Julia Flötotto
This chapter describes CGAL's interpolation package which implements natural neighbor coordinate functions as well as different methods for scattered data interpolation most of which are based on natural neighbor coordinates. The functions for computing natural neighbor coordinates in Euclidean space are described in Section , the functions concerning the coordinate and neighbor computation on surfaces are discussed in Section . In Section , we describe the different interpolation functions.
Scattered data interpolation solves the following problem: given measures of a function on a set of discrete data points, the task is to interpolate this function on an arbitrary query point. More formally, let be a set of points in or and be a scalar function defined inside the convex hull of . We assume that the function values are known at the points of , i.e. to each , we associate . Sometimes, the gradient of is also known at . It is denoted . The interpolation is carried out for an arbitrary query point . Except for interpolation on surfaces, must lie inside the convex hull of .
Figure: example: has five natural neighbors . The natural neighbor coordinate is the ratio of the area of the pink polygon, , over the area of the total highlighted zone.
Let denote the volume of the potential Voronoi cell of and denote the volume of the sub-cell that would be stolen from the cell of by the cell of . The natural neighbor coordinate of with respect to the data point is defined by
A two-dimensional example is depicted in Figure .
Various papers ([Sib80], [Far90], [Pip93], [Bro97],[HS00]) show that the natural neighbor coordinates have the following properties:
The interpolation package of CGAL provides functions to compute natural neighbor coordinates for and points with respect to Voronoi diagrams as well as with respect to power diagrams (only ), i.e. for weighted points. Refer to the reference pages natural_neighbor_coordinates_2, natural_neighbor_coordinates_3 and regular_neighbor_coordinates_2.
In addition, the package provides functions to compute natural neighbor coordinates on well sampled point set surfaces. See Section and the reference page surface_neighbor_coordinates_3 for further information.
// //file: examples/Interpolation/nn_coordinates_2.C // #include <CGAL/basic.h> #include <utility> #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <CGAL/natural_neighbor_coordinates_2.h> struct K : CGAL::Exact_predicates_inexact_constructions_kernel {}; typedef CGAL::Delaunay_triangulation_2<K> Delaunay_triangulation; typedef std::vector< std::pair< K::Point_2, K::FT > > Point_coordinate_vector; int main() { Delaunay_triangulation dt; for (int y=0 ; y<3 ; y++) for (int x=0 ; x<3 ; x++) dt.insert(K::Point_2(x,y)); //coordinate computation K::Point_2 p(1.2, 0.7); Point_coordinate_vector coords; CGAL::Triple< std::back_insert_iterator<Point_coordinate_vector>, K::FT, bool> result = CGAL::natural_neighbor_coordinates_2(dt, p, std::back_inserter(coords)); if(!result.third){ std::cout << "The coordinate computation was not successful." << std::endl; std::cout << "The point (" <<p << ") lies outside the convex hull." << std::endl; } K::FT norm = result.second; std::cout << "Coordinate computation successful." << std::endl; std::cout << "Normalization factor: " <<norm << std::endl; return 0; }
// //file: examples/Interpolation/rn_coordinates_2.C // #include <CGAL/basic.h> #include <utility> #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Regular_triangulation_2.h> #include <CGAL/Regular_triangulation_euclidean_traits_2.h> #include <CGAL/regular_neighbor_coordinates_2.h> struct K : CGAL::Exact_predicates_inexact_constructions_kernel {}; typedef CGAL::Regular_triangulation_euclidean_traits_2<K> Gt; typedef CGAL::Regular_triangulation_2<Gt> Regular_triangulation; typedef Regular_triangulation::Weighted_point Weighted_point; typedef std::vector< std::pair< Weighted_point, K::FT > > Point_coordinate_vector; int main() { Regular_triangulation rt; for (int y=0 ; y<3 ; y++) for (int x=0 ; x<3 ; x++) rt.insert(Weighted_point(K::Point_2(x,y), 0)); //coordinate computation Weighted_point wp(K::Point_2(1.2, 0.7),2); Point_coordinate_vector coords; CGAL::Triple< std::back_insert_iterator<Point_coordinate_vector>, K::FT, bool> result = CGAL::regular_neighbor_coordinates_2(rt, wp, std::back_inserter(coords)); if(!result.third){ std::cout << "The coordinate computation was not successful." << std::endl; std::cout << "The point (" <<wp.point() << ") lies outside the convex hull." << std::endl; } K::FT norm = result.second; std::cout << "Coordinate computation successful." << std::endl; std::cout << "Normalization factor: " <<norm << std::endl; return 0; }For surface neighbor coordinates, the surface normal at the query point must be provided, see Section .
This section introduces the functions to compute natural neighbor coordinates and surface neighbors associated to a set of sample points issued from a surface and given a query point on . We suppose that is a closed and compact surface of , and let be an -sample of (refer to Amenta and Bern [AB99]). The concepts are based on the definition of Boissonnat and Flötotto [BF02], [Flö03]. Both references contain a thorough description of the requirements and the mathematical properties.
Two observations lead to the definition of surface neighbors and surface neighbor coordinates: First, it is clear that the tangent plane of the surface at the point approximates in the neighborhood of . It has been shown in [BF02] that, if the surface is well sampled with respect to the curvature and the local thickness of , i.e. it is an -sample, the intersection of the tangent plane with the Voronoi cell of in the Voronoi diagram of has a small diameter. Consequently, inside this Voronoi cell, the tangent plane is a reasonable approximation of . Furthermore, the second observation allows to compute this intersection diagram easily: one can show using Pythagoras' theorem that the intersection of a three-dimensional Voronoi diagram with a plane is a two-dimensional power diagram. The points defining the power diagram are the projections of the points in onto , each point weighted with its negative square distance to . Algorithms for the computation of power diagrams via the dual regular triangulation are well known and for example provided by CGAL in the class Regular_triangulation_2<Gt, Tds>.
In CGAL, the regular triangulation dual to the intersection of a Voronoi diagram with a plane can be computed by instantiating the Regular_triangulation_2<Gt, Tds> class with the traits class Voronoi_intersection_2_traits_3<K>. This traits class contains a point and a vector as class member which define the plane . All predicates and constructions used by Regular_triangulation_2<Gt, Tds> are replaced by the corresponding operators on three-dimensional points. For example, the power test predicate (which takes three weighted points , , of the regular triangulation and tests the power distance of a fourth point with respect to the power circle orthogonal to , , ) is replaced by a Side_of_plane_centered_sphere_2_3 predicate that tests the position of a point with respect to the sphere centered on the plane passing through the points , , . This approach allows to avoid the explicit constructions of the projected points and the weights which are very prone to rounding errors.
The computation of natural neighbor coordinates on surfaces is based upon the computation of regular neighbor coordinates with respect to the regular triangulation that is dual to , the intersection of and the Voronoi diagram of , via the function regular_neighbor_coordinates_2.
Of course, we might introduce all data points into this regular triangulation. However, this is not necessary because we are only interested in the cell of . It is sufficient to guarantee that all surface neighbors of the query point are among the input points that are passed as argument to the function. The sample points can be filtered for example by distance, e.g. using range search or -nearest neighbor queries, or with the help of the Delaunay triangulation since the surface neighbors are necessarily a subset of the natural neighbors of the query point in this triangulation. CGAL provides a function that encapsulates the filtering based on the Delaunay triangulation. For input points filtered by distance, functions are provided that indicate whether or not points that lie outside the input range (i.e. points that are further from than the furthest input point) can still influence the result. This allows to iteratively enlarge the set of input points until the range is sufficient to certify the result.
The surface neighbors of the query point are its neighbors in the regular triangulation that is dual to , the intersection of and the Voronoi diagram of . As for surface neighbor coordinates, this regular triangulation is computed and the same kind of filtering of the data points as well as the certification described above is provided.
//file: examples/Interpolation/surface_neighbor_coordinates_3.C // example with random points on a sphere #include <CGAL/basic.h> #include <utility> #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/point_generators_3.h> #include <CGAL/copy_n.h> #include <CGAL/Origin.h> #include <CGAL/surface_neighbor_coordinates_3.h> struct K : CGAL::Exact_predicates_inexact_constructions_kernel {}; typedef K::FT Coord_type; typedef K::Point_3 Point_3; typedef K::Vector_3 Vector_3; typedef std::vector< std::pair< Point_3, K::FT > > Point_coordinate_vector; int main() { int n=100; std::vector< Point_3> points; points.reserve(n); std::cout << "Generate " << n << " random points on a sphere." << std::endl; CGAL::Random_points_on_sphere_3<Point_3> g(1); CGAL::copy_n( g, n, std::back_inserter(points)); Point_3 p(1, 0,0); Vector_3 normal(p-CGAL::ORIGIN); std::cout << "Compute surface neighbor coordinates for " << p << std::endl; Point_coordinate_vector coords; CGAL::Triple< std::back_insert_iterator<Point_coordinate_vector>, K::FT, bool> result = CGAL::surface_neighbor_coordinates_3(points.begin(), points.end(), p, normal, std::back_inserter(coords), K()); if(!result.third){ //Undersampling: std::cout << "The coordinate computation was not successful." << std::endl; return 0; } K::FT norm = result.second; std::cout << "Testing the barycentric property " << std::endl; Point_3 b(0, 0,0); for(std::vector< std::pair< Point_3, Coord_type > >::const_iterator it = coords.begin(); it!=coords.end(); ++it) b = b + (it->second/norm)* (it->first - CGAL::ORIGIN); std::cout <<" weighted barycenter: " << b <<std::endl; std::cout << " squared distance: " << CGAL::squared_distance(p,b) <<std::endl; return 0; }
Sibson [Sib81] defines a very simple interpolant that re-produces linear functions exactly. The interpolation of is given as the linear combination of the neighbors' function values weighted by the coordinates:
Indeed, if for all natural neighbors of , we have
by the barycentric coordinate property. The first example in Subsection shows how the function is called.
Sibson's interpolant is a combination of the linear interpolant and an interpolant which is the weighted sum of the first degree functions
Sibson observed that the combination of and reconstructs exactly a spherical quadric if they are mixed as follows:
where in Sibson's original work, .
CGAL contains a second implementation with which is less demanding on the number type because it avoids the square-root computation needed to compute the distance . The theoretical guarantees are the same (see [Flö03]). Simply, the smaller the slope of around , the faster the interpolant approaches as .
Farin [Far90] extended Sibson's work and realizes a continuous interpolant by embedding natural neighbor coordinates in the Bernstein-Bézier representation of a cubic simplex. If the gradient of at the data points is known, this interpolant reproduces quadratic functions exactly. The function gradient can be approximated from the function values by Sibson's method [Sib81] (see Section ) which is exact only for spherical quadrics.
Knowing the gradient for all , we formulate a very simple interpolant that reproduces exactly quadratic functions. This interpolant is not continuous in general. It is defined as follows:
where is the natural neighbor coordinate of with respect to associated to . This method works only for points inside the convex hull of the data points because, for a point on the convex hull, is not defined. For spherical quadrics, the result is exact.
CGAL provides functions to approximate the gradients of all data points that are inside the convex hull. There is one function for each type of natural neighbor coordinate (i.e. natural_neighbor_coordinates_2, regular_neighbor_coordinates_2).
// //file: examples/Interpolation/linear_interoplation_2.C // #include <CGAL/basic.h> #include <utility> #include <CGAL/Exact_predicates_inexact_constructions_kernel.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> struct K : CGAL::Exact_predicates_inexact_constructions_kernel {}; typedef CGAL::Delaunay_triangulation_2<K> Delaunay_triangulation; typedef CGAL::Interpolation_traits_2<K> Traits; typedef K::FT Coord_type; typedef K::Point_2 Point; int main() { Delaunay_triangulation T; std::map<Point, Coord_type, K::Less_xy_2> function_values; typedef CGAL::Data_access< std::map<Point, Coord_type, K::Less_xy_2 > > Value_access; Coord_type a(0.25), bx(1.3), by(-0.7); for (int y=0 ; y<3 ; y++) for (int x=0 ; x<3 ; x++){ K::Point_2 p(x,y); T.insert(p); function_values.insert(std::make_pair(p,a + bx* x+ by*y)); } //coordinate computation K::Point_2 p(1.3,0.34); std::vector< std::pair< Point, Coord_type > > coords; Coord_type norm = CGAL::natural_neighbor_coordinates_2 (T, p,std::back_inserter(coords)).second; Coord_type res = CGAL::linear_interpolation(coords.begin(), coords.end(), norm, Value_access(function_values)); std::cout << " Tested interpolation on " << p << " interpolation: " << res << " exact: " << a + bx* p.x()+ by* p.y()<< std::endl; return 0; }
// //file: examples/Interpolation/sibson_interpolation_2.C // #include <CGAL/basic.h> #include <utility> #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <CGAL/natural_neighbor_coordinates_2.h> #include <CGAL/Interpolation_gradient_fitting_traits_2.h> #include <CGAL/sibson_gradient_fitting.h> #include <CGAL/interpolation_functions.h> struct K : CGAL::Exact_predicates_inexact_constructions_kernel {}; typedef CGAL::Delaunay_triangulation_2<K> Delaunay_triangulation; typedef CGAL::Interpolation_gradient_fitting_traits_2<K> Traits; typedef K::FT Coord_type; typedef K::Point_2 Point; typedef std::map<Point, Coord_type, K::Less_xy_2> Point_value_map ; typedef std::map<Point, K::Vector_2 , K::Less_xy_2 > Point_vector_map; int main() { Delaunay_triangulation T; Point_value_map function_values; Point_vector_map function_gradients; //parameters for spherical function: Coord_type a(0.25), bx(1.3), by(-0.7), c(0.2); for (int y=0 ; y<4 ; y++) for (int x=0 ; x<4 ; x++){ K::Point_2 p(x,y); T.insert(p); function_values.insert(std::make_pair(p,a + bx* x+ by*y + c*(x*x+y*y))); } sibson_gradient_fitting_nn_2(T,std::inserter(function_gradients, function_gradients.begin()), CGAL::Data_access<Point_value_map> (function_values), Traits()); //coordiante computation K::Point_2 p(1.6,1.4); std::vector< std::pair< Point, Coord_type > > coords; Coord_type norm = CGAL::natural_neighbor_coordinates_2(T, p,std::back_inserter (coords)).second; //Sibson interpolant: version without sqrt: std::pair<Coord_type, bool> res = CGAL::sibson_c1_interpolation_square (coords.begin(), coords.end(),norm,p, CGAL::Data_access<Point_value_map>(function_values), CGAL::Data_access<Point_vector_map>(function_gradients), Traits()); if(res.second) std::cout << " Tested interpolation on " << p << " interpolation: " << res.first << " exact: " << a + bx * p.x()+ by * p.y()+ c*(p.x()*p.x()+p.y()*p.y()) << std::endl; else std::cout << "C^1 Interpolation not successful." << std::endl << " not all function_gradients are provided." << std::endl << " You may resort to linear interpolation." << std::endl; return 0; };
An additional example compares numerically the errors of the different interpolation functions with respect to a known function. It is distributed in the examples directory.