(InputIterator first,
InputIterator beyond,
typename K::Line_2& line, // best fit line
typename K::Point_2& c, // centroid
const typename K::Iso_rectangle_2*,// used for indirection
const K&, // kernel
const CGAL::Dimension_tag<1>& tag)
{
// types
typedef typename K::Iso_rectangle_2 Iso_rectangle;
typedef typename K::Segment_2 Segment_2;
// precondition: at least one element in the container.
CGAL_precondition(first != beyond);
std::list<Segment_2> segments;
for(InputIterator it = first;
it != beyond;
it++)
{
const Iso_rectangle& t = *it;
segments.push_back(Segment_2(t[0],t[1]));
segments.push_back(Segment_2(t[1],t[2]));
segments.push_back(Segment_2(t[2],t[3]));
segments.push_back(Segment_2(t[3],t[0]));
}
return linear_least_squares_fitting_2(segments.begin(),segments.end(),line,c,K(),tag);
} // end linear_least_squares_fitting_2 for rectangle set with 1D tag
template < typename InputIterator,
typename K >
typename K::FT
linear_least_squares_fitting_2(InputIterator first,
InputIterator beyond,
typename K::Line_2& line, // best fit line
typename K::Point_2& c, // centroid
const typename K::Iso_rectangle_2*,// used for indirection
const K&, // kernel
const CGAL::Dimension_tag<0>& tag)
{
// types
typedef typename K::Iso_rectangle_2 Iso_rectangle;
typedef typename K::Point_2 Point_2;
// precondition: at least one element in the container.
CGAL_precondition(first != beyond);
std::list<Point_2> points;
for(InputIterator it = first;
it != beyond;
it++)
{
const Iso_rectangle& t = *it;
points.push_back(Point_2(t[0]));
points.push_back(Point_2(t[1]));
points.push_back(Point_2(t[2]));
points.push_back(Point_2(t[3]));
}
return linear_least_squares_fitting_2(points.begin(),points.end(),line,c,K(),tag);
} // end linear_least_squares_fitting_2 for rectangle set with 0D tag
} // end namespace internal
CGAL_END_NAMESPACE
#endif // CGAL_LINEAR_LEAST_SQUARES_FITTING_RECTANGLES_2_H
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