CGAL 5.4 - Quadtrees, Octrees, and Orthtrees
User Manual

# Introduction

Quadtrees are tree data structures in which each node encloses a square section of space, and each internal node has exactly 4 children. Octrees are a similar data structure in 3D in which each node encloses a cubic section of space, and each internal node has exactly 8 children.

We call the generalization of such data structure "orthtrees", as orthants are generalizations of quadrants and octants. The term "hyperoctree" can also be found in literature to name such data structures in dimensions 4 and higher.

This package provides a general data structure Orthtree along with aliases for Quadtree and Octree. These trees can be constructed with custom point ranges and split predicates, and iterated on with various traversal methods.

Figure 91.1 Building an orthtree in 3D (octree) from a point cloud.

# Building

An orthtree is created using a set of points. The points are not copied: the provided point range is used directly and is rearranged by the orthtree. Altering the point range after creating the orthtree might leave it in an invalid state. The constructor returns a tree with a single (root) node that contains all the points.

The method refine() must be called to subdivide space further. This method uses a split predicate which takes a node as input and returns true if this node should be split, false otherwise: this enables users to choose on what criterion should the orthtree be refined. Predefined predicates are provided such as Maximum_depth or Maximum_number_of_inliers.

The simplest way to create an orthtree is using a vector of points. The constructor generally expects a separate point range and map, but the point map defaults to Identity_property_map if none is provided.

The split predicate is a user-defined functor that determines whether a node needs to be split. Custom predicates can easily be defined if the existing ones do not match users' needs.

## Building a Quadtree

The Orthtree class may be templated with Orthtree_traits_2 and thus behave as a quadtree. For convenience, the alias Quadtree is provided.

The following example shows how to create a quadtree object from a vector of Point_2 objects and refine it, which means constructing the tree's space subdivision itself, using a maximum depth of 10 and a maximum number of inliers per node (bucket size) of 5. The refinement is stopped as soon as one of the conditions is violated: if a node has more inliers than bucket_size but is already at max_depth, it is not split. Similarly, a node that is at a depth smaller than max_depth but already has fewer inliers than bucket_size is not split.

#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Random.h>
// Type Declarations
typedef Kernel::Point_2 Point_2;
typedef std::vector<Point_2> Point_vector;
int main()
{
CGAL::Random r;
Point_vector points_2d;
for (std::size_t i = 0; i < 5; ++ i)
points_2d.emplace_back(r.get_double(-1., 1.),
r.get_double(-1., 1.));
return EXIT_SUCCESS;
}

## Building an Octree

The Orthtree class may be templated with Orthtree_traits_3 and thus behave as an octree. For convenience, the alias Octree is provided.

The following example shows how to create an octree from a vector of Point_3 objects:

#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef std::list<Point> Point_vector;
int main() {
// Here, our point set is a vector
Point_vector points;
// Add a few points to the vector
points.emplace_back(1, 1, 1);
points.emplace_back(2, 1, -11);
points.emplace_back(2, 1, 1);
points.emplace_back(1, -2, 1);
points.emplace_back(1, 1, 1);
points.emplace_back(-1, 1, 1);
// Create an octree from the points
Octree octree(points);
// Build the octree
octree.refine(10, 20);
// Print out the tree
std::cout << octree;
return EXIT_SUCCESS;
}

## Building an Octree from a Point_set_3

Some data structures such as Point_set_3 require a non-default point map type and object. This example illustrates how to create an octree from a Point_set_3 loaded from a file. It also shows a more explicit way of setting the split predicate when refining the tree.

An octree is constructed from the point set and its map. The tree is refined with a maximum depth (deepest node allowed) of 10, and a bucket size (maximum number of points contained by a single node) of 20. The tree is then written to the standard output.

The split predicate is manually constructed and passed to the refine method.

#include <fstream>
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
#include <CGAL/Point_set_3.h>
#include <CGAL/Point_set_3/IO.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef CGAL::Point_set_3<Point> Point_set;
typedef Point_set::Point_map Point_map;
int main(int argc, char **argv) {
// Point set will be used to hold our points
Point_set points;
// Load points from a file.
std::ifstream stream((argc > 1) ? argv[1] : CGAL::data_file_path("points_3/cube.pwn"));
stream >> points;
if (0 == points.number_of_points()) {
std::cerr << "Error: cannot read file" << std::endl;
return EXIT_FAILURE;
}
std::cout << "loaded " << points.number_of_points() << " points\n" << std::endl;
// Create an octree from the points
Octree octree(points, points.point_map());
// Build the octree with a small bucket size, using a more verbose method
// Print out the tree
std::cout << octree;
return EXIT_SUCCESS;
}

## Building an Octree with a Custom Split Predicate

The following example illustrates how to refine an octree using a split predicate that isn't provided by default. This particular predicate sets a node's bucket size as a ratio of its depth. For example, for a ratio of 2, a node at depth 2 can hold 4 points, a node at depth 7 can hold 14.

#include <fstream>
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
#include <CGAL/Point_set_3.h>
#include <CGAL/Point_set_3/IO.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef Kernel::FT FT;
typedef CGAL::Point_set_3<Point> Point_set;
typedef Point_set::Point_map Point_map;
// Split Predicate
// The predicate is a functor which returns a boolean value, whether a node needs to be split or not
struct Split_by_ratio {
std::size_t ratio;
Split_by_ratio(std::size_t ratio) : ratio(ratio) {}
template<class Node>
bool operator()(const Node &n) const {
return n.size() > (ratio * n.depth());
}
};
int main(int argc, char **argv) {
// Point set will be used to hold our points
Point_set points;
// Load points from a file.
std::ifstream stream((argc > 1) ? argv[1] : CGAL::data_file_path("points_3/cube.pwn"));
stream >> points;
if (0 == points.number_of_points()) {
std::cerr << "Error: cannot read file" << std::endl;
return EXIT_FAILURE;
}
std::cout << "loaded " << points.number_of_points() << " points" << std::endl;
// Create an octree from the points
Octree octree(points, points.point_map());
// Build the octree using our custom split predicate
octree.refine(Split_by_ratio(2));
// Print out the tree
std::cout << octree;
return EXIT_SUCCESS;
}

## Building an Orthtree

The following example shows how to build an generalized orthtree in dimension 4. A std::vector<Point_d> is manually filled with points. The vector is used as the point set, an Identity_property_map is automatically set as the orthtree's map type, so a map does not need to be provided.

#include <iostream>
#include <CGAL/Epick_d.h>
#include <CGAL/Orthtree.h>
#include <CGAL/Orthtree_traits_d.h>
#include <CGAL/Random.h>
// Type Declarations
typedef CGAL::Dimension_tag<4> Dimension;
typedef CGAL::Epick_d<Dimension> Kernel;
typedef Kernel::Point_d Point_d;
typedef std::vector<Point_d> Point_vector;
int main()
{
CGAL::Random r;
Point_vector points_dd;
for (std::size_t i = 0; i < 5; ++ i)
{
std::array<double, Dimension::value> init;
for (double& v : init)
v = r.get_double(-1., 1.);
points_dd.emplace_back (init.begin(), init.end());
}
Orthtree orthtree(points_dd);
orthtree.refine(10, 5);
return EXIT_SUCCESS;
}

# Traversal

Note
For simplicity, the rest of the user manual will only use octrees, but all the presented features also apply to quadtrees and higher dimension orthtrees.

Traversal is the act of navigating among the nodes of the tree. The Orthtree and Node classes provide a number of different solutions for traversing the tree.

## Manual Traversal

Because our orthtree is a form of connected acyclic undirected graph, it is possible to navigate between any two nodes. What that means in practice, is that given a node on the tree, it is possible to access any other node using the right set of operations. The Node class provides functions that enable the user to access each of its children, as well as its parent (if it exists).

The following example demonstrates ways of accessing different nodes of a tree, given a reference to one.

From the root node, children can be accessed using the subscript operator CGAL::Orthtree::Node::operator[](). For an octree, values from 0-7 provide access to the different children.

For non-root nodes, it is possible to access parent nodes using the parent() accessor.

These accessors and operators can be chained to access any node in the tree in a single line of code, as shown in the following example:

#include <fstream>
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
#include <CGAL/Point_set_3.h>
#include <CGAL/Point_set_3/IO.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef CGAL::Point_set_3<Point> Point_set;
typedef Point_set::Point_map Point_map;
int main(int argc, char **argv) {
// Point set will be used to hold our points
Point_set points;
// Load points from a file.
std::ifstream stream((argc > 1) ? argv[1] : CGAL::data_file_path("points_3/cube.pwn"));
stream >> points;
if (0 == points.number_of_points()) {
std::cerr << "Error: cannot read file" << std::endl;
return EXIT_FAILURE;
}
std::cout << "loaded " << points.number_of_points() << " points\n" << std::endl;
// Create an octree from the points
Octree octree(points, points.point_map());
// Build the octree using the default arguments
octree.refine();
// Print out a few nodes
std::cout << "Navigation relative to the root node" << std::endl;
std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl;
std::cout << "the root node: " << std::endl;
std::cout << octree.root() << std::endl;
std::cout << "the first child of the root node: " << std::endl;
std::cout << octree.root()[0] << std::endl;
std::cout << "the fifth child: " << std::endl;
std::cout << octree.root()[4] << std::endl;
std::cout << "the fifth child, accessed without the root keyword: " << std::endl;
std::cout << octree[4] << std::endl;
std::cout << "the second child of the fourth child: " << std::endl;
std::cout << octree.root()[4][1] << std::endl;
std::cout << "the second child of the fourth child, accessed without the root keyword: " << std::endl;
std::cout << octree[4][1] << std::endl;
std::cout << std::endl;
// Retrieve one of the deeper children
Octree::Node cur = octree[3][2];
std::cout << "Navigation relative to a child node" << std::endl;
std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl;
std::cout << "the third child of the fourth child: " << std::endl;
std::cout << cur << std::endl;
std::cout << "the third child: " << std::endl;
std::cout << cur.parent() << std::endl;
std::cout << "the next sibling of the third child of the fourth child: " << std::endl;
std::cout << cur.parent()[cur.local_coordinates().to_ulong() + 1] << std::endl;
return EXIT_SUCCESS;
}

## Preorder Traversal

It is often useful to be able to iterate over the nodes of the tree in a particular order. For example, the stream operator << uses a traversal to print out each node. A few traversals are provided, among them Preorder_traversal and Postorder_traversal. To traverse a tree in preorder is to visit each parent immediately followed by its children, whereas in postorder, traversal the children are visited first.

The following example illustrates how to use the provided traversals.

A tree is constructed, and a traversal is used to create a range that can be iterated over using a for-each loop. The default output operator for the orthtree uses the preorder traversal to do a pretty-print of the tree structure. In this case, we print out the nodes of the tree without indentation instead.

#include <fstream>
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
#include <CGAL/Point_set_3.h>
#include <CGAL/Point_set_3/IO.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef CGAL::Point_set_3<Point> Point_set;
typedef Point_set::Point_map Point_map;
typedef CGAL::Orthtrees::Preorder_traversal Preorder_traversal;
int main(int argc, char **argv) {
// Point set will be used to hold our points
Point_set points;
// Load points from a file.
std::ifstream stream((argc > 1) ? argv[1] : CGAL::data_file_path("points_3/cube.pwn"));
stream >> points;
if (0 == points.number_of_points()) {
std::cerr << "Error: cannot read file" << std::endl;
return EXIT_FAILURE;
}
std::cout << "loaded " << points.number_of_points() << " points\n" << std::endl;
// Create an octree from the points
Octree octree(points, points.point_map());
// Build the octree
octree.refine();
// Print out the octree using preorder traversal
for (Octree::Node node : octree.traverse<Preorder_traversal>()) {
std::cout << node << std::endl;
}
return EXIT_SUCCESS;
}

## Custom Traversal

Users can define their own traversal methods by creating models of the OrthtreeTraversal concept. The following example shows how to define a custom traversal that only traverses the first branch of the octree:

#include <fstream>
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
#include <CGAL/Point_set_3.h>
#include <CGAL/Point_set_3/IO.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef CGAL::Point_set_3<Point> Point_set;
typedef Point_set::Point_map Point_map;
typedef Octree::Node Node;
struct First_branch_traversal
{
Node first (Node root) const
{
return root;
}
Node next (Node n) const
{
if (n.is_leaf())
return Node();
return n[0];
}
};
int main(int argc, char **argv) {
// Point set will be used to hold our points
Point_set points;
// Load points from a file.
std::ifstream stream((argc > 1) ? argv[1] : CGAL::data_file_path("points_3/cube.pwn"));
stream >> points;
if (0 == points.number_of_points()) {
std::cerr << "Error: cannot read file" << std::endl;
return EXIT_FAILURE;
}
std::cout << "loaded " << points.number_of_points() << " points\n" << std::endl;
// Create an octree from the points
Octree octree(points, points.point_map());
// Build the octree
octree.refine();
// Print out the first branch using custom traversal
for (auto &node : octree.traverse<First_branch_traversal>())
std::cout << node << std::endl;
return EXIT_SUCCESS;
}

## of Traversals

Figure Figure 91.2 shows in which order nodes are visited depending on the traversal method used.

Figure 91.2 Quadtree visualized as a graph. Each node is labelled according to the order in which it is visited by the traversal. When using leaves and level traversals, the quadtree is only partially traversed.

# Acceleration of Common Tasks

Once an orthtree is built, its structure can be used to accelerate different tasks.

## Finding the Nearest Neighbor of a Point

The naive way of finding the nearest neighbor of a point requires finding the distance to every other point. An orthtree can be used to perform the same task in significantly less time. For large numbers of points, this can be a large enough difference to outweigh the time spent building the tree.

Note that a kd-tree is expected to outperform the orthtree for this task, it should be preferred unless features specific to the orthtree are needed.

The following example illustrates how to use an octree to accelerate the search for points close to a location.

Points are loaded from a file and an octree is built. The nearest neighbor method is invoked for several input points. A k value of 1 is used to find the single closest point. Results are put in a vector, and then printed.

#include <fstream>
#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
#include <CGAL/Point_set_3.h>
#include <CGAL/Point_set_3/IO.h>
#include <boost/iterator/function_output_iterator.hpp>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef CGAL::Point_set_3<Point> Point_set;
typedef Point_set::Point_map Point_map;
int main(int argc, char **argv) {
// Point set will be used to hold our points
Point_set points;
// Load points from a file.
std::ifstream stream((argc > 1) ? argv[1] : CGAL::data_file_path("points_3/cube.pwn"));
stream >> points;
if (0 == points.number_of_points()) {
std::cerr << "Error: cannot read file" << std::endl;
return EXIT_FAILURE;
}
std::cout << "loaded " << points.number_of_points() << " points" << std::endl;
// Create an octree from the points
Octree octree(points, points.point_map());
// Build the octree
octree.refine(10, 20);
// Find the nearest points to a few locations
std::vector<Point> points_to_find = {
{0, 0, 0},
{1, 1, 1},
{-1, -1, -1},
{-0.46026, -0.25353, 0.32051},
{-0.460261, -0.253533, 0.320513}
};
for (const Point& p : points_to_find)
octree.nearest_neighbors
(p, 1, // k=1 to find the single closest point
boost::make_function_output_iterator
([&](const Point& nearest)
{
std::cout << "the nearest point to (" << p <<
") is (" << nearest << ")" << std::endl;
}));
return EXIT_SUCCESS;
}

An orthtree is graded if the difference of depth between two adjacent leaves is at most 1 for every pair of leaves.

Figure 91.3 Quadtree before and after being graded.

The following example demonstrates how to use the grade method to eliminate large jumps in depth within the orthtree.

A tree is created such that one node is split many more times than those it borders. grade() splits the octree's nodes so that adjacent nodes never have a difference in depth greater than one. The tree is printed before and after grading, so that the differences are visible.

#include <iostream>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Octree.h>
// Type Declarations
typedef Kernel::Point_3 Point;
typedef std::vector<Point> Point_vector;
int main() {
// Here, our point set is a vector
Point_vector points;
// Add a few points to the vector, most of which are in one region
points.emplace_back(1, 1, 1);
points.emplace_back(2, 1, -11);
points.emplace_back(2, 1, 1);
points.emplace_back(1, -2, 1);
points.emplace_back(1, 1, 1);
points.emplace_back(-1, 1, 1);
points.emplace_back(-1.1, 1, 1);
points.emplace_back(-1.01, 1, 1);
points.emplace_back(-1.001, 1, 1);
points.emplace_back(-1.0001, 1, 1);
points.emplace_back(-1.0001, 1, 1);
// Create an octree from the points
Octree octree(points);
// Build the octree with a small bucket size, so we get a deep node
octree.refine(10, 2);
// Print out the tree
std::cout << "\nUn-graded tree" << std::endl;
std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl;
std::cout << octree << std::endl;
// Grade the tree to eliminate large jumps in depth
// Print out the tree again
std::cout << "\nGraded tree" << std::endl;
std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl;
std::cout << octree << std::endl;
return EXIT_SUCCESS;
}

# Performance

## Tree Construction

Tree construction benchmarks were conducted by randomly generating a collection of points, and then timing the process of creating a fully refined tree which contains them.

Because of its simplicity, an octree can be constructed faster than a kd-tree.

Figure 91.4 Plot of the time to construct a tree.

## Nearest Neighbors

Orthtree nodes are uniform, so orthtrees will tend to have deeper hierarchies than equivalent kd-trees. As a result, orthtrees will generally perform worse for nearest neighbor searches. Both nearest neighbor algorithms have a theoretical complexity of O(log(n)), but the orthtree can generally be expected to have a higher coefficient.

Figure 91.5 Plot of the time to find the 10 nearest neighbors of a random point using a pre-constructed tree.

The performance difference between the two trees is large, but both algorithms compare very favorably to the linear complexity of the naive approach, which involves comparing every point to the search point.

Using the orthtree for nearest neighbor computations instead of the kd-tree can be justified either when few queries are needed (as the construction is faster) or when the orthtree is also needed for other purposes.

Figure 91.6 Plot of the time to find nearest neighbors using tree methods and a naive approach.

For nontrivial point counts, the naive approach's calculation time dwarfs that of either the orthtree or kd-tree.

# History

A prototype code was implemented by Pierre Alliez and improved by Tong Zhao and Cédric Portaneri. From this prototype code, the package was developed by Jackson Campolatarro as part of the Google Summer of Code

1. Simon Giraudot, supervisor of the GSoC internship, completed and finalized the package for integration in CGAL 5.3. Pierre Alliez provided kind help and advice all the way through.