cgal/Orthtree/doc/Orthtree/Orthtree.txt

namespace CGAL {

/*!
\mainpage User Manual
\anchor Chapter_Orthtree

\cgalAutoToc
\authors Jackson Campolattaro, Simon Giraudot, Cédric Portaneri, Tong Zhao, Pierre Alliez

\section Section_Orthtree_Introduction 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.

\cgalFigureBegin{Orthtree_fig, orthtree.png}
Building an %orthtree in 3D (%octree) from a point cloud.
\cgalFigureEnd


\section Section_Orthtree_Building 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()](@ref CGAL::Orthtree::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](@ref CGAL::Orthtrees::Maximum_depth) or [Maximum_number_of_inliers](@ref CGAL::Orthtrees::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.

\subsection Section_Orthtree_Quadtree 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.


\cgalExample{Orthtree/quadtree_build_from_point_vector.cpp}

\subsection Section_Orthtree_Point_Vector 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:

\cgalExample{Orthtree/octree_build_from_point_vector.cpp}

\subsection Section_Orthtree_Point_Set 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.

\cgalExample{Orthtree/octree_build_from_point_set.cpp}

\subsection Section_Orthtree_Custom_Split_Precicate 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.

\cgalExample{Orthtree/octree_build_with_custom_split.cpp}

\subsection Section_Orthtree_Orthtree_Point_Vector 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.

\cgalExample{Orthtree/orthtree_build.cpp}

\section Section_Orthtree_Traversal 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](@ref CGAL::Orthtree::Node) classes provide a
number of different solutions for traversing the tree.

\subsection Section_Orthtree_Manual_Traveral 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()](@ref CGAL::Orthtree::Node::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:

\cgalExample{Orthtree/octree_traversal_manual.cpp}

\subsection Section_Orthtree_Preorder_Traversal 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](@ref CGAL::Orthtrees::Preorder_traversal) and [Postorder_traversal](@ref CGAL::Orthtrees::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.

\cgalExample{Orthtree/octree_traversal_preorder.cpp}

\subsection Section_Orthtree_Custom_Traversal 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:

\cgalExample{Orthtree/octree_traversal_custom.cpp}

\subsection Comparison of Traversals

Figure \cgalFigureRef{Orthtree_traversal_fig} shows in which order
nodes are visited depending on the traversal method used.

\cgalFigureBegin{Orthtree_traversal_fig, quadtree_traversal.png}
%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.
\cgalFigureEnd

\section Section_Orthtree_Acceleration Acceleration of Common Tasks

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

\subsection Section_Orthtree_Nearest_Neighbor 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.

\cgalExample{Orthtree/octree_find_nearest_neighbor.cpp}

\subsection Section_Orthtree_Grade Grading

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

\cgalFigureBegin{Orthtree_quadree_graded_fig, quadtree_graded.png}
%Quadtree before and after being graded.
\cgalFigureEnd

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()](@ref CGAL::Orthtree::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.

\cgalExample{Orthtree/octree_grade.cpp}

\section Section_Orthtree_Performance Performance

\subsection Section_Orthtree_Performance_Construction 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.

\cgalFigureBegin{Orthtree_construction_benchmark_fig, construction_benchmark.png}
%Plot of the time to construct a tree.
\cgalFigureEnd

\subsection Section_Orthtree_Performance_Nearest_Neighbors 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.

\cgalFigureBegin{Orthtree_nearest_neighbor_benchmark_fig, nearest_neighbor_benchmark.png}
%Plot of the time to find the 10 nearest neighbors of a random point using a pre-constructed tree.
\cgalFigureEnd

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.

\cgalFigureBegin{Orthtree_nearest_neighbor_benchmark_with_naive_fig, nearest_neighbor_benchmark_with_naive.png}
%Plot of the time to find nearest neighbors using tree methods and a naive approach.
\cgalFigureEnd

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

\section Section_Orthtree_History 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
2020. 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.

*/

}