libcgal-dev 4.11-2build1 / usr / include / CGAL / Poisson_reconstruction_function.h

// Copyright (c) 2007-09  INRIA (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)     : Laurent Saboret, Pierre Alliez

#ifndef CGAL_POISSON_RECONSTRUCTION_FUNCTION_H
#define CGAL_POISSON_RECONSTRUCTION_FUNCTION_H

#include <CGAL/license/Poisson_surface_reconstruction_3.h>


#ifndef CGAL_DIV_NORMALIZED
#  ifndef CGAL_DIV_NON_NORMALIZED
#    define CGAL_DIV_NON_NORMALIZED 1
#  endif
#endif

#include <vector>
#include <deque>
#include <algorithm>
#include <cmath>
#include <iterator>

#include <CGAL/trace.h>
#include <CGAL/Reconstruction_triangulation_3.h>
#include <CGAL/spatial_sort.h>
#ifdef CGAL_EIGEN3_ENABLED
#include <CGAL/Eigen_solver_traits.h>
#else
#endif
#include <CGAL/centroid.h>
#include <CGAL/property_map.h>
#include <CGAL/surface_reconstruction_points_assertions.h>
#include <CGAL/poisson_refine_triangulation.h>
#include <CGAL/Robust_circumcenter_filtered_traits_3.h>
#include <CGAL/compute_average_spacing.h>
#include <CGAL/Timer.h>

#include <boost/shared_ptr.hpp>
#include <boost/array.hpp>
#include <boost/type_traits/is_convertible.hpp>
#include <boost/utility/enable_if.hpp>

/*! 
  \file Poisson_reconstruction_function.h
*/

namespace CGAL {

  namespace internal {
template <class RT>
bool
invert(
       const RT& a0,  const RT& a1,  const RT& a2,
       const RT& a3,  const RT& a4,  const RT& a5,
       const RT& a6,  const RT& a7,  const RT& a8,
       RT& i0,   RT& i1,   RT& i2,
       RT& i3,   RT& i4,   RT& i5,
       RT& i6,   RT& i7,   RT& i8)
{
    // Compute the adjoint.
    i0 = a4*a8 - a5*a7;
    i1 = a2*a7 - a1*a8;
    i2 = a1*a5 - a2*a4;
    i3 = a5*a6 - a3*a8;
    i4 = a0*a8 - a2*a6;
    i5 = a2*a3 - a0*a5;
    i6 = a3*a7 - a4*a6;
    i7 = a1*a6 - a0*a7;
    i8 = a0*a4 - a1*a3;

    RT det = a0*i0 + a1*i3 + a2*i6;

    if(det != 0) {
      RT idet = (RT(1.0))/det;
      i0 *= idet;
      i1 *= idet;
      i2 *= idet;
      i3 *= idet;
      i4 *= idet;
      i5 *= idet;
      i6 *= idet;
      i7 *= idet;
      i8 *= idet;
      return true;
    }

    return false;
}

  }


/// \cond SKIP_IN_MANUAL
struct Poisson_visitor {
  void before_insertion() const
  {}
};

struct Poisson_skip_vertices { 
  double ratio;
  Random& m_random;
  Poisson_skip_vertices(const double ratio, Random& random)
    : ratio(ratio), m_random(random) {}

  template <typename Iterator>
  bool operator()(Iterator) const {
    return m_random.get_double() < ratio;
  }
};

// Given f1 and f2, two sizing fields, that functor wrapper returns
//   max(f1, f2*f2)
// The wrapper stores only pointers to the two functors.
template <typename F1, typename F2>
struct Special_wrapper_of_two_functions_keep_pointers {
  F1 *f1;
  F2 *f2;
  Special_wrapper_of_two_functions_keep_pointers(F1* f1, F2* f2) 
    : f1(f1), f2(f2) {}

  template <typename X>
  double operator()(const X& x) const {
    return (std::max)((*f1)(x), CGAL::square((*f2)(x)));
  }

  template <typename X>
  double operator()(const X& x) {
    return (std::max)((*f1)(x), CGAL::square((*f2)(x)));
  }
}; // end struct Special_wrapper_of_two_functions_keep_pointers<F1, F2>
/// \endcond 


/*!
\ingroup PkgPoissonSurfaceReconstruction

\brief Implementation of the Poisson Surface Reconstruction method.
  
Given a set of 3D points with oriented normals sampled on the boundary
of a 3D solid, the Poisson Surface Reconstruction method \cgalCite{Kazhdan06} 
solves for an approximate indicator function of the inferred
solid, whose gradient best matches the input normals. The output
scalar function, represented in an adaptive octree, is then
iso-contoured using an adaptive marching cubes.

`Poisson_reconstruction_function` implements a variant of this
algorithm which solves for a piecewise linear function on a 3D
Delaunay triangulation instead of an adaptive octree.

\tparam Gt Geometric traits class. 

\cgalModels `ImplicitFunction`

*/
template <class Gt>
class Poisson_reconstruction_function
{
// Public types
public:

  /// \name Types 
  /// @{

  typedef Gt Geom_traits; ///< Geometric traits class
  /// \cond SKIP_IN_MANUAL
  typedef Reconstruction_triangulation_3<Robust_circumcenter_filtered_traits_3<Gt> >
                                                   Triangulation;
  /// \endcond
  typedef typename Triangulation::Cell_handle   Cell_handle;

  // Geometric types
  typedef typename Geom_traits::FT FT; ///< number type.
  typedef typename Geom_traits::Point_3 Point; ///< point type.
  typedef typename Geom_traits::Vector_3 Vector; ///< vector type.
  typedef typename Geom_traits::Sphere_3 Sphere; 

  /// @}

// Private types
private:

  // Internal 3D triangulation, of type Reconstruction_triangulation_3.
  // Note: poisson_refine_triangulation() requires a robust circumcenter computation.

  // Repeat Triangulation types
  typedef typename Triangulation::Triangulation_data_structure Triangulation_data_structure;
  typedef typename Geom_traits::Ray_3 Ray;
  typedef typename Geom_traits::Plane_3 Plane;
  typedef typename Geom_traits::Segment_3 Segment;
  typedef typename Geom_traits::Triangle_3 Triangle;
  typedef typename Geom_traits::Tetrahedron_3 Tetrahedron;
  typedef typename Triangulation::Vertex_handle Vertex_handle;
  typedef typename Triangulation::Cell   Cell;
  typedef typename Triangulation::Vertex Vertex;
  typedef typename Triangulation::Facet  Facet;
  typedef typename Triangulation::Edge   Edge;
  typedef typename Triangulation::Cell_circulator  Cell_circulator;
  typedef typename Triangulation::Facet_circulator Facet_circulator;
  typedef typename Triangulation::Cell_iterator    Cell_iterator;
  typedef typename Triangulation::Facet_iterator   Facet_iterator;
  typedef typename Triangulation::Edge_iterator    Edge_iterator;
  typedef typename Triangulation::Vertex_iterator  Vertex_iterator;
  typedef typename Triangulation::Point_iterator   Point_iterator;
  typedef typename Triangulation::Finite_vertices_iterator Finite_vertices_iterator;
  typedef typename Triangulation::Finite_cells_iterator    Finite_cells_iterator;
  typedef typename Triangulation::Finite_facets_iterator   Finite_facets_iterator;
  typedef typename Triangulation::Finite_edges_iterator    Finite_edges_iterator;
  typedef typename Triangulation::All_cells_iterator       All_cells_iterator;
  typedef typename Triangulation::Locate_type Locate_type;

// Data members.
// Warning: the Surface Mesh Generation package makes copies of implicit functions,
// thus this class must be lightweight and stateless.
private:

  // operator() is pre-computed on vertices of *m_tr by solving
  // the Poisson equation Laplacian(f) = divergent(normals field).
  boost::shared_ptr<Triangulation> m_tr;

  mutable boost::shared_ptr<std::vector<boost::array<double,9> > > m_Bary;
  mutable std::vector<Point> Dual;
  mutable std::vector<Vector> Normal;

  // contouring and meshing
  Point m_sink; // Point with the minimum value of operator()
  mutable Cell_handle m_hint; // last cell found = hint for next search

  FT average_spacing;


  /// function to be used for the different constructors available that are
  /// doing the same thing but with default template parameters
  template <typename InputIterator,
            typename PointPMap,
            typename NormalPMap,
            typename Visitor
  >
  void forward_constructor(
    InputIterator first,
    InputIterator beyond,
    PointPMap point_pmap,
    NormalPMap normal_pmap,
    Visitor visitor)
  {
    CGAL::Timer task_timer; task_timer.start();
    CGAL_TRACE_STREAM << "Creates Poisson triangulation...\n";

    // Inserts points in triangulation
    m_tr->insert(
      first,beyond,
      point_pmap,
      normal_pmap,
      visitor);

    // Prints status
    CGAL_TRACE_STREAM << "Creates Poisson triangulation: " << task_timer.time() << " seconds, "
                                                           << std::endl;
  }


// Public methods
public:

  /// \name Creation 
  /// @{


  /*! 
    Creates a Poisson implicit function from the  range of points `[first, beyond)`. 

    \tparam InputIterator iterator over input points. 

    \tparam PointPMap is a model of `ReadablePropertyMap` with
      a `value_type = Point`.  It can be omitted if `InputIterator`
      `value_type` is convertible to `Point`. 
    
    \tparam NormalPMap is a model of `ReadablePropertyMap`
      with a `value_type = Vector`.
  */ 
  template <typename InputIterator,
            typename PointPMap,
            typename NormalPMap
  >
  Poisson_reconstruction_function(
    InputIterator first,  ///< iterator over the first input point.
    InputIterator beyond, ///< past-the-end iterator over the input points.
    PointPMap point_pmap, ///< property map: `value_type of InputIterator` -> `Point` (the position of an input point).
    NormalPMap normal_pmap ///< property map: `value_type of InputIterator` -> `Vector` (the *oriented* normal of an input point).
  )
    : m_tr(new Triangulation), m_Bary(new std::vector<boost::array<double,9> > )
    , average_spacing(CGAL::compute_average_spacing<CGAL::Sequential_tag>(first, beyond, point_pmap, 6))
  {
    forward_constructor(first, beyond, point_pmap, normal_pmap, Poisson_visitor());
  }

  /// \cond SKIP_IN_MANUAL
  template <typename InputIterator,
            typename PointPMap,
            typename NormalPMap,
            typename Visitor
  >
  Poisson_reconstruction_function(
    InputIterator first,  ///< iterator over the first input point.
    InputIterator beyond, ///< past-the-end iterator over the input points.
    PointPMap point_pmap, ///< property map: `value_type of InputIterator` -> `Point` (the position of an input point).
    NormalPMap normal_pmap, ///< property map: `value_type of InputIterator` -> `Vector` (the *oriented* normal of an input point).
    Visitor visitor)
    : m_tr(new Triangulation), m_Bary(new std::vector<boost::array<double,9> > )
    , average_spacing(CGAL::compute_average_spacing<CGAL::Sequential_tag>(first, beyond, point_pmap, 6))
  {
    forward_constructor(first, beyond, point_pmap, normal_pmap, visitor);
  }

  // This variant creates a default point property map = Identity_property_map and Visitor=Poisson_visitor
  template <typename InputIterator,
            typename NormalPMap
  >
  Poisson_reconstruction_function(
    InputIterator first,  ///< iterator over the first input point.
    InputIterator beyond, ///< past-the-end iterator over the input points.
    NormalPMap normal_pmap, ///< property map: `value_type of InputIterator` -> `Vector` (the *oriented* normal of an input point).
    typename boost::enable_if<
      boost::is_convertible<typename std::iterator_traits<InputIterator>::value_type, Point>
    >::type* = 0
  )
  : m_tr(new Triangulation), m_Bary(new std::vector<boost::array<double,9> > )
  , average_spacing(CGAL::compute_average_spacing<CGAL::Sequential_tag>(first, beyond, 6))
  {
    forward_constructor(first, beyond, 
      make_identity_property_map(
      typename std::iterator_traits<InputIterator>::value_type()),
      normal_pmap, Poisson_visitor());
    CGAL::Timer task_timer; task_timer.start();
  }
  /// \endcond

  /// @}

  /// \name Operations
  /// @{

  /// Returns a sphere bounding the inferred surface.
  Sphere bounding_sphere() const
  {
    return m_tr->bounding_sphere();
  }
  
  /// \cond SKIP_IN_MANUAL
  const Triangulation& tr() const {
    return *m_tr;
  }
  
  // This variant requires all parameters.
  template <class SparseLinearAlgebraTraits_d,
            class Visitor>
  bool compute_implicit_function(
                                 SparseLinearAlgebraTraits_d solver,// = SparseLinearAlgebraTraits_d(),
                                 Visitor visitor,
                                 double approximation_ratio = 0,
                                 double average_spacing_ratio = 5) 
  {
    CGAL::Timer task_timer; task_timer.start();
    CGAL_TRACE_STREAM << "Delaunay refinement...\n";

    // Delaunay refinement
    const FT radius_edge_ratio_bound = 2.5;
    const unsigned int max_vertices = (unsigned int)1e7; // max 10M vertices
    const FT enlarge_ratio = 1.5;
    const FT radius = sqrt(bounding_sphere().squared_radius()); // get triangulation's radius
    const FT cell_radius_bound = radius/5.; // large

    internal::Poisson::Constant_sizing_field<Triangulation> sizing_field(CGAL::square(cell_radius_bound));

    std::vector<int> NB; 

    NB.push_back( delaunay_refinement(radius_edge_ratio_bound,sizing_field,max_vertices,enlarge_ratio));

    while(m_tr->insert_fraction(visitor)){

      NB.push_back( delaunay_refinement(radius_edge_ratio_bound,sizing_field,max_vertices,enlarge_ratio));
    }

    if(approximation_ratio > 0. && 
       approximation_ratio * std::distance(m_tr->input_points_begin(),
                                           m_tr->input_points_end()) > 20) {

      // Add a pass of Delaunay refinement.
      //
      // In that pass, the sizing field, of the refinement process of the
      // triangulation, is based on the result of a poisson function with a
      // sample of the input points. The ratio is 'approximation_ratio'.
      //
      // For optimization reasons, the cell criteria of the refinement
      // process uses two sizing fields:
      //
      //   - the minimum of the square of 'coarse_poisson_function' and the
      // square of the constant field equal to 'average_spacing',
      //
      //   - a second sizing field that is constant, and equal to:
      //
      //         average_spacing*average_spacing_ratio
      //
      // If a given cell is smaller than the constant second sizing field,
      // then the cell is considered as small enough, and the first sizing
      // field, more costly, is not evaluated.

      typedef Filter_iterator<typename Triangulation::Input_point_iterator,
                              Poisson_skip_vertices> Some_points_iterator;
      //make it deterministic
      Random random(0);
      Poisson_skip_vertices skip(1.-approximation_ratio,random);
      
      CGAL_TRACE_STREAM << "SPECIAL PASS that uses an approximation of the result (approximation ratio: "
                << approximation_ratio << ")" << std::endl;
      CGAL::Timer approximation_timer; approximation_timer.start();

      CGAL::Timer sizing_field_timer; sizing_field_timer.start();
      Poisson_reconstruction_function<Geom_traits> 
        coarse_poisson_function(Some_points_iterator(m_tr->input_points_end(),
                                                     skip,
                                                     m_tr->input_points_begin()),
                                Some_points_iterator(m_tr->input_points_end(),
                                                     skip),
                                Normal_of_point_with_normal_pmap<Geom_traits>() );
      coarse_poisson_function.compute_implicit_function(solver, Poisson_visitor(),
                                                        0.);
      internal::Poisson::Constant_sizing_field<Triangulation> 
        min_sizing_field(CGAL::square(average_spacing));
      internal::Poisson::Constant_sizing_field<Triangulation> 
        sizing_field_ok(CGAL::square(average_spacing*average_spacing_ratio));

      Special_wrapper_of_two_functions_keep_pointers<
        internal::Poisson::Constant_sizing_field<Triangulation>,
        Poisson_reconstruction_function<Geom_traits> > sizing_field2(&min_sizing_field,
                                                                     &coarse_poisson_function);
        
      sizing_field_timer.stop();
      std::cerr << "Construction time of the sizing field: " << sizing_field_timer.time() 
                << " seconds" << std::endl;

      NB.push_back( delaunay_refinement(radius_edge_ratio_bound,
                                        sizing_field2,
                                        max_vertices,
                                        enlarge_ratio,
                                        sizing_field_ok) );
      approximation_timer.stop();
      CGAL_TRACE_STREAM << "SPECIAL PASS END (" << approximation_timer.time() <<  " seconds)" << std::endl;
    }

    
    // Prints status
    CGAL_TRACE_STREAM << "Delaunay refinement: " << "added ";
    for(std::size_t i = 0; i < NB.size()-1; i++){
      CGAL_TRACE_STREAM << NB[i] << " + "; 
    } 
    CGAL_TRACE_STREAM << NB.back() << " Steiner points, "
                      << task_timer.time() << " seconds, "
                      << std::endl;
    task_timer.reset();

#ifdef CGAL_DIV_NON_NORMALIZED
    CGAL_TRACE_STREAM << "Solve Poisson equation with non-normalized divergence...\n";
#else
    CGAL_TRACE_STREAM << "Solve Poisson equation with normalized divergence...\n";
#endif

    // Computes the Poisson indicator function operator()
    // at each vertex of the triangulation.
    double lambda = 0.1;
    if ( ! solve_poisson(solver, lambda) )
    {
      std::cerr << "Error: cannot solve Poisson equation" << std::endl;
      return false;
    }

    // Shift and orient operator() such that:
    // - operator() = 0 on the input points,
    // - operator() < 0 inside the surface.
    set_contouring_value(median_value_at_input_vertices());

    // Prints status
    CGAL_TRACE_STREAM << "Solve Poisson equation: " << task_timer.time() << " seconds, "
                                                    << std::endl;
    task_timer.reset();

    return true;
  }
  /// \endcond

  /*!
    This function must be called after the
    insertion of oriented points. It computes the piecewise linear scalar
    function operator() by: applying Delaunay refinement, solving for
    operator() at each vertex of the triangulation with a sparse linear
    solver, and shifting and orienting operator() such that it is 0 at all
    input points and negative inside the inferred surface.

    \tparam SparseLinearAlgebraTraits_d Symmetric definite positive sparse linear solver.
    If \ref thirdpartyEigen "Eigen" 3.1 (or greater) is available and `CGAL_EIGEN3_ENABLED`
    is defined, an overload with \link Eigen_solver_traits <tt>Eigen_solver_traits<Eigen::ConjugateGradient<Eigen_sparse_symmetric_matrix<double>::EigenType> ></tt> \endlink
    as default solver is provided.
  
    \param solver sparse linear solver.
    \param smoother_hole_filling controls if the Delaunay refinement is done for the input points, or for an approximation of the surface obtained from a first pass of the algorithm on a sample of the points.

    \return `false` if the linear solver fails. 
  */ 
  template <class SparseLinearAlgebraTraits_d>
  bool compute_implicit_function(SparseLinearAlgebraTraits_d solver, bool smoother_hole_filling = false)
  {
    if (smoother_hole_filling)
      return compute_implicit_function<SparseLinearAlgebraTraits_d,Poisson_visitor>(solver,Poisson_visitor(),0.02,5);
    else
      return compute_implicit_function<SparseLinearAlgebraTraits_d,Poisson_visitor>(solver,Poisson_visitor());
  }

  /// \cond SKIP_IN_MANUAL
#ifdef CGAL_EIGEN3_ENABLED
  // This variant provides the default sparse linear traits class = Eigen_solver_traits.
  bool compute_implicit_function(bool smoother_hole_filling = false)
  {
    typedef Eigen_solver_traits<Eigen::ConjugateGradient<Eigen_sparse_symmetric_matrix<double>::EigenType> > Solver;
    return compute_implicit_function<Solver>(Solver(), smoother_hole_filling);
  }
#endif

  boost::tuple<FT, Cell_handle, bool> special_func(const Point& p) const
  {
    m_hint = m_tr->locate(p  ,m_hint  ); // no hint when we use hierarchy

    if(m_tr->is_infinite(m_hint)) {
      int i = m_hint->index(m_tr->infinite_vertex());
      return boost::make_tuple(m_hint->vertex((i+1)&3)->f(),
                               m_hint, true);
    }

    FT a,b,c,d;
    barycentric_coordinates(p,m_hint,a,b,c,d);
    return boost::make_tuple(a * m_hint->vertex(0)->f() +
                             b * m_hint->vertex(1)->f() +
                             c * m_hint->vertex(2)->f() +
                             d * m_hint->vertex(3)->f(),
                             m_hint, false);
  }
  /// \endcond

  /*! 
    `ImplicitFunction` interface: evaluates the implicit function at a 
    given 3D query point. The function `compute_implicit_function()` must be 
    called before the first call to `operator()`. 
  */ 
  FT operator()(const Point& p) const
  {
    m_hint = m_tr->locate(p ,m_hint); 

    if(m_tr->is_infinite(m_hint)) {
      int i = m_hint->index(m_tr->infinite_vertex());
      return m_hint->vertex((i+1)&3)->f();
    }

    FT a,b,c,d;
    barycentric_coordinates(p,m_hint,a,b,c,d);
    return a * m_hint->vertex(0)->f() +
           b * m_hint->vertex(1)->f() +
           c * m_hint->vertex(2)->f() +
           d * m_hint->vertex(3)->f();
  }
  
  /// \cond SKIP_IN_MANUAL
  void initialize_cell_indices()
  {
    int i=0;
    for(Finite_cells_iterator fcit = m_tr->finite_cells_begin();
        fcit != m_tr->finite_cells_end();
        ++fcit){
      fcit->info()= i++;
    }
  }

  void initialize_barycenters() const
  {
    m_Bary->resize(m_tr->number_of_cells());

    for(std::size_t i=0; i< m_Bary->size();i++){
      (*m_Bary)[i][0]=-1;
    }
  }

  void initialize_cell_normals() const
  {
    Normal.resize(m_tr->number_of_cells());
    int i = 0;
    int N = 0;
    for(Finite_cells_iterator fcit = m_tr->finite_cells_begin();
        fcit != m_tr->finite_cells_end();
        ++fcit){
      Normal[i] = cell_normal(fcit);
      if(Normal[i] == NULL_VECTOR){
        N++;
      }
      ++i;
    }
    std::cerr << N << " out of " << i << " cells have NULL_VECTOR as normal" << std::endl;
  }

  void initialize_duals() const
  {
    Dual.resize(m_tr->number_of_cells());    
    int i = 0;
    for(Finite_cells_iterator fcit = m_tr->finite_cells_begin();
        fcit != m_tr->finite_cells_end();
        ++fcit){
      Dual[i++] = m_tr->dual(fcit);
    }
  }

  void clear_duals() const
  {
    Dual.clear();
  }

  void clear_normals() const
  {
    Normal.clear();
  }

  void initialize_matrix_entry(Cell_handle ch) const
  {
    boost::array<double,9> & entry = (*m_Bary)[ch->info()];
    const Point& pa = ch->vertex(0)->point();
    const Point& pb = ch->vertex(1)->point();
    const Point& pc = ch->vertex(2)->point();
    const Point& pd = ch->vertex(3)->point();
    
    Vector va = pa - pd;
    Vector vb = pb - pd;
    Vector vc = pc - pd;
    
    internal::invert(va.x(), va.y(), va.z(),
           vb.x(), vb.y(), vb.z(),
           vc.x(), vc.y(), vc.z(),
           entry[0],entry[1],entry[2],entry[3],entry[4],entry[5],entry[6],entry[7],entry[8]);
  }
  /// \endcond
  
  /// Returns a point located inside the inferred surface.
  Point get_inner_point() const
  {
    // Gets point / the implicit function is minimum
    return m_sink;
  }

  /// @}

// Private methods:
private:

  /// Delaunay refinement (break bad tetrahedra, where
  /// bad means badly shaped or too big). The normal of
  /// Steiner points is set to zero.
  /// Returns the number of vertices inserted.

  template <typename Sizing_field>
  unsigned int delaunay_refinement(FT radius_edge_ratio_bound, ///< radius edge ratio bound (ignored if zero)
                                   Sizing_field sizing_field, ///< cell radius bound (ignored if zero)
                                   unsigned int max_vertices, ///< number of vertices bound
                                   FT enlarge_ratio) ///< bounding box enlarge ratio
  {
    return delaunay_refinement(radius_edge_ratio_bound,
                               sizing_field,
                               max_vertices,
                               enlarge_ratio,
                               internal::Poisson::Constant_sizing_field<Triangulation>());
  }

  template <typename Sizing_field, 
            typename Second_sizing_field>
  unsigned int delaunay_refinement(FT radius_edge_ratio_bound, ///< radius edge ratio bound (ignored if zero)
                                   Sizing_field sizing_field, ///< cell radius bound (ignored if zero)
                                   unsigned int max_vertices, ///< number of vertices bound
                                   FT enlarge_ratio, ///< bounding box enlarge ratio
                                   Second_sizing_field second_sizing_field)
  {
    Sphere elarged_bsphere = enlarged_bounding_sphere(enlarge_ratio);
    unsigned int nb_vertices_added = poisson_refine_triangulation(*m_tr,radius_edge_ratio_bound,sizing_field,second_sizing_field,max_vertices,elarged_bsphere);

    return nb_vertices_added;
  }

  /// Poisson reconstruction.
  /// Returns false on error.
  ///
  /// @commentheading Template parameters:
  /// @param SparseLinearAlgebraTraits_d Symmetric definite positive sparse linear solver.
  template <class SparseLinearAlgebraTraits_d>
  bool solve_poisson(
    SparseLinearAlgebraTraits_d solver, ///< sparse linear solver
    double lambda)
  {
    CGAL_TRACE("Calls solve_poisson()\n");

    double time_init = clock();

    double duration_assembly = 0.0;
    double duration_solve = 0.0;


    initialize_cell_indices();
    initialize_barycenters();

    // get #variables
    constrain_one_vertex_on_convex_hull();
    m_tr->index_unconstrained_vertices();
    unsigned int nb_variables = static_cast<unsigned int>(m_tr->number_of_vertices()-1);

    CGAL_TRACE("  Number of variables: %ld\n", (long)(nb_variables));

    // Assemble linear system A*X=B
    typename SparseLinearAlgebraTraits_d::Matrix A(nb_variables); // matrix is symmetric definite positive
    typename SparseLinearAlgebraTraits_d::Vector X(nb_variables), B(nb_variables);

    initialize_duals();
#ifndef CGAL_DIV_NON_NORMALIZED
    initialize_cell_normals();
#endif
    Finite_vertices_iterator v, e;
    for(v = m_tr->finite_vertices_begin(),
        e = m_tr->finite_vertices_end();
        v != e;
        ++v)
    {
      if(!m_tr->is_constrained(v)) {
#ifdef CGAL_DIV_NON_NORMALIZED
        B[v->index()] = div(v); // rhs -> divergent
#else // not defined(CGAL_DIV_NORMALIZED)
        B[v->index()] = div_normalized(v); // rhs -> divergent
#endif // not defined(CGAL_DIV_NORMALIZED)
        assemble_poisson_row<SparseLinearAlgebraTraits_d>(A,v,B,lambda);
      }
    }

    clear_duals();
    clear_normals();
    duration_assembly = (clock() - time_init)/CLOCKS_PER_SEC;
    CGAL_TRACE("  Creates matrix: done (%.2lf s)\n", duration_assembly);

    CGAL_TRACE("  Solve sparse linear system...\n");

    // Solve "A*X = B". On success, solution is (1/D) * X.
    time_init = clock();
    double D;
    if(!solver.linear_solver(A, B, X, D))
      return false;
    CGAL_surface_reconstruction_points_assertion(D == 1.0);
    duration_solve = (clock() - time_init)/CLOCKS_PER_SEC;

    CGAL_TRACE("  Solve sparse linear system: done (%.2lf s)\n", duration_solve);

    // copy function's values to vertices
    unsigned int index = 0;
    for (v = m_tr->finite_vertices_begin(), e = m_tr->finite_vertices_end(); v!= e; ++v)
      if(!m_tr->is_constrained(v))
        v->f() = X[index++];

    CGAL_TRACE("End of solve_poisson()\n");

    return true;
  }

  /// Shift and orient the implicit function such that:
  /// - the implicit function = 0 for points / f() = contouring_value,
  /// - the implicit function < 0 inside the surface.
  ///
  /// Returns the minimum value of the implicit function.
  FT set_contouring_value(FT contouring_value)
  {
    // median value set to 0.0
    shift_f(-contouring_value);

     // Check value on convex hull (should be positive): if more than
    // half the vertices of the convex hull are negative, we flip the
    // sign (this is particularly useful if the surface is open, then
    // it is closed using the smallest part of the sphere).
    std::vector<Vertex_handle> convex_hull;
    m_tr->adjacent_vertices (m_tr->infinite_vertex (),
			     std::back_inserter (convex_hull));
    unsigned int nb_negative = 0;
    for (std::size_t i = 0; i < convex_hull.size (); ++ i)
      if (convex_hull[i]->f() < 0.0)
        ++ nb_negative;
    
    if(nb_negative > convex_hull.size () / 2)
      flip_f();

    // Update m_sink
    FT sink_value = find_sink();
    return sink_value;
  }


/// Gets median value of the implicit function over input vertices.
  FT median_value_at_input_vertices() const
  {
    std::deque<FT> values;
    Finite_vertices_iterator v, e;
    for(v = m_tr->finite_vertices_begin(),
        e= m_tr->finite_vertices_end();
        v != e; 
        v++)
      if(v->type() == Triangulation::INPUT)
        values.push_back(v->f());

    std::size_t size = values.size();
    if(size == 0)
    {
      std::cerr << "Contouring: no input points\n";
      return 0.0;
    }

    std::sort(values.begin(),values.end());
    std::size_t index = size/2;
    // return values[size/2];
    return 0.5 * (values[index] + values[index+1]); // avoids singular cases
  }

  void barycentric_coordinates(const Point& p,
                               Cell_handle cell,
                               FT& a,
                               FT& b,
                               FT& c,
                               FT& d) const
  {

    //    const Point& pa = cell->vertex(0)->point();
    // const Point& pb = cell->vertex(1)->point();
    // const Point& pc = cell->vertex(2)->point();
    const Point& pd = cell->vertex(3)->point();
#if 1
    //Vector va = pa - pd;
    //Vector vb = pb - pd;
    //Vector vc = pc - pd;
    Vector vp = p - pd;

    //FT i00, i01, i02, i10, i11, i12, i20, i21, i22;
    //internal::invert(va.x(), va.y(), va.z(),
    //       vb.x(), vb.y(), vb.z(),
    //       vc.x(), vc.y(), vc.z(),
    //       i00, i01, i02, i10, i11, i12, i20, i21, i22);
    const boost::array<double,9> & i = (*m_Bary)[cell->info()];
    if(i[0]==-1){
      initialize_matrix_entry(cell);
    }
    //    UsedBary[cell->info()] = true;
    a = i[0] * vp.x() + i[3] * vp.y() + i[6] * vp.z();
    b = i[1] * vp.x() + i[4] * vp.y() + i[7] * vp.z();
    c = i[2] * vp.x() + i[5] * vp.y() + i[8] * vp.z();
    d = 1 - ( a + b + c);
#else
    FT v = volume(pa,pb,pc,pd);
    a = std::fabs(volume(pb,pc,pd,p) / v);
    b = std::fabs(volume(pa,pc,pd,p) / v);
    c = std::fabs(volume(pb,pa,pd,p) / v);
    d = std::fabs(volume(pb,pc,pa,p) / v);

    std::cerr << "_________________________________\n";
    std::cerr << aa << "  " << bb << "  " << cc << "  " << dd << std::endl;
    std::cerr << a << "  " << b << "  " << c << "  " << d << std::endl;

#endif
  }

  FT find_sink()
  {
    m_sink = CGAL::ORIGIN;
    FT min_f = 1e38;
    Finite_vertices_iterator v, e;
    for(v = m_tr->finite_vertices_begin(),
        e= m_tr->finite_vertices_end();
        v != e;
        v++)
    {
      if(v->f() < min_f)
      {
        m_sink = v->point();
        min_f = v->f();
      }
    }
    return min_f;
  }

  void shift_f(const FT shift)
  {
    Finite_vertices_iterator v, e;
    for(v = m_tr->finite_vertices_begin(),
        e = m_tr->finite_vertices_end();
        v!= e;
        v++)
      v->f() += shift;
  }

  void flip_f()
  {
    Finite_vertices_iterator v, e;
    for(v = m_tr->finite_vertices_begin(),
          e = m_tr->finite_vertices_end();
        v != e;
        v++)
      v->f() = -v->f();
  }

  Vertex_handle any_vertex_on_convex_hull()
  {
    Cell_handle ch = m_tr->infinite_vertex()->cell();
    return  ch->vertex( (ch->index( m_tr->infinite_vertex())+1)%4);
  }


  void constrain_one_vertex_on_convex_hull(const FT value = 0.0)
  {
    Vertex_handle v = any_vertex_on_convex_hull();
    m_tr->constrain(v);
    v->f() = value;
  }

  // TODO: Some entities are computed too often
  // - nn and area should not be computed for the face and its opposite face
  // 
  // divergent
  FT div_normalized(Vertex_handle v)
  {
    std::vector<Cell_handle> cells;
    cells.reserve(32);
    m_tr->incident_cells(v,std::back_inserter(cells));
  
    FT div = 0;
    typename std::vector<Cell_handle>::iterator it;
    for(it = cells.begin(); it != cells.end(); it++)
    {
      Cell_handle cell = *it;
      if(m_tr->is_infinite(cell))
        continue;

      // compute average normal per cell
      Vector n = get_cell_normal(cell);

      // zero normal - no need to compute anything else
      if(n == CGAL::NULL_VECTOR)
        continue;


      // compute n'
      int index = cell->index(v);
      const Point& x = cell->vertex(index)->point();
      const Point& a = cell->vertex((index+1)%4)->point();
      const Point& b = cell->vertex((index+2)%4)->point();
      const Point& c = cell->vertex((index+3)%4)->point();
      Vector nn = (index%2==0) ? CGAL::cross_product(b-a,c-a) : CGAL::cross_product(c-a,b-a);
      nn = nn / std::sqrt(nn*nn); // normalize
      Vector p = a - x;
      Vector q = b - x;
      Vector r = c - x;
      FT p_n = std::sqrt(p*p);
      FT q_n = std::sqrt(q*q);
      FT r_n = std::sqrt(r*r);
      FT solid_angle = p*(CGAL::cross_product(q,r));
      solid_angle = std::abs(solid_angle / (p_n*q_n*r_n + (p*q)*r_n + (q*r)*p_n + (r*p)*q_n));

      FT area = std::sqrt(squared_area(a,b,c));
      FT length = p_n + q_n + r_n;
      div += n * nn * area / length ;
    }
    return div * FT(3.0);
  }

  FT div(Vertex_handle v)
  {
    std::vector<Cell_handle> cells;
    cells.reserve(32);
    m_tr->incident_cells(v,std::back_inserter(cells));
  
    FT div = 0.0;
    typename std::vector<Cell_handle>::iterator it;
    for(it = cells.begin(); it != cells.end(); it++)
    {
      Cell_handle cell = *it;
      if(m_tr->is_infinite(cell))
        continue;
      
      const int index = cell->index(v);
      const Point& a = cell->vertex(m_tr->vertex_triple_index(index, 0))->point();
      const Point& b = cell->vertex(m_tr->vertex_triple_index(index, 1))->point();
      const Point& c = cell->vertex(m_tr->vertex_triple_index(index, 2))->point();
      const Vector nn = CGAL::cross_product(b-a,c-a);

      div+= nn * (//v->normal() + 
                  cell->vertex((index+1)%4)->normal() +
                  cell->vertex((index+2)%4)->normal() +
                  cell->vertex((index+3)%4)->normal());
    }
    return div;
  }

  Vector get_cell_normal(Cell_handle cell)
  {
    return Normal[cell->info()];
  }

  Vector cell_normal(Cell_handle cell) const
  {
    const Vector& n0 = cell->vertex(0)->normal();
    const Vector& n1 = cell->vertex(1)->normal();
    const Vector& n2 = cell->vertex(2)->normal();
    const Vector& n3 = cell->vertex(3)->normal();
    Vector n = n0 + n1 + n2 + n3;
    if(n != NULL_VECTOR){
      FT sq_norm = n*n;
      if(sq_norm != 0.0){
        return n / std::sqrt(sq_norm); // normalize
      }
    }
    return NULL_VECTOR;
  }

  // cotan formula as area(voronoi face) / len(primal edge)
  FT cotan_geometric(Edge& edge)
  {
    Cell_handle cell = edge.first;
    Vertex_handle vi = cell->vertex(edge.second);
    Vertex_handle vj = cell->vertex(edge.third);

    // primal edge
    const Point& pi = vi->point();
    const Point& pj = vj->point();
    Vector primal = pj - pi;
    FT len_primal = std::sqrt(primal * primal);
    return area_voronoi_face(edge) / len_primal;
  }

  // spin around edge
  // return area(voronoi face)
  FT area_voronoi_face(Edge& edge)
  {
    // circulate around edge
    Cell_circulator circ = m_tr->incident_cells(edge);
    Cell_circulator done = circ;
    std::vector<Point> voronoi_points;
    voronoi_points.reserve(9);
    do
    {
      Cell_handle cell = circ;
      if(!m_tr->is_infinite(cell))
        voronoi_points.push_back(Dual[cell->info()]);
      else // one infinite tet, switch to another calculation
        return area_voronoi_face_boundary(edge);
      circ++;
    }
    while(circ != done);

    if(voronoi_points.size() < 3)
    {
      CGAL_surface_reconstruction_points_assertion(false);
      return 0.0;
    }

    // sum up areas
    FT area = 0.0;
    const Point& a = voronoi_points[0];
    std::size_t nb_triangles = voronoi_points.size() - 1;
    for(std::size_t i=1;i<nb_triangles;i++)
    {
      const Point& b = voronoi_points[i];
      const Point& c = voronoi_points[i+1];
      area += std::sqrt(squared_area(a,b,c));
    }
    return area;
  }

  // approximate area when a cell is infinite
  FT area_voronoi_face_boundary(Edge& edge)
  {
    FT area = 0.0;
    Vertex_handle vi = edge.first->vertex(edge.second);
    Vertex_handle vj = edge.first->vertex(edge.third);

    const Point& pi = vi->point();
    const Point& pj = vj->point();
    Point m = CGAL::midpoint(pi,pj);

    // circulate around each incident cell
    Cell_circulator circ = m_tr->incident_cells(edge);
    Cell_circulator done = circ;
    do
    {
      Cell_handle cell = circ;
      if(!m_tr->is_infinite(cell))
      {
        // circumcenter of cell
        Point c = Dual[cell->info()];
        Tetrahedron tet = m_tr->tetrahedron(cell);

        int i = cell->index(vi);
        int j = cell->index(vj);
        int k =  Triangulation_utils_3::next_around_edge(i,j);
        int l =  Triangulation_utils_3::next_around_edge(j,i);

        Vertex_handle vk = cell->vertex(k);
        Vertex_handle vl = cell->vertex(l);

        const Point& pk = vk->point();
        const Point& pl = vl->point();

        // if circumcenter is outside tet
        // pick barycenter instead
        if(tet.has_on_unbounded_side(c))
        {
          Point cell_points[4] = {pi,pj,pk,pl};
          c = CGAL::centroid(cell_points, cell_points+4);
        }

        Point ck = CGAL::circumcenter(pi,pj,pk);
        Point cl = CGAL::circumcenter(pi,pj,pl);

        area += std::sqrt(squared_area(m,c,ck));
        area += std::sqrt(squared_area(m,c,cl));
      }
      circ++;
    }
    while(circ != done);
    return area;
  }

  /// Assemble vi's row of the linear system A*X=B
  ///
  /// @commentheading Template parameters:
  /// @param SparseLinearAlgebraTraits_d Symmetric definite positive sparse linear solver.
  template <class SparseLinearAlgebraTraits_d>
  void assemble_poisson_row(typename SparseLinearAlgebraTraits_d::Matrix& A,
                            Vertex_handle vi,
                            typename SparseLinearAlgebraTraits_d::Vector& B,
                            double lambda)
  {
    // for each vertex vj neighbor of vi
    std::vector<Edge> edges;
    m_tr->incident_edges(vi,std::back_inserter(edges));

    double diagonal = 0.0;

    for(typename std::vector<Edge>::iterator it = edges.begin();
        it != edges.end();
        it++)
      {
        Vertex_handle vj = it->first->vertex(it->third);
        if(vj == vi){
          vj = it->first->vertex(it->second);
        }
        if(m_tr->is_infinite(vj))
          continue;

        // get corresponding edge
        Edge edge( it->first, it->first->index(vi), it->first->index(vj));
        if(vi->index() < vj->index()){
          std::swap(edge.second,  edge.third);
        }

        double cij = cotan_geometric(edge);

        if(m_tr->is_constrained(vj)){
          if(! is_valid(vj->f())){
            std::cerr << "vj->f() = " << vj->f() << " is not valid" << std::endl;
          }
          B[vi->index()] -= cij * vj->f(); // change rhs
          if(! is_valid( B[vi->index()])){
            std::cerr << " B[vi->index()] = " <<  B[vi->index()] << " is not valid" << std::endl;
          }

        } else {
          if(! is_valid(cij)){
            std::cerr << "cij = " << cij << " is not valid" << std::endl;
          }
          A.set_coef(vi->index(),vj->index(), -cij, true /*new*/); // off-diagonal coefficient
        }

        diagonal += cij;
      }
    // diagonal coefficient
    if (vi->type() == Triangulation::INPUT){
      A.set_coef(vi->index(),vi->index(), diagonal + lambda, true /*new*/) ;
    } else{
      A.set_coef(vi->index(),vi->index(), diagonal, true /*new*/);
    }
  }
  

  /// Computes enlarged geometric bounding sphere of the embedded triangulation.
  Sphere enlarged_bounding_sphere(FT ratio) const
  {
    Sphere bsphere = bounding_sphere(); // triangulation's bounding sphere
    return Sphere(bsphere.center(), bsphere.squared_radius() * ratio*ratio);
  }

}; // end of Poisson_reconstruction_function


} //namespace CGAL

#endif // CGAL_POISSON_RECONSTRUCTION_FUNCTION_H