void restricted_laguerre_edges (const T &t,
const RT &rt,
std::vector<Segment> &edges)
{
typedef RT::Vertex_handle Vertex_handle_RT;
typedef MA::Voronoi_intersection_traits<K> Traits;
typedef typename MA::Tri_intersector<T,RT,Traits> Tri_isector;
typedef typename Tri_isector::Pgon Pgon;
Tri_isector isector;
MA::voronoi_triangulation_intersection_raw
(t,rt, [&] (const Pgon &pgon, typename T::Face_handle f, Vertex_handle_RT v)
{
for (size_t i = 0; i < pgon.size(); ++i)
{
size_t iprev = (i+pgon.size()-1)%pgon.size();
size_t inext = (i+1)%pgon.size();
Point p = isector.vertex_to_point(pgon[i], pgon[iprev]);
Point q = isector.vertex_to_point(pgon[i], pgon[inext]);
if (pgon[i].type != Tri_isector::EDGE_DT)
continue;
edges.push_back(Segment(p,q));
}
});
}
double kantorovich (const T &densityT,
const Functions &densityF,
const Matrix &X,
const Vector &weights,
Vector &g,
SparseMatrix &h)
{
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Polygon_2<K> Polygon;
typedef K::FT FT;
typedef CGAL::Regular_triangulation_filtered_traits_2<K> RT_Traits;
typedef CGAL::Regular_triangulation_vertex_base_2<RT_Traits> Vbase;
typedef CGAL::Triangulation_vertex_base_with_info_2
<size_t, RT_Traits, Vbase> Vb;
typedef CGAL::Regular_triangulation_face_base_2<RT_Traits> Cb;
typedef CGAL::Triangulation_data_structure_2<Vb,Cb> Tds;
typedef CGAL::Regular_triangulation_2<RT_Traits, Tds> RT;
typedef RT::Vertex_handle Vertex_handle_RT;
typedef RT::Weighted_point Weighted_point;
typedef typename CGAL::Point_2<K> Point;
size_t N = X.rows();
assert(weights.rows() == N);
assert(weights.cols() == 1);
assert(X.cols() == 2);
// insert points with indices in the regular triangulation
std::vector<std::pair<Weighted_point,size_t> > Xw(N);
for (size_t i = 0; i < N; ++i)
{
Xw[i] = std::make_pair(Weighted_point(Point(X(i,0), X(i,1)),
weights(i)), i);
}
RT dt (Xw.begin(), Xw.end());
dt.infinite_vertex()->info() = -1;
// compute the quadratic part
typedef MA::Voronoi_intersection_traits<K> Traits;
typedef typename MA::Tri_intersector<T,RT,Traits> Tri_isector;
typedef typename Tri_isector::Pgon Pgon;
typedef Eigen::Triplet<FT> Triplet;
std::vector<Triplet> htri;
FT total(0), fval(0), total_area(0);
g = Vector::Zero(N);
MA::voronoi_triangulation_intersection_raw
(densityT,dt,
[&] (const Pgon &pgon,
typename T::Face_handle f,
Vertex_handle_RT v)
{
Tri_isector isector;
Polygon p;
std::vector<Vertex_handle_RT> adj;
for (size_t i = 0; i < pgon.size(); ++i)
{
size_t ii = (i==0)?(pgon.size()-1):(i-1);
//size_t ii = (i+1)%pgon.size();
p.push_back(isector.vertex_to_point(pgon[i], pgon[ii]));
adj.push_back((pgon[i].type == Tri_isector::EDGE_DT) ?
pgon[i].edge_dt.second : 0);
}
size_t idv = v->info();
auto fit = densityF.find(f);
assert(fit != densityF.end());
auto fv = fit->second; // function to integrate
// compute hessian
for (size_t i = 0; i < p.size(); ++i)
{
if (adj[i] == 0)
continue;
Vertex_handle_RT w = adj[i];
size_t idw = w->info();
FT r = MA::integrate_1<FT>(p.edge(i), fv);
FT d = 2*sqrt(CGAL::squared_distance(v->point(),
w->point()));
htri.push_back(Triplet(idv, idw, -r/d));
htri.push_back(Triplet(idv, idv, +r/d));
}
// compute value and gradient
FT warea = MA::integrate_1<FT>(p, FT(0), fv);
FT intg = MA::integrate_3<FT>(p, FT(0), [&](Point p)
{
return fv(p) *
CGAL::squared_distance(p,
v->point());
});
fval = fval + warea * weights[idv] - intg;
g[idv] = g[idv] + warea;
total += warea;
total_area += p.area();
});
h = SparseMatrix(N,N);
h.setFromTriplets(htri.begin(), htri.end());
h.makeCompressed();
return fval;
}
