AABB Tree

官方文档链接：CGAL 5.5 – 3D Fast Intersection and Distance Computation (AABB Tree): User Manual

1 介绍

AABB树提供了一个静态的数据结构和算法，能够对有限3D几何对象集合进行高效的相交和距离查询。

相交查询可以是任何类型，前提是在traits类中实现了相应的交集谓词和构造函数。
距离查询仅限于点的查询。

AABB树的数据结构将几何数据的迭代器范围作为输入，然后将其转换为primitives（图元）。在这些primitives中，构造了一个轴对齐边界框（axis-aligned bounding boxes）（AABB）的层次结构，用于加速相交和距离查询。每个图元都能访问一个输入几何对象（datum）和该对象的参考id。例如，一个图元将3D triangle作为datum，多面体表面的face handle作为id。而通过AABB tree进行相交和距离查询时，返回值中就包含了相交对象/最近点和相交图元id/最近图元id。

左图为表面三角网格模型，右图为其构建的AABB树。

2 接口

相交：

AABB_tree::do_intersect()
AABB_tree::number_of_intersected_primitives()
AABB_tree::all_intersected_primitives()
AABB_tree::any_intersected_primitive()
AABB_tree::first_intersected_primitive()

以上函数不会构造相交对象，仅做测试。

AABB_tree::all_intersections()
AABB_tree::any_intersection()
AABB_tree::first_intersection()

以上函数会构造相交的对象。

距离：

AABB_tree::closest_point()
AABB_tree::closest_point_and_primitive()
AABB_tree::accelerate_distance_queries()

注意，在AABB Tree中应避免出现退化的图元，防止算法出错。

3 几个栗子

下面例子中，三维三角形集合以list的形式存储。AABB图元将三角形（triangle）作为datum（数据），list里的迭代器作为id。程序中实现了射线与三角形集合的相交查询，点与三角形集合的最近点查询和距离计算。

// Author(s) : Camille Wormser, Pierre Alliez
#include <iostream>
#include <list>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/AABB_tree.h>
#include <CGAL/AABB_traits.h>
#include <CGAL/AABB_triangle_primitive.h>
typedef CGAL::Simple_cartesian<double> K;//内核定义
typedef K::FT FT;
typedef K::Ray_3 Ray;
typedef K::Line_3 Line;
typedef K::Point_3 Point;
typedef K::Triangle_3 Triangle;
typedef std::list<Triangle>::iterator Iterator; //三角形list迭代器
typedef CGAL::AABB_triangle_primitive<K, Iterator> Primitive;
typedef CGAL::AABB_traits<K, Primitive> AABB_triangle_traits;
typedef CGAL::AABB_tree<AABB_triangle_traits> Tree;
int main()
{
    Point a(1.0, 0.0, 0.0);
    Point b(0.0, 1.0, 0.0);
    Point c(0.0, 0.0, 1.0);
    Point d(0.0, 0.0, 0.0);
    std::list<Triangle> triangles;
    triangles.push_back(Triangle(a,b,c));
    triangles.push_back(Triangle(a,b,d));
    triangles.push_back(Triangle(a,d,c));
    // constructs AABB tree
    Tree tree(triangles.begin(),triangles.end());
    // counts #intersections
    Ray ray_query(a,b);
    std::cout << tree.number_of_intersected_primitives(ray_query)
        << " intersections(s) with ray query" << std::endl;
    // compute closest point and squared distance
    Point point_query(2.0, 2.0, 2.0);
    Point closest_point = tree.closest_point(point_query);
    std::cerr << "closest point is: " << closest_point << std::endl;
    FT sqd = tree.squared_distance(point_query);
    std::cout << "squared distance: " << sqd << std::endl;
    return EXIT_SUCCESS;
}

在下面这个例子中，将创建一个多面体三角面片的AABB树。其中，AABB图元将三角形面片句柄包装为id，对应的面片作为几何对象（datum）。

// Author(s) : Camille Wormser, Pierre Alliez
#include <iostream>
#include <list>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/AABB_tree.h>
#include <CGAL/AABB_traits.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/AABB_face_graph_triangle_primitive.h>
typedef CGAL::Simple_cartesian<double> K;
typedef K::Point_3 Point;
typedef K::Plane_3 Plane;
typedef K::Vector_3 Vector;
typedef K::Segment_3 Segment;
typedef K::Ray_3 Ray;
typedef CGAL::Polyhedron_3<K> Polyhedron;
typedef CGAL::AABB_face_graph_triangle_primitive<Polyhedron> Primitive;
typedef CGAL::AABB_traits<K, Primitive> Traits;
typedef CGAL::AABB_tree<Traits> Tree;
typedef boost::optional< Tree::Intersection_and_primitive_id<Segment>::Type > Segment_intersection;
typedef boost::optional< Tree::Intersection_and_primitive_id<Plane>::Type > Plane_intersection;
typedef Tree::Primitive_id Primitive_id;
int main()
{
    Point p(1.0, 0.0, 0.0);
    Point q(0.0, 1.0, 0.0);
    Point r(0.0, 0.0, 1.0);
    Point s(0.0, 0.0, 0.0);
    Polyhedron polyhedron;
    polyhedron.make_tetrahedron(p, q, r, s);
    // constructs AABB tree
    Tree tree(faces(polyhedron).first, faces(polyhedron).second, polyhedron);
    // constructs segment query
    Point a(-0.2, 0.2, -0.2);
    Point b(1.3, 0.2, 1.3);
    Segment segment_query(a,b);
    // tests intersections with segment query
    if(tree.do_intersect(segment_query))
        std::cout << "intersection(s)" << std::endl;
    else
        std::cout << "no intersection" << std::endl;
    // computes #intersections with segment query
    std::cout << tree.number_of_intersected_primitives(segment_query)
        << " intersection(s)" << std::endl;
    // computes first encountered intersection with segment query
    // (generally a point)
    Segment_intersection intersection =
        tree.any_intersection(segment_query);
    if(intersection)
    {
        // gets intersection object
      const Point* p = boost::get<Point>(&(intersection->first));
      if(p)
        std::cout << "intersection object is a point " << *p << std::endl;
    }
    // computes all intersections with segment query (as pairs object - primitive_id)
    std::list<Segment_intersection> intersections;
    tree.all_intersections(segment_query, std::back_inserter(intersections));
    // computes all intersected primitives with segment query as primitive ids
    std::list<Primitive_id> primitives;
    tree.all_intersected_primitives(segment_query, std::back_inserter(primitives));
    // constructs plane query
    Vector vec(0.0,0.0,1.0);
    Plane plane_query(a,vec);
    // computes first encountered intersection with plane query
    // (generally a segment)
    Plane_intersection plane_intersection = tree.any_intersection(plane_query);
    if(plane_intersection)
    {
      if(boost::get<Segment>(&(plane_intersection->first)))
            std::cout << "intersection object is a segment" << std::endl;
    }
    return EXIT_SUCCESS;
}

下面例子先读取一个闭合的多面体表面，然后以每个face的重心为起始点和垂直于face向模型内部的方向作射线，进行一个ray shooting query。

#include <CGAL/Simple_cartesian.h>
#include <CGAL/Surface_mesh.h>
#include <CGAL/AABB_tree.h>
#include <CGAL/AABB_traits.h>
#include <CGAL/AABB_face_graph_triangle_primitive.h>
#include <CGAL/Polygon_mesh_processing/compute_normal.h>
#include <CGAL/Polygon_mesh_processing/orientation.h>
#include <iostream>
#include <fstream>
typedef CGAL::Simple_cartesian<double> K;
typedef K::FT FT;
typedef K::Point_3 Point;
typedef K::Vector_3 Vector;
typedef K::Ray_3 Ray;
typedef CGAL::Surface_mesh<Point> Mesh;
typedef boost::graph_traits<Mesh>::face_descriptor face_descriptor;
typedef boost::graph_traits<Mesh>::halfedge_descriptor halfedge_descriptor;
typedef CGAL::AABB_face_graph_triangle_primitive<Mesh> Primitive;
typedef CGAL::AABB_traits<K, Primitive> Traits;
typedef CGAL::AABB_tree<Traits> Tree;
typedef boost::optional<Tree::Intersection_and_primitive_id<Ray>::Type> Ray_intersection;
struct Skip
{
  face_descriptor fd;
  Skip(const face_descriptor fd)
    : fd(fd)
  {}
  bool operator()(const face_descriptor& t) const
  { if(t == fd){
      std::cerr << "ignore " << t  <<std::endl;
    };
    return(t == fd);
  }
};
int main(int argc, char* argv[])
{
  const std::string filename = (argc > 1) ? argv[1] : CGAL::data_file_path("meshes/tetrahedron.off");
  Mesh mesh;
  if(!CGAL::IO::read_polygon_mesh(filename, mesh))
  {
    std::cerr << "Invalid input." << std::endl;
    return 1;
  }
  Tree tree(faces(mesh).first, faces(mesh).second, mesh);
  double d = CGAL::Polygon_mesh_processing::is_outward_oriented(mesh)?-1:1;
  for(face_descriptor fd : faces(mesh))
  {
    halfedge_descriptor hd = halfedge(fd,mesh);
    Point p = CGAL::centroid(mesh.point(source(hd,mesh)),
                             mesh.point(target(hd,mesh)),
                             mesh.point(target(next(hd,mesh),mesh)));
    Vector v = CGAL::Polygon_mesh_processing::compute_face_normal(fd,mesh);
    Ray ray(p,d * v);
    Skip skip(fd);
    Ray_intersection intersection = tree.first_intersection(ray, skip);
    if(intersection)
    {
      if(boost::get<Point>(&(intersection->first))){
        const Point* p =  boost::get<Point>(&(intersection->first) );
        std::cout <<  *p << std::endl;
      }
    }
  }
  std::cerr << "done" << std::endl;
  return 0;
}

因为重心计算属于浮点运算，因此射线第一个击中的面可能是起点质心所在的面。为了避免此状况，这里需要给first_intersection()传入一个skip functor将此面忽略。

上个例子是计算的射线与mesh的相交，下面这个例子展示如何查询一个点到mesh的squared distance和closest point及其所在的triangle。


// Author(s) : Pierre Alliez
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/AABB_tree.h>
#include <CGAL/AABB_traits.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/AABB_face_graph_triangle_primitive.h>
typedef CGAL::Simple_cartesian<double> K;
typedef K::FT FT;
typedef K::Point_3 Point;
typedef K::Segment_3 Segment;
typedef CGAL::Polyhedron_3<K> Polyhedron;
typedef CGAL::AABB_face_graph_triangle_primitive<Polyhedron> Primitive;
typedef CGAL::AABB_traits<K, Primitive> Traits;
typedef CGAL::AABB_tree<Traits> Tree;
typedef Tree::Point_and_primitive_id Point_and_primitive_id;
int main()
{
    Point p(1.0, 0.0, 0.0);
    Point q(0.0, 1.0, 0.0);
    Point r(0.0, 0.0, 1.0);
    Point s(0.0, 0.0, 0.0);
    Polyhedron polyhedron;
    polyhedron.make_tetrahedron(p, q, r, s);
    // constructs AABB tree and computes internal KD-tree
    // data structure to accelerate distance queries
    Tree tree(faces(polyhedron).first, faces(polyhedron).second, polyhedron);
    // query point
    Point query(0.0, 0.0, 3.0);
    // computes squared distance from query
    FT sqd = tree.squared_distance(query);
    std::cout << "squared distance: " << sqd << std::endl;
    // computes closest point
    Point closest = tree.closest_point(query);
    std::cout << "closest point: " << closest << std::endl;
    // computes closest point and primitive id
    Point_and_primitive_id pp = tree.closest_point_and_primitive(query);
    Point closest_point = pp.first;
    Polyhedron::Face_handle f = pp.second; // closest primitive id
    std::cout << "closest point: " << closest_point << std::endl;
    std::cout << "closest triangle: ( "
              << f->halfedge()->vertex()->point() << " , "
              << f->halfedge()->next()->vertex()->point() << " , "
              << f->halfedge()->next()->next()->vertex()->point()
              << " )" << std::endl;
    return EXIT_SUCCESS;
}

最后一个例子，是对于AABB树中图元的增量插入。虽然AABB树是一个静态的数据结构，但是它允许插入primitives（图元）。

#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/AABB_tree.h>
#include <CGAL/AABB_traits.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/AABB_face_graph_triangle_primitive.h>
typedef CGAL::Simple_cartesian<double> K;
typedef K::FT FT;
typedef K::Point_3 Point;
typedef K::Segment_3 Segment;
typedef CGAL::Polyhedron_3<K> Polyhedron;
typedef CGAL::AABB_face_graph_triangle_primitive<Polyhedron, CGAL::Default, CGAL::Tag_false> Primitive;
typedef CGAL::AABB_traits<K, Primitive> Traits;
typedef CGAL::AABB_tree<Traits> Tree;
typedef Tree::Point_and_primitive_id Point_and_primitive_id;
int main()
{
    Point p(1.0, 0.0, 0.0);
    Point q(0.0, 1.0, 0.0);
    Point r(0.0, 0.0, 1.0);
    Point s(0.0, 0.0, 0.0);
    Polyhedron polyhedron1;
    polyhedron1.make_tetrahedron(p, q, r, s);
    Point p2(11.0, 0.0, 0.0);
    Point q2(10.0, 1.0, 0.0);
    Point r2(10.0, 0.0, 1.0);
    Point s2(10.0, 0.0, 0.0);
    Polyhedron polyhedron2;
    polyhedron2.make_tetrahedron(p2, q2, r2, s2);
    // constructs AABB tree and computes internal KD-tree
    // data structure to accelerate distance queries
    Tree tree(faces(polyhedron1).first, faces(polyhedron1).second, polyhedron1);
    tree.insert(faces(polyhedron2).first, faces(polyhedron2).second, polyhedron2);
    // query point
    Point query(0.0, 0.0, 3.0);
    // computes squared distance from query
    FT sqd = tree.squared_distance(query);
    std::cout << "squared distance: " << sqd << std::endl;
    return EXIT_SUCCESS;
}