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cgal/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h

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// Copyright (c) 2005 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, Bruno Levy
#ifndef CGAL_LSCM_PARAMETERIZER_3_H
#define CGAL_LSCM_PARAMETERIZER_3_H
#include <CGAL/license/Surface_mesh_parameterization.h>
#include <CGAL/internal/Surface_mesh_parameterization/Containers_filler.h>
#include <CGAL/Parameterizer_traits_3.h>
#include <CGAL/Two_vertices_parameterizer_3.h>
#include <CGAL/Polygon_mesh_processing/connected_components.h>
#include <CGAL/circulator.h>
#include <CGAL/Timer.h>
#ifdef CGAL_EIGEN3_ENABLED
#include <CGAL/Eigen_solver_traits.h>
#endif
#include <CGAL/OpenNL/linear_solver.h>
#include <boost/foreach.hpp>
#include <boost/unordered_set.hpp>
#include <iostream>
#include <vector>
/// \file LSCM_parameterizer_3.h
namespace CGAL {
// ------------------------------------------------------------------------------------
// Declaration
// ------------------------------------------------------------------------------------
/// \ingroup PkgSurfaceParameterizationMethods
///
/// The class `LSCM_parameterizer_3` implements the
/// *Least Squares Conformal Maps (LSCM)* parameterization \cgalCite{cgal:lprm-lscm-02}.
///
/// This is a conformal parameterization, i.e. it attempts to preserve angles.
///
/// This is a free border parameterization. There is no need to map the border
/// of the surface onto a convex polygon (only two pinned vertices are needed
/// to ensure a unique solution), but a one-to-one mapping is *not* guaranteed.
///
/// \cgalModels `ParameterizerTraits_3`
///
/// \tparam TriangleMesh a model of `FaceGraph`
/// \tparam BorderParameterizer_3 Strategy to parameterize the surface border. <br>
/// Note: The minimum is to parameterize two vertices.
/// \tparam SparseLinearAlgebraTraits_d Traits class to solve a sparse linear system. <br>
/// Note: We may use a symmetric definite positive solver because LSCM
/// solves the system in the least squares sense.
///
/// \sa `CGAL::Parameterizer_traits_3<TriangleMesh>`
/// \sa `CGAL::Fixed_border_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Barycentric_mapping_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
/// \sa `CGAL::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
template
<
class TriangleMesh,
class BorderParameterizer_3
= Two_vertices_parameterizer_3<TriangleMesh>,
class SparseLinearAlgebraTraits_d
#if defined(CGAL_EIGEN3_ENABLED) || defined(DOXYGEN_RUNNING)
= Eigen_solver_traits<Eigen::SimplicialLDLT<Eigen_sparse_symmetric_matrix<double>::EigenType> >
#else
= OpenNL::SymmetricLinearSolverTraits<typename TriangleMesh::NT>
#endif
>
class LSCM_parameterizer_3
: public Parameterizer_traits_3<TriangleMesh>
{
// Private types
private:
// Superclass
typedef Parameterizer_traits_3<TriangleMesh> Base;
// Public types
public:
// We have to repeat the types exported by superclass
/// @cond SKIP_IN_MANUAL
typedef typename Base::Error_code Error_code;
/// @endcond
/// Export BorderParameterizer_3 template parameter.
typedef BorderParameterizer_3 Border_param;
// Private types
private:
typedef SparseLinearAlgebraTraits_d Sparse_LA;
// Private types
private:
typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor vertex_descriptor;
typedef typename boost::graph_traits<TriangleMesh>::halfedge_descriptor halfedge_descriptor;
typedef typename boost::graph_traits<TriangleMesh>::face_descriptor face_descriptor;
typedef typename boost::graph_traits<TriangleMesh>::face_iterator face_iterator;
typedef typename boost::graph_traits<TriangleMesh>::vertex_iterator vertex_iterator;
// Mesh_Adaptor_3 subtypes:
typedef Parameterizer_traits_3<TriangleMesh> Traits;
typedef typename Traits::NT NT;
typedef typename Traits::Point_2 Point_2;
typedef typename Traits::Point_3 Point_3;
typedef typename Traits::Vector_2 Vector_2;
typedef typename Traits::Vector_3 Vector_3;
// SparseLinearAlgebraTraits_d subtypes:
typedef typename Sparse_LA::Vector Vector;
typedef typename Sparse_LA::Matrix Matrix;
typedef typename OpenNL::LinearSolver<Sparse_LA> LeastSquaresSolver;
// Public operations
public:
/// Constructor
LSCM_parameterizer_3(Border_param border_param = Border_param(),
///< %Object that maps the surface's border to 2D space
Sparse_LA sparse_la = Sparse_LA())
///< Traits object to access a sparse linear system
: m_borderParameterizer(border_param), m_linearAlgebra(sparse_la)
{ }
// Default copy constructor and operator =() are fine
/// Compute a one-to-one mapping from a triangular 3D surface mesh
/// to a piece of the 2D space.
/// The mapping is piecewise linear (linear in each triangle).
/// The result is the (u,v) pair image of each vertex of the 3D surface.
///
/// \tparam VertexUVmap must be a property map that associates a %Point_2
/// (type deduced by `Parameterized_traits_3`) to a `vertex_descriptor`
/// (type deduced by the graph traits of `TriangleMesh`).
/// \tparam VertexIndexMap must be a property map that associates a unique integer index
/// to a `vertex_descriptor` (type deduced by the graph traits of `TriangleMesh`).
/// \tparam VertexParameterizedMap must be a property map that associates a boolean
/// to a `vertex_descriptor` (type deduced by the graph traits of `TriangleMesh`).
///
/// \param mesh a triangulated surface.
/// \param bhd an halfedge descriptor on the boundary of `mesh`.
/// \param uvmap an instanciation of the class `VertexUVmap`.
/// \param vimap an instanciation of the class `VertexIndexMap`.
/// \param vpmap an instanciation of the class `VertexParameterizedMap`.
///
/// \pre `mesh` must be a surface with one connected component.
/// \pre `mesh` must be a triangular mesh.
/// \pre The vertices must be indexed (`vimap` must be initialized).
///
template <typename VertexUVmap, typename VertexIndexMap, typename VertexParameterizedMap>
Error_code parameterize(TriangleMesh& mesh,
halfedge_descriptor bhd,
VertexUVmap uvmap,
VertexIndexMap vimap,
VertexParameterizedMap vpmap)
{
// Count vertices
int nbVertices = num_vertices(mesh);
// Compute (u,v) for (at least two) border vertices
// and mark them as "parameterized"
Error_code status = get_border_parameterizer().parameterize_border(mesh,bhd,uvmap,vpmap);
if(status != Base::OK)
return status;
// Create sparse linear system "A*X = B" of size 2*nbVertices x 2*nbVertices
// (in fact, we need only 2 lines per triangle x 1 column per vertex)
LeastSquaresSolver solver(2 * nbVertices);
solver.set_least_squares(true) ;
// Initialize the "A*X = B" linear system after
// (at least two) border vertices parameterization
initialize_system_from_mesh_border(solver, mesh, uvmap, vimap, vpmap);
// Fill the matrix for the other vertices
solver.begin_system();
boost::unordered_set<vertex_descriptor> ccvertices;
std::vector<face_descriptor> ccfaces;
CGAL::internal::Parameterization::Containers_filler<TriangleMesh>
fc(mesh, ccvertices, &ccfaces);
CGAL::Polygon_mesh_processing::connected_component(
face(opposite(bhd, mesh), mesh),
mesh,
boost::make_function_output_iterator(fc));
BOOST_FOREACH(face_descriptor fd, ccfaces){
// Create two lines in the linear system per triangle (one for u, one for v)
status = setup_triangle_relations(solver, mesh, fd, vimap);
if (status != Base::OK)
return status;
}
solver.end_system();
// Solve the "A*X = B" linear system in the least squares sense
if(! solver.solve())
status = Base::ERROR_CANNOT_SOLVE_LINEAR_SYSTEM;
if(status != Base::OK)
return status;
// Copy X coordinates into the (u,v) pair of each vertex
//set_mesh_uv_from_system(mesh, solver, uvmap);
BOOST_FOREACH(vertex_descriptor vd, ccvertices){
int index = get(vimap,vd);
NT u = solver.variable(2 * index).value();
NT v = solver.variable(2 * index + 1).value();
put(uvmap, vd, Point_2(u,v));
}
return status;
}
// Private operations
private:
/// Initialize "A*X = B" linear system after
/// (at least two) border vertices are parameterized.
///
/// \pre Vertices must be indexed.
/// \pre X and B must be allocated and empty.
/// \pre At least 2 border vertices must be parameterized.
template <typename UVmap, typename VertexIndexMap, typename VertexParameterizedMap>
void initialize_system_from_mesh_border(LeastSquaresSolver& solver,
const TriangleMesh& mesh,
UVmap uvmap,
VertexIndexMap vimap,
VertexParameterizedMap vpmap)
{
BOOST_FOREACH(vertex_descriptor v, vertices(mesh)){
// Get vertex index in sparse linear system
int index = get(vimap, v);
// Get vertex (u,v) (meaningless if vertex is not parameterized)
Point_2 uv = get(uvmap, v);
// TODO: it is meaningless but must it be called for non-border vertices??
// Write (u,v) in X (meaningless if vertex is not parameterized)
// Note : 2*index --> u
// 2*index + 1 --> v
solver.variable(2 * index ).set_value(uv.x());
solver.variable(2 * index + 1).set_value(uv.y());
// Copy (u,v) in B if vertex is parameterized
if (get(vpmap,v)) {
solver.variable(2 * index ).lock();
solver.variable(2 * index + 1).lock();
}
}
}
/// Utility for setup_triangle_relations():
/// Computes the coordinates of the vertices of a triangle
/// in a local 2D orthonormal basis of the triangle's plane.
void project_triangle(const Point_3& p0, const Point_3& p1, const Point_3& p2, // in
Point_2& z0, Point_2& z1, Point_2& z2); // out
/// Create two lines in the linear system per triangle (one for u, one for v).
///
/// \pre vertices must be indexed.
template <typename VertexIndexMap >
Error_code setup_triangle_relations(LeastSquaresSolver& solver,
const TriangleMesh& mesh,
face_descriptor facet,
VertexIndexMap vimap);
// Private accessors
private:
/// Get the object that maps the surface's border onto a 2D space.
Border_param& get_border_parameterizer() { return m_borderParameterizer; }
/// Get the sparse linear algebra (traits object to access the linear system).
Sparse_LA& get_linear_algebra_traits() { return m_linearAlgebra; }
// Fields
private:
/// %Object that maps (at least two) border vertices onto a 2D space
Border_param m_borderParameterizer;
/// Traits object to solve a sparse linear system
Sparse_LA m_linearAlgebra;
};
// Utility for setup_triangle_relations():
// Computes the coordinates of the vertices of a triangle
// in a local 2D orthonormal basis of the triangle's plane.
template<class TriangleMesh, class Border_param, class Sparse_LA>
inline
void
LSCM_parameterizer_3<TriangleMesh, Border_param, Sparse_LA>::
project_triangle(const Point_3& p0, const Point_3& p1, const Point_3& p2, // in
Point_2& z0, Point_2& z1, Point_2& z2) // out
{
Vector_3 X = p1 - p0;
NT X_norm = std::sqrt(X * X);
if(X_norm != 0.0)
X = X / X_norm;
Vector_3 Z = CGAL::cross_product(X, p2 - p0);
NT Z_norm = std::sqrt(Z * Z);
if(Z_norm != 0.0)
Z = Z / Z_norm;
Vector_3 Y = CGAL::cross_product(Z, X);
const Point_3& O = p0;
NT x0 = 0;
NT y0 = 0;
NT x1 = std::sqrt( (p1 - O)*(p1 - O) );
NT y1 = 0;
NT x2 = (p2 - O) * X;
NT y2 = (p2 - O) * Y;
z0 = Point_2(x0, y0);
z1 = Point_2(x1, y1);
z2 = Point_2(x2, y2);
}
/// Create two lines in the linear system per triangle (one for u, one for v).
///
/// \pre vertices of `mesh` must be indexed.
// Implementation note: LSCM equation is:
// (Z1 - Z0)(U2 - U0) = (Z2 - Z0)(U1 - U0)
// where Uk = uk + i.v_k is the complex number corresponding to (u,v) coords
// Zk = xk + i.yk is the complex number corresponding to local (x,y) coords
// cool: no divide with this expression; makes it more numerically stable
// in presence of degenerate triangles
template<class TriangleMesh, class Border_param, class Sparse_LA>
template <typename VertexIndexMap>
inline
typename LSCM_parameterizer_3<TriangleMesh, Border_param, Sparse_LA>::Error_code
LSCM_parameterizer_3<TriangleMesh, Border_param, Sparse_LA>::
setup_triangle_relations(LeastSquaresSolver& solver,
const TriangleMesh& mesh,
face_descriptor facet,
VertexIndexMap vimap)
{
typedef typename boost::property_map<TriangleMesh, boost::vertex_point_t>::const_type PPmap;
PPmap ppmap = get(vertex_point, mesh);
// Get the 3 vertices of the triangle
vertex_descriptor v0, v1, v2;
halfedge_descriptor h0 = halfedge(facet, mesh);
v0 = target(h0, mesh);
halfedge_descriptor h1 = next(h0, mesh);
v1 = target(h1, mesh);
halfedge_descriptor h2 = next(h1, mesh);
v2 = target(h2, mesh);
// Get the vertices index
int id0 = get(vimap, v0);
int id1 = get(vimap, v1);
int id2 = get(vimap, v2);
// Get the vertices position
const Point_3& p0 = get(ppmap, v0);
const Point_3& p1 = get(ppmap, v1);
const Point_3& p2 = get(ppmap, v2);
// Computes the coordinates of the vertices of a triangle
// in a local 2D orthonormal basis of the triangle's plane.
Point_2 z0, z1, z2;
project_triangle(p0, p1, p2, //in
z0, z1, z2); // out
Vector_2 z01 = z1 - z0;
Vector_2 z02 = z2 - z0;
NT a = z01.x();
NT b = z01.y();
NT c = z02.x();
NT d = z02.y();
CGAL_assertion(b == 0.0);
// Create two lines in the linear system per triangle (one for u, one for v)
// LSCM equation is:
// (Z1 - Z0)(U2 - U0) = (Z2 - Z0)(U1 - U0)
// where Uk = uk + i.v_k is the complex number corresponding to (u,v) coords
// Zk = xk + i.yk is the complex number corresponding to local (x,y) coords
//
// Note : 2*index --> u
// 2*index + 1 --> v
int u0_id = 2*id0 ;
int v0_id = 2*id0 + 1;
int u1_id = 2*id1 ;
int v1_id = 2*id1 + 1;
int u2_id = 2*id2 ;
int v2_id = 2*id2 + 1;
// Real part
// Note: b = 0
solver.begin_row();
solver.add_coefficient(u0_id, -a+c);
solver.add_coefficient(v0_id, b-d);
solver.add_coefficient(u1_id, -c);
solver.add_coefficient(v1_id, d);
solver.add_coefficient(u2_id, a);
solver.end_row();
//
// Imaginary part
// Note: b = 0
solver.begin_row();
solver.add_coefficient(u0_id, -b+d);
solver.add_coefficient(v0_id, -a+c);
solver.add_coefficient(u1_id, -d);
solver.add_coefficient(v1_id, -c);
solver.add_coefficient(v2_id, a);
solver.end_row();
return Base::OK;
}
} // namespace CGAL
#endif // CGAL_LSCM_PARAMETERIZER_3_H
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