The Voronoi extensions of the Boost Polygon library provide functionality to
construct a Voronoi diagram of a set of points
and linear segments in 2D space with the following set of limitations:
Координаты входных точек и конечных точек входных сегментов должны иметь интегральный тип. Тип данных int32 поддерживается реализацией по умолчанию. Поддержка других типов данных (например, int64) может быть достигнута за счет конфигурации признаков типа координат (Voronoi Advanced tutorial).
Точки входа и сегменты не должны пересекаться, кроме их конечных точек. Это означает, что входная точка не должна находиться внутри входного сегмента, и входные сегменты не должны пересекаться, кроме своих конечных точек.
While the first restriction is permanent (it
allows to give the exact warranties about the output precision and
algorithm execution flow),
the second one may be resolved using the Boost.Polygon segment utils.
The strong sides of the
library and the main benefits comparing to the other implementations are
discussed in the following paragraphs.
Fully Functional with Segments
There are just a few implementations of the Voronoi diagram
construction
algorithm that can
handle input data sets that contain linear segments, even considering
the commercial
libraries.
Support of the
segments allows to discretize any input geometry (sampled
floating-point coordinates can be scaled and snapped to the integer
grid): circle, ellipse,
parabola. This functionality allows to compute
the medial axis transform of the arbitrary set of input geometries,
with direct applications in the computer vision
projects.
Robustness and Efficiency
Robustness issues can be divided onto the two main categories: memory
management
issues and numeric stability issues. The implementation avoids the
first type of the issues using pure STL data structures, thus there is
no
presence of the new operator in the code. The second category of
the problems is
resolved using the multiprecision geometric
predicates.
Even for the commercial libraries, usage of such predicates
results in a vast performance slowdown. The library implementation
overcomes this by avoiding the multiprecision
computations in the 95% of the cases by
using the efficient, floating-point based predicates. Such predicates
don't
produce the correct result always, however the library embeds the
relative
error arithmetic apparatus to identify such situations and switch
to the
higher precision predicates when appropriate. As the result, the
implementation has a solid performance comparing to the other known
libraries (more details in the benchmarks).
Precision of the Output Structures
The implementation guaranties, that the relative error of the
coordinates of the output
geometries is at most 64 machine epsilons (6
bits of mantissa, for the IEEE-754 floating-point type), while on
average it's slightly lower. This means, that the precision of the
output
geometries can be increased simply by using a floating-point type with
the larger mantissa. The practical point of this statements is
explained in the following table:
During the finalization step the implementation unites Voronoi vertices whose both
coordinates are located within the relative error range equal to 128
machine epsilons and removes any Voronoi edges between those. This is
the only case, that might cause differences between the algorithm output
topology and theoretically precise one, and practically means the
following: for a Voronoi diagram of a set of solid bodies inside the
Solar System (radius 242 metres) and the long double (64 bit
mantissa) output coordinate type the maximum absolute error within the
Solar System rectangle will be equal to 2-64 * 242
* 26 = 2-18 metres; as the result, vertices with
both coordinates that are within 2-18 metres (8
micrometres or the size of a bacteria) will be considered
equal and united.
Simple Interface
The boost/polygon/voronoi.hpp
library header defines the following static functions to integrate the
Voronoi extensions functionality with the Boost.Polygon interfaces:
template
<typename Point, typename VB>
size_t insert(const Point
&point, VB *vb)
Вставляет точку в структуру данных конструктора Voronoi. Тип точки должен моделировать концепцию точки. Индекс возврата вставленного сайта.
Построение диаграммы Вороноя набора точек. Соответствующий тип точки должен моделировать концепцию точки. Сложность: O(N * log N), использование памяти: O(N), где N - общее количество точек ввода.
Построение диаграммы Вороноя набора сегментов. Соответствующий тип сегмента должен моделировать концепцию сегмента. Сложность: O(N * log N), использование памяти: O(N), где N - общее количество входных сегментов.
Построение диаграммы Вороноя набора точек и сегментов. Соответствующий тип точки должен моделировать концепцию точки. Соответствующий тип сегмента должен моделировать концепцию сегмента. Сложность: O(N* log N), использование памяти: O(N), где N - общее количество точек ввода и сегментов.
The following two lines of code construct a Voronoi diagram of a set of
points (as long as the corresponding input geometry type satisfies the
Boost.Polygon concept model):
The library provides the clear interfaces to associate the user data
with the
output geometries and efficiently traverse
the
Voronoi graph.
More details on those topics are covered in the basic Voronoi tutorial. Advanced
usage of the library with the configuration of the coordinate
types is explained in the advanced
Voronoi tutorial.
The library allows users to implement their own Voronoi diagram /
Delaunay triangulation construction routines based on the Voronoi builder API.
No Third Party Dependencies
The Voronoi extensions of the Boost Polygon library doesn't depend on
any 3rd party code
and contains single dependency on the Boost libraries:
boost/cstdint.hpp.
All the required multiprecision types and related functionality are
encapsulated as
part of the implementation. The library is fast to compile (3 public
and 4 private heades), has strong cohesion between its components and
is clearly modularized from the rest of the Boost Polygon library, with
the optional integration through the voronoi.hpp header.
Extensible for the User Provided Coordinate Types
The implementation is coordinate type agnostic. As long
as the user provided types satisfy the set of the requirements of the Voronoi builder coordinate type traits, no additional changes are required
neither to the algorithm, nor to the implementation of the predicates.
For example, it's possible to
construct the Voronoi diagram with the 256-bit integer input coordinate
type and 512-bit output floating-point type without making any changes to the
library.
Future Development
Below one may find the list of the possible directions for the future
development of the library.
Реализация структуры данных Делаунайской триангуляции.
Реализация структуры данных медиальной оси.
Утилиты сериализации для структуры данных диаграммы Вороноя.
Опустить ограничение на непересекающиеся входные геометрии.
Поддержка различных типов метрик расстояния.
Theoretical Research
The Voronoi extensions were developed as part of the Google Summer of Code 2010.
The library was actively maintained for the last three years and involved deep
mathematical research in the field of algorithms, data structures, relative
error arithmetic and numerical robustness. Upon the community request,
more details on the theoretical aspects of the implementation will be published.
The authors would like to acknowledge the Steven Fortune's article
"A Sweepline algorithm for Voronoi diagrams",
that covers fundamental ideas of the current implementation.
