CGAL 6.1.1 - 2D and Surface Function Interpolation

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Interpolation/interpolation_2.cpp
// Compares the result of several interpolation methods
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Random.h>
#include <CGAL/squared_distance_2.h>
#include <CGAL/Delaunay_triangulation_2.h>
#include <CGAL/Interpolation_traits_2.h>
#include <CGAL/natural_neighbor_coordinates_2.h>
#include <CGAL/interpolation_functions.h>
#include <CGAL/point_generators_2.h>
#include <CGAL/algorithm.h>
#include <CGAL/Origin.h>
#include <cassert>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef K::FT Coord_type;
typedef K::Vector_2 Vector;
typedef K::Point_2 Point;
typedef CGAL::Delaunay_triangulation_2<K> Delaunay_triangulation;
typedef CGAL::Interpolation_traits_2<K> Traits;
typedef std::vector< std::pair<Point, Coord_type> > Coordinate_vector;
typedef std::map<Point, Coord_type, K::Less_xy_2> Point_value_map;
typedef std::map<Point, Vector, K::Less_xy_2 > Point_vector_map;
int main()
{
//number of sample points:
int n = 24;
//number of interpolation points:
int m = 20;
std::vector<Point> points;
points.reserve(n+m);
//put four bounding box points:
points.push_back(Point(-3,-3));
points.push_back(Point(3,-3));
points.push_back(Point(-3,3));
points.push_back(Point(3,3));
// Create n+m-4 points within a disc of radius 2
double r_d = 3;
CGAL::Random rng(1513114263);
CGAL::Random_points_in_disc_2<Point> g(r_d,rng );
std::copy_n( g, n+m, std::back_inserter(points));
Delaunay_triangulation T;
Point_value_map values;
Point_vector_map gradients;
//parameters for quadratic function:
Coord_type alpha = Coord_type(1.0),
beta1 = Coord_type(2.0),
beta2 = Coord_type(1.0),
gamma1 = Coord_type(0.3),
gamma2 = Coord_type(0.0),
gamma3 = Coord_type(0.0),
gamma4 = Coord_type(0.3);
for(int j=0; j<n ; j++){
T.insert(points[j]);
//determine function value/gradient:
Coord_type x(points[j].x());
Coord_type y(points[j].y());
Coord_type value = alpha + beta1*x + beta2*y + gamma1*(x*x) +
gamma4*(y*y) + (gamma2+ gamma3) *(x*y);
Vector gradient(beta1+ (gamma2+ gamma3)*y + Coord_type(2)*(gamma1*x),
beta2+ (gamma2+ gamma3)*x + Coord_type(2)*(gamma4*y));
values.insert(std::make_pair(points[j], value));
gradients.insert(std::make_pair(points[j], gradient));
}
//variables for statistics:
std::pair<Coord_type, bool> res;
Coord_type error, l_total = Coord_type(0),
q_total(l_total), f_total(l_total), s_total(l_total),
ssquare_total(l_total), l_max(l_total),
q_max(l_total), f_max(l_total), s_max(l_total),
ssquare_max(l_total),
total_value(l_total), l_value(l_total);
int failure(0);
//interpolation + error statistics
for(int i=n;i<n+m;i++){
Coord_type x(points[i].x());
Coord_type y(points[i].y());
Coord_type exact_value = alpha + beta1*x + beta2*y + gamma1*(x*x) +
gamma4*(y*y) + (gamma2+ gamma3) *(x*y);
total_value += exact_value;
//Coordinate_vector:
std::vector< std::pair< Point, Coord_type > > coords;
Coord_type norm =
CGAL::natural_neighbor_coordinates_2(T, points[i],
std::back_inserter(coords)).second;
assert(norm>0);
//linear interpolant:
l_value = CGAL::linear_interpolation(coords.begin(), coords.end(),
norm,
CGAL::Data_access<Point_value_map>(values));
error = CGAL_NTS abs(l_value - exact_value);
l_total += error;
if (error > l_max) l_max = error;
//Farin interpolant:
res = CGAL::farin_c1_interpolation(coords.begin(),
coords.end(), norm,points[i],
CGAL::Data_access<Point_value_map>(values),
CGAL::Data_access<Point_vector_map>
(gradients),
Traits());
if(res.second){
error = CGAL_NTS abs(res.first - exact_value);
f_total += error;
if (error > f_max) f_max = error;
} else ++failure;
//quadratic interpolant:
res = CGAL::quadratic_interpolation(coords.begin(), coords.end(),
norm,points[i],
CGAL::Data_access<Point_value_map>
(values),
CGAL::Data_access<Point_vector_map>
(gradients),
Traits());
if(res.second){
error = CGAL_NTS abs(res.first - exact_value);
q_total += error;
if (error > q_max) q_max = error;
} else ++failure;
//Sibson interpolant: version without sqrt:
res = CGAL::sibson_c1_interpolation_square(coords.begin(),
coords.end(), norm,
points[i],
CGAL::Data_access<Point_value_map>
(values),
CGAL::Data_access<Point_vector_map>
(gradients),
Traits());
//error statistics
if(res.second){
error = CGAL_NTS abs(res.first - exact_value);
ssquare_total += error;
if (error > ssquare_max) ssquare_max = error;
} else ++failure;
//with sqrt(the traditional):
res = CGAL::sibson_c1_interpolation(coords.begin(),
coords.end(), norm,
points[i],
CGAL::Data_access<Point_value_map>
(values),
CGAL::Data_access<Point_vector_map>
(gradients),
Traits());
//error statistics
if(res.second){
error = CGAL_NTS abs(res.first - exact_value);
s_total += error;
if (error > s_max) s_max = error;
} else ++failure;
}
/************** end of Interpolation: dump statistics **************/
std::cout << "Result: -----------------------------------" << std::endl;
std::cout << "Interpolation of '" << alpha <<" + "
<< beta1<<" x + "
<< beta2 << " y + " << gamma1 <<" x^2 + " << gamma2+ gamma3
<<" xy + " << gamma4 << " y^2'" << std::endl;
std::cout << "Knowing " << m << " sample points. Interpolation on "
<< n <<" random points. "<< std::endl;
std::cout <<"Average function value "
<< (1.0/n) * CGAL::to_double(total_value)
<< ", nb of failures "<< failure << std::endl;
std::cout << "linear interpolant mean error "
<< CGAL::to_double(l_total)/n << " max "
<< CGAL::to_double(l_max) <<std::endl;
std::cout << "quadratic interpolant mean error "
<< CGAL::to_double(q_total)/n << " max "
<< CGAL::to_double(q_max) << std::endl;
std::cout << "Farin interpolant mean error "
<< CGAL::to_double(f_total)/n << " max "
<< CGAL::to_double(f_max) << std::endl;
std::cout << "Sibson interpolant(classic) mean error "
<< CGAL::to_double(s_total)/n << " max "
<< CGAL::to_double(s_max) << std::endl;
std::cout << "Sibson interpolant(square_dist) mean error "
<< CGAL::to_double(ssquare_total)/n << " max "
<< CGAL::to_double(ssquare_max) << std::endl;
return EXIT_SUCCESS;
}
CGAL::Delaunay_triangulation_2
CGAL::Exact_predicates_inexact_constructions_kernel
CGAL::Interpolation_traits_2
Interpolation_traits_2 is a model for the concept InterpolationTraits and can be used to instantiate ...
Definition: Interpolation_traits_2.h:22
CGAL::to_double
double to_double(const NT &x)
CGAL::sibson_c1_interpolation_square
ValueFunctor::result_type sibson_c1_interpolation_square(CoordinateInputIterator first, CoordinateInputIterator beyond, const typename std::iterator_traits< CoordinateInputIterator >::value_type::second_type &norm, const Point &p, ValueFunctor value_function, GradFunctor gradient_function, const Traits &traits)
Same as sibson_c1_interpolation(), except that no square root operation is required for the number ty...
CGAL::linear_interpolation
ValueFunctor::result_type::first_type linear_interpolation(CoordinateInputIterator first, CoordinateInputIterator beyond, const typename std::iterator_traits< CoordinateInputIterator >::value_type::second_type &norm, ValueFunctor value_function)
The function linear_interpolation() computes the weighted sum of the function values which must be pr...
CGAL::farin_c1_interpolation
ValueFunctor::result_type farin_c1_interpolation(CoordinateInputIterator first, CoordinateInputIterator beyond, const typename std::iterator_traits< CoordinateInputIterator >::value_type::second_type &norm, const Point &p, ValueFunctor value_function, GradFunctor gradient_function, const Traits &traits)
Generates the interpolated function value computed by Farin's interpolant.
CGAL::quadratic_interpolation
ValueFunctor::result_type quadratic_interpolation(CoordinateInputIterator first, CoordinateInputIterator beyond, const typename std::iterator_traits< CoordinateInputIterator >::value_type::second_type &norm, const Point &p, ValueFunctor value_function, GradFunctor gradient_function, const Traits &traits)
The function quadratic_interpolation() generates the interpolated function value as the weighted sum ...
CGAL::sibson_c1_interpolation
std::pair< typename ValueFunctor::result_type, bool > sibson_c1_interpolation(CoordinateInputIterator first, CoordinateInputIterator beyond, const typename std::iterator_traits< CoordinateInputIterator >::value_type::second_type &norm, const Point &p, ValueFunctor value_function, GradFunctor gradient_function, const Traits &traits)
The function sibson_c1_interpolation() generates the interpolated function value at the point p,...
CGAL::natural_neighbor_coordinates_2
CGAL::Triple< CoordinateOutputIterator, typename Dt::Geom_traits::FT, bool > natural_neighbor_coordinates_2(const Dt &dt, const typename Dt::Geom_traits::Point_2 &p, CoordinateOutputIterator out, OutputFunctor fct, typename Dt::Face_handle start=typename Dt::Face_handle())
Computes the natural neighbor coordinates for p with respect to the points in the two-dimensional Del...
CGAL::Data_access
The struct Data_access implements a functor that allows to retrieve data from an associative containe...
Definition: interpolation_functions.h:26