// Copyright (c) 2005  INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// 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.
//
// $URL$
// $Id$
//
// 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/centroid.h>
#include <CGAL/eigen_2.h>
#include <CGAL/eigen.h>
#include <CGAL/Linear_algebraCd.h>
#include <CGAL/PCA_util.h>

#include <iterator>
#include <list>
#include <cmath>

namespace CGAL {

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::Vector_2 Vector;
  typedef typename K::Iso_rectangle_2 Iso_rectangle;
  typedef typename CGAL::Linear_algebraCd<FT> LA;
  typedef typename LA::Matrix Matrix;

  // 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

} //namespace CGAL

#endif // CGAL_LINEAR_LEAST_SQUARES_FITTING_RECTANGLES_2_H
