cgal/Orthtree/doc/Orthtree/Orthtree.txt

namespace CGAL {

/*!
\mainpage User Manual
\anchor Chapter_Orthtree

\cgalAutoToc
\authors Jackson Campolattaro, Simon Giraudot, Cedric 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 contents 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

A common purpose for an orthtree is to subdivide a collection of points,
and the Orthtree package provides a traits class for this purpose.
The points are not copied: the provided point range is used directly and rearranged by
the orthtree. Altering the point range after creating the orthtree
may leave the tree 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 for the point-set orthtree,
including [Maximum_depth](@ref CGAL::Orthtrees::Maximum_depth)
and [Maximum_number_of_inliers](@ref CGAL::Orthtrees::Maximum_number_of_inliers).

The simplest way to create a point-set orthtree is from 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.

The following example shows the construction of an Octree from a vector of points.
`octree.refine(10, 1)` uses the default split predicate, which
enforces a max-depth and a maximum number of inliers per node ("bucket size").
Nodes are split if their depth is less than 10, and they contain more than one inlier.

\cgalExample{Orthtree/octree_build_from_point_vector.cpp}

\subsection Section_Orthtree_Quadtree Building a Quadtree

The `Orthtree` class may be templated with `Orthtree_traits_point<>`
with a 2d dimension tag and thus behave as a %quadtree.
For convenience, the alias `Quadtree` is provided.

The following example shows the construction of %quadtree from a vector of `Point_2` objects.
`quadtree.refine(10, 5)` uses the default split predicate, which
enforces a max-depth and a maximum number of inliers per node ("bucket size").
Nodes are split if their depth is less than 10, and they contain more than 5 inliers.


\cgalExample{Orthtree/quadtree_build_from_point_vector.cpp}

\subsection Section_Orthtree_Point_Vector Building an Octree

`Orthtree_traits_point<>` can also be templated with a 3d dimension tag 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.
As with the %quadtree example, we use the default split predicate.
In this case the max depth is 10, and the bucket size is 1.


\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 corresponding map,
and then written to the standard output.

The split predicate is manually constructed and passed to the refine method.
In this case, we use a maximum number of inliers with no corresponding max depth,
this means that nodes will continue to be split until none contain more than 10 points.

\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

An orthtree can also be used with an arbitrary number of dimensions.
The `Orthtree_traits_point` template can infer the arbitrary dimension count from the d-dimensional kernel.

The following example shows how to build a generalized orthtree in dimension 4.
As `std::vector<Point_d>` is manually filled with 4-dimensional points.
The vector is used as the point set, and 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` class provides 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_index` type provides a handle on a node, and the `orthree` class provides methods
that enable the user to retrieve the indices of each of its children as well as its parent (if they exist).

Given any node index `node`, the `n`th child of that node can be found with `orthtree.child(node, n)`.
For an octree, values of `n` from 0-7 provide access to the different children.
For non-root nodes, it is possible to access parent nodes using the `orthtree.parent()` accessor.

To access grandchildren, it isn't necessary to use nested `orthtree.child()` calls.
Instead, the `orthtree.descendant(node, a, b, ...)` convenience method is provided.
This retrieves the `b`th child of the `a`th child, to any depth.

In most cases, we want to find the descendants of the root node.
For this case, there is another convenience method `orthtree.node(a, b, c, ...)`.
This retrieves the node specified by the descent `a`, `b`, `c`.
It is equivalent to `orthtree.descendant(orthtree.root(), a, b, c, ...)`.

The following example demonstrates the use of several of these accessors.

\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 custom traversal must provide a method which returns the starting point of the traversal (`first_index()`)
and another method which returns the next node in the sequence (`next_index()`).
`next_index()` returns an empty optional when the end of the traversal is reached.
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}

Not all octrees are compatible with nearest neighbor functionality,
as the idea of a nearest neighbor may not make sense for some tree contents.
For the nearest neighbor functions to work, the traits class must implement the
`CollectionPartitioningOrthtreeTraits` concept.

\subsection Section_Orthtree_Grade Grading

An orthtree is considered "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 \cgalBigO{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 Cedric 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.

*/

}