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// 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.
//
// $URL$
// $Id$
// SPDX-License-Identifier: GPL-3.0+
//
// 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>
#include <CGAL/disable_warnings.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>
(CGAL::make_range(first, beyond), 6,
CGAL::parameters::point_map(point_pmap)))
{
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>(CGAL::make_range(first, beyond), 6,
CGAL::parameters::point_map(point_pmap)))
{
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>(CGAL::make_range(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_map<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
#include <CGAL/enable_warnings.h>
#endif // CGAL_POISSON_RECONSTRUCTION_FUNCTION_H
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