\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.11.2 - Point Set Processing
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Collection of algorithms of point set processing (smoothing, simplification, etc.).

Classes

class  CGAL::Point_set_with_structure< Traits >
 A 3D point set with structure information based on a set of detected planes. More...
 

Functions

template<class FT , class VCMTraits >
bool CGAL::vcm_is_on_feature_edge (cpp11::array< FT, 6 > &cov, double threshold, VCMTraits)
 determines if a point is on a sharp feature edge from a point set for which the Voronoi covariance Measures have been computed. More...
 
template<class ForwardIterator , class PointPMap , class Kernel >
void CGAL::compute_vcm (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, std::vector< cpp11::array< typename Kernel::FT, 6 > > &ccov, double offset_radius, double convolution_radius, const Kernel &kernel)
 computes the Voronoi Covariance Measure (VCM) of a point cloud, a construction that can be used for normal estimation and sharp feature detection. More...
 
template<typename ForwardIterator , typename PointPMap , typename NormalPMap , typename VCMTraits >
void CGAL::vcm_estimate_normals (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, NormalPMap normal_pmap, double offset_radius, double convolution_radius, VCMTraits)
 Estimates normal directions of the points in the range [first, beyond) using the Voronoi Covariance Measure with a radius for the convolution. More...
 
template<typename ForwardIterator , typename PointPMap , typename NormalPMap , typename VCMTraits >
void CGAL::vcm_estimate_normals (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, NormalPMap normal_pmap, double offset_radius, unsigned int k, VCMTraits)
 Estimates normal directions of the points in the range [first, beyond) using the Voronoi Covariance Measure with a number of neighbors for the convolution. More...
 
template<typename Traits , typename OutputIterator >
OutputIterator CGAL::structure_point_set (typename Traits::Input_range::iterator first, typename Traits::Input_range::iterator beyond, typename Traits::Point_map point_map, typename Traits::Normal_map normal_map, OutputIterator output, Shape_detection_3::Efficient_RANSAC< Traits > &shape_detection, double epsilon, double attraction_factor=3.)
 This is an implementation of the Point Set Structuring algorithm. More...
 
template<typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
ForwardIterator CGAL::mst_orient_normals (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, NormalPMap normal_pmap, unsigned int k, const Kernel &kernel)
 Orients the normals of the [first, beyond) range of points using the propagation of a seed orientation through a minimum spanning tree of the Riemannian graph [Hoppe92]. More...
 
template<typename Concurrency_tag , typename OutputIterator , typename RandomAccessIterator , typename PointPMap , typename Kernel >
OutputIterator CGAL::wlop_simplify_and_regularize_point_set (RandomAccessIterator first, RandomAccessIterator beyond, OutputIterator output, PointPMap point_pmap, double select_percentage, double radius, unsigned int iter_number, bool require_uniform_sampling, const Kernel &)
 This is an implementation of the Weighted Locally Optimal Projection (WLOP) simplification algorithm. More...
 
template<typename Concurrency_tag , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
double CGAL::bilateral_smooth_point_set (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, NormalPMap normal_pmap, unsigned int k, typename Kernel::FT sharpness_angle, const Kernel &)
 This function smooths an input point set by iteratively projecting each point onto the implicit surface patch fitted over its k nearest neighbors. More...
 
template<typename Concurrency_tag , typename OutputIterator , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
OutputIterator CGAL::edge_aware_upsample_point_set (ForwardIterator first, ForwardIterator beyond, OutputIterator output, PointPMap point_pmap, NormalPMap normal_pmap, const typename Kernel::FT sharpness_angle, typename Kernel::FT edge_sensitivity, typename Kernel::FT neighbor_radius, const std::size_t number_of_output_points, const Kernel &)
 This method progressively upsamples the point set while approaching the edge singularities (detected by normal variation), which generates a denser point set from an input point set. More...
 
template<typename ForwardIterator , typename PointPMap , typename DiagonalizeTraits , typename Kernel >
ForwardIterator CGAL::hierarchy_simplify_point_set (ForwardIterator begin, ForwardIterator end, PointPMap point_pmap, const unsigned int size, const double var_max, const DiagonalizeTraits &, const Kernel &)
 Recursively split the point set in smaller clusters until the clusters have less than size elements or until their variation factor is below var_max. More...
 
template<typename ForwardIterator , typename PointPMap , typename Kernel >
ForwardIterator CGAL::random_simplify_point_set (ForwardIterator first, ForwardIterator beyond, PointPMap, double removed_percentage, const Kernel &)
 Randomly deletes a user-specified fraction of the input points. More...
 
template<typename Concurrency_tag , typename InputIterator , typename PointPMap , typename Kernel , typename SvdTraits >
void CGAL::jet_smooth_point_set (InputIterator first, InputIterator beyond, PointPMap point_pmap, unsigned int k, const Kernel &, unsigned int degree_fitting=2, unsigned int degree_monge=2)
 Smoothes the [first, beyond) range of points using jet fitting on the k nearest neighbors and reprojection onto the jet. More...
 
template<typename Concurrency_tag , typename InputIterator , typename PointPMap , typename Kernel >
Kernel::FT CGAL::compute_average_spacing (InputIterator first, InputIterator beyond, PointPMap point_pmap, unsigned int k, const Kernel &)
 Computes average spacing from k nearest neighbors. More...
 
template<typename Concurrency_tag , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel , typename SvdTraits >
void CGAL::jet_estimate_normals (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, NormalPMap normal_pmap, unsigned int k, const Kernel &, unsigned int degree_fitting=2)
 Estimates normal directions of the [first, beyond) range of points using jet fitting on the k nearest neighbors. More...
 
template<typename SamplesInputIterator , typename SamplesPointPMap , typename QueriesInputIterator , typename QueriesPointPMap , typename OutputIterator , typename Kernel >
OutputIterator CGAL::estimate_local_k_neighbor_scales (SamplesInputIterator first, SamplesInputIterator beyond, SamplesPointPMap samples_pmap, QueriesInputIterator first_query, QueriesInputIterator beyond_query, QueriesPointPMap queries_pmap, OutputIterator output, const Kernel &)
 Estimates the local scale in a K nearest neighbors sense on a set of user-defined query points. More...
 
template<typename InputIterator , typename PointPMap , typename Kernel >
std::size_t CGAL::estimate_global_k_neighbor_scale (InputIterator first, InputIterator beyond, PointPMap point_pmap, const Kernel &kernel)
 Estimates the global scale in a K nearest neighbors sense. More...
 
template<typename SamplesInputIterator , typename SamplesPointPMap , typename QueriesInputIterator , typename QueriesPointPMap , typename OutputIterator , typename Kernel >
OutputIterator CGAL::estimate_local_range_scales (SamplesInputIterator first, SamplesInputIterator beyond, SamplesPointPMap samples_pmap, QueriesInputIterator first_query, QueriesInputIterator beyond_query, QueriesPointPMap queries_pmap, OutputIterator output, const Kernel &)
 Estimates the local scale in a range sense on a set of user-defined query points. More...
 
template<typename InputIterator , typename PointPMap , typename Kernel >
Kernel::FT CGAL::estimate_global_range_scale (InputIterator first, InputIterator beyond, PointPMap point_pmap, const Kernel &kernel)
 Estimates the global scale in a range sense. More...
 
template<typename Concurrency_tag , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
void CGAL::pca_estimate_normals (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, NormalPMap normal_pmap, unsigned int k, const Kernel &)
 Estimates normal directions of the [first, beyond) range of points by linear least squares fitting of a plane over the k nearest neighbors. More...
 
template<typename InputIterator , typename PointPMap , typename Kernel >
InputIterator CGAL::remove_outliers (InputIterator first, InputIterator beyond, PointPMap point_pmap, unsigned int k, double threshold_percent, double threshold_distance, const Kernel &)
 Removes outliers: More...
 
template<typename ForwardIterator , typename PointPMap , typename Kernel >
ForwardIterator CGAL::grid_simplify_point_set (ForwardIterator first, ForwardIterator beyond, PointPMap point_pmap, double epsilon, const Kernel &)
 Merges points which belong to the same cell of a grid of cell size = epsilon. More...
 

Function Documentation

template<typename Concurrency_tag , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
double CGAL::bilateral_smooth_point_set ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
unsigned int  k,
typename Kernel::FT  sharpness_angle,
const Kernel  
)

This function smooths an input point set by iteratively projecting each point onto the implicit surface patch fitted over its k nearest neighbors.

Bilateral projection preserves sharp features according to the normal (gradient) information. Both point positions and normals will be modified. For more details, please see section 4 in [5].

A parallel version of this function is provided and requires the executable to be linked against the Intel TBB library. To control the number of threads used, the user may use the tbb::task_scheduler_init class. See the TBB documentation for more details.

Precondition
Normals must be unit vectors
k >= 2
Template Parameters
Concurrency_tagenables sequential versus parallel algorithm. Possible values are Sequential_tag and Parallel_tag.
ForwardIteratoriterator over input points.
PointPMapis a model of ReadWritePropertyMap with the value type of ForwardIterator as key and Kernel::Point_3 as value type. It can be omitted if the value type of ForwardIterator is convertible to Kernel::Point_3.
NormalPMapis a model of ReadWritePropertyMap with the value type of ForwardIterator as key and Kernel::Vector_3 as value type.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap using Kernel_traits.
Returns
Average point movement error. It's a convergence criterium for the algorithm. This value can help the user to decide how many iterations are sufficient.
Parameters
firstforward iterator on the first input point.
beyondpast-the-end iterator.
point_pmappoint property map.
normal_pmapnormal property map.
ksize of the neighborhood for the implicit surface patch fitting. The larger the value is, the smoother the result will be.
sharpness_anglecontrols the sharpness of the result. The larger the value is, the smoother the result will be. The range of possible value is [0, 90].

#include <CGAL/bilateral_smooth_point_set.h>

template<typename Concurrency_tag , typename InputIterator , typename PointPMap , typename Kernel >
Kernel::FT CGAL::compute_average_spacing ( InputIterator  first,
InputIterator  beyond,
PointPMap  point_pmap,
unsigned int  k,
const Kernel  
)

Computes average spacing from k nearest neighbors.

Precondition
k >= 2.
Template Parameters
Concurrency_tagenables sequential versus parallel algorithm. Possible values are Sequential_tag and Parallel_tag.
InputIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of InputIterator is convertible to Point_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Returns
average spacing (scalar).
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of InputIterator -> Point_3
knumber of neighbors.

#include <CGAL/compute_average_spacing.h>

template<class ForwardIterator , class PointPMap , class Kernel >
void CGAL::compute_vcm ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
std::vector< cpp11::array< typename Kernel::FT, 6 > > &  ccov,
double  offset_radius,
double  convolution_radius,
const Kernel kernel 
)

computes the Voronoi Covariance Measure (VCM) of a point cloud, a construction that can be used for normal estimation and sharp feature detection.

The VCM associates to each point the covariance matrix of its Voronoi cell intersected with the ball of radius offset_radius. In addition, if the second radius convolution_radius is positive, the covariance matrices are smoothed via a convolution process. More specifically, each covariance matrix is replaced by the average of the matrices of the points located at a distance at most convolution_radius. The choice for parameter offset_radius should refer to the geometry of the underlying surface while the choice for parameter convolution_radius should refer to the noise level in the point cloud. For example, if the point cloud is a uniform and noise-free sampling of a smooth surface, offset_radius should be set to the minimum local feature size of the surface, while convolution_radius can be set to zero.

The Voronoi covariance matrix of each vertex is stored in an array a of length 6 and is as follow:

\( \begin{bmatrix} a[0] & a[1] & a[2] \\ a[1] & a[3] & a[4] \\ a[2] & a[4] & a[5] \\ \end{bmatrix}\)
Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with a value_type = Kernel::Point_3.
CovariancePMapis a model of ReadWritePropertyMap with a value_type = cpp11::array<Kernel::FT,6>.
KernelGeometric traits class.
See Also
CGAL::vcm_is_on_feature_edge()
CGAL::vcm_estimate_normals()

#include <CGAL/vcm_estimate_normals.h>

Examples:
Point_set_processing_3/edges_example.cpp.
template<typename Concurrency_tag , typename OutputIterator , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
OutputIterator CGAL::edge_aware_upsample_point_set ( ForwardIterator  first,
ForwardIterator  beyond,
OutputIterator  output,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
const typename Kernel::FT  sharpness_angle,
typename Kernel::FT  edge_sensitivity,
typename Kernel::FT  neighbor_radius,
const std::size_t  number_of_output_points,
const Kernel  
)

This method progressively upsamples the point set while approaching the edge singularities (detected by normal variation), which generates a denser point set from an input point set.

This has applications in point-based rendering, hole filling, and sparse surface reconstruction. Normals of points are required as input. For more details, please refer to [5].

Template Parameters
Concurrency_tagenables sequential versus parallel versions of compute_average_spacing() (called internally). Possible values are Sequential_tag and Parallel_tag.
OutputIteratorType of the output iterator. The type of the objects put in it is std::pair<Kernel::Point_3, Kernel::Vector_3>. Note that the user may use a function_output_iterator to match specific needs.
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with the value type of ForwardIterator as key and Kernel::Point_3 as value type. It can be omitted if the value type of ForwardIterator is convertible to Kernel::Point_3.
NormalPMapis a model of ReadablePropertyMap with the value type of ForwardIterator as key and Kernel::Vector_3 as value type.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap using Kernel_traits.
Parameters
firstforward iterator on the first input point.
beyondpast-the-end iterator.
outputoutput iterator where output points and normals are put.
point_pmappoint property map.
normal_pmapvector property map.
sharpness_anglecontrols the preservation of sharp features. The larger the value is, the smoother the result will be. The range of possible values is [0, 90]. See section Parameter: sharpness_angle for an example.
edge_sensitivitylarger values of edge-sensitivity give higher priority to inserting points along sharp features. The range of possible values is [0, 1]. See section Parameter: edge_sensitivity for an example.
neighbor_radiusindicates the radius of the largest hole that should be filled. The default value is set to 3 times the average spacing of the point set. If the value given by user is smaller than the average spacing, the function will use the default value instead.
number_of_output_pointsnumber of output points to generate.

#include <CGAL/edge_aware_upsample_point_set.h>

template<typename InputIterator , typename PointPMap , typename Kernel >
std::size_t CGAL::estimate_global_k_neighbor_scale ( InputIterator  first,
InputIterator  beyond,
PointPMap  point_pmap,
const Kernel kernel 
)

Estimates the global scale in a K nearest neighbors sense.

The computed scale corresponds to the smallest scale such that the K subsets of points have the appearance of a surface in 3D or the appearance of a curve in 2D (see Automatic Scale Estimation).

Template Parameters
InputIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel> or Point_2<Kernel>. It can be omitted if the value type of InputIterator is convertible to Point_3<Kernel> or to Point_2<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Note
This function accepts both 2D and 3D points.
Returns
The estimated scale in the K nearest neighbors sense.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of InputIterator -> Point_3 or Point_2
kernelgeometric traits.

#include <CGAL/estimate_scale.h>

Examples:
Point_set_processing_3/scale_estimation_example.cpp.
template<typename InputIterator , typename PointPMap , typename Kernel >
Kernel::FT CGAL::estimate_global_range_scale ( InputIterator  first,
InputIterator  beyond,
PointPMap  point_pmap,
const Kernel kernel 
)

Estimates the global scale in a range sense.

The computed scale corresponds to the smallest scale such that the subsets of points inside the sphere range have the appearance of a surface in 3D or the appearance of a curve in 2D (see Automatic Scale Estimation).

Template Parameters
InputIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel> or Point_2<Kernel>. It can be omitted if the value type of InputIterator is convertible to Point_3<Kernel> or to Point_2<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Note
This function accepts both 2D and 3D points.
Returns
The estimated scale in the range sense.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of InputIterator -> Point_3 or Point_3
kernelgeometric traits.

#include <CGAL/estimate_scale.h>

Examples:
Point_set_processing_3/scale_estimation_example.cpp.
template<typename SamplesInputIterator , typename SamplesPointPMap , typename QueriesInputIterator , typename QueriesPointPMap , typename OutputIterator , typename Kernel >
OutputIterator CGAL::estimate_local_k_neighbor_scales ( SamplesInputIterator  first,
SamplesInputIterator  beyond,
SamplesPointPMap  samples_pmap,
QueriesInputIterator  first_query,
QueriesInputIterator  beyond_query,
QueriesPointPMap  queries_pmap,
OutputIterator  output,
const Kernel  
)

Estimates the local scale in a K nearest neighbors sense on a set of user-defined query points.

The computed scales correspond to the smallest scales such that the K subsets of points have the appearance of a surface in 3D or the appearance of a curve in 2D (see Automatic Scale Estimation).

Template Parameters
SamplesInputIteratoriterator over input sample points.
SamplesPointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel> or Point_2<Kernel>. It can be omitted if the value type of SamplesInputIterator is convertible to Point_3<Kernel> or to Point_2<Kernel>.
QueriesInputIteratoriterator over points where scale should be computed.
QueriesInputIteratoris a model of ReadablePropertyMap with value type Point_3<Kernel> or Point_2<Kernel>. It can be omitted if the value type of QueriesInputIterator is convertible to Point_3<Kernel> or to Point_2<Kernel>.
OutputIteratoris used to store the computed scales. It accepts values of type std::size_t.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of SamplesPointPMap.
Note
This function accepts both 2D and 3D points, but sample points and query must have the same dimension.
Parameters
firstiterator over the first input sample.
beyondpast-the-end iterator over the input samples.
samples_pmapproperty map: value_type of InputIterator -> Point_3 or Point_2
first_queryiterator over the first point where scale must be estimated
beyond_querypast-the-end iterator over the points where scale must be estimated
queries_pmapproperty map: value_type of InputIterator -> Point_3 or Point_2
outputoutput iterator to store the computed scales

#include <CGAL/estimate_scale.h>

Examples:
Point_set_processing_3/scale_estimation_2d_example.cpp.
template<typename SamplesInputIterator , typename SamplesPointPMap , typename QueriesInputIterator , typename QueriesPointPMap , typename OutputIterator , typename Kernel >
OutputIterator CGAL::estimate_local_range_scales ( SamplesInputIterator  first,
SamplesInputIterator  beyond,
SamplesPointPMap  samples_pmap,
QueriesInputIterator  first_query,
QueriesInputIterator  beyond_query,
QueriesPointPMap  queries_pmap,
OutputIterator  output,
const Kernel  
)

Estimates the local scale in a range sense on a set of user-defined query points.

The computed scales correspond to the smallest scales such that the subsets of points included in the sphere range have the appearance of a surface in 3D or the appearance of a curve in 2D (see Automatic Scale Estimation).

Template Parameters
SamplesInputIteratoriterator over input sample points.
SamplesPointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel> or Point_2<Kernel>. It can be omitted if the value type of SamplesInputIterator is convertible to Point_3<Kernel> or to Point_2<Kernel>.
QueriesInputIteratoriterator over points where scale should be computed.
QueriesInputIteratoris a model of ReadablePropertyMap with value type Point_3<Kernel> or Point_2<Kernel>. It can be omitted if the value type of QueriesInputIterator is convertible to Point_3<Kernel> or to Point_2<Kernel>.
OutputIteratoris used to store the computed scales. It accepts values of type Kernel::FT.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of SamplesPointPMap.
Note
This function accepts both 2D and 3D points, but sample points and query must have the same dimension.
Parameters
firstiterator over the first input sample.
beyondpast-the-end iterator over the input samples.
samples_pmapproperty map: value_type of InputIterator -> Point_3 or Point_2
first_queryiterator over the first point where scale must be estimated
beyond_querypast-the-end iterator over the points where scale must be estimated
queries_pmapproperty map: value_type of InputIterator -> Point_3 or Point_2
outputoutput iterator to store the computed scales

#include <CGAL/estimate_scale.h>

template<typename ForwardIterator , typename PointPMap , typename Kernel >
ForwardIterator CGAL::grid_simplify_point_set ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
double  epsilon,
const Kernel  
)

Merges points which belong to the same cell of a grid of cell size = epsilon.

This method modifies the order of input points so as to pack all remaining points first, and returns an iterator over the first point to remove (see erase-remove idiom). For this reason it should not be called on sorted containers.

Precondition
epsilon > 0
Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of ForwardIterator is convertible to Point_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Returns
iterator over the first point to remove.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of ForwardIterator -> Point_3
epsilontolerance value when merging 3D points.

#include <CGAL/grid_simplify_point_set.h>

Examples:
Point_set_processing_3/grid_simplification_example.cpp, Point_set_processing_3/grid_simplify_indices.cpp, and Point_set_processing_3/scale_estimation_example.cpp.
template<typename ForwardIterator , typename PointPMap , typename DiagonalizeTraits , typename Kernel >
ForwardIterator CGAL::hierarchy_simplify_point_set ( ForwardIterator  begin,
ForwardIterator  end,
PointPMap  point_pmap,
const unsigned int  size,
const double  var_max,
const DiagonalizeTraits ,
const Kernel  
)

Recursively split the point set in smaller clusters until the clusters have less than size elements or until their variation factor is below var_max.

This method modifies the order of input points so as to pack all remaining points first, and returns an iterator over the first point to remove (see erase-remove idiom). For this reason it should not be called on sorted containers.

Precondition
0 < var_max < 1/3
size > 0
Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of ForwardIterator is convertible to Point_3<Kernel>.
DiagonalizeTraitsis a model of DiagonalizeTraits. It can be omitted: if Eigen 3 (or greater) is available and CGAL_EIGEN3_ENABLED is defined then an overload using Eigen_diagonalize_traits is provided. Otherwise, the internal implementation Internal_diagonalize_traits is used.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Returns
iterator over the first point to remove.

#include <CGAL/hierarchy_simplify_point_set.h>

Examples:
Point_set_processing_3/hierarchy_simplification_example.cpp.
template<typename Concurrency_tag , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel , typename SvdTraits >
void CGAL::jet_estimate_normals ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
unsigned int  k,
const Kernel ,
unsigned int  degree_fitting = 2 
)

Estimates normal directions of the [first, beyond) range of points using jet fitting on the k nearest neighbors.

The output normals are randomly oriented.

Precondition
k >= 2
Template Parameters
Concurrency_tagenables sequential versus parallel algorithm. Possible values are Sequential_tag and Parallel_tag.
ForwardIteratoriterator model of the concept of the same name over input points and able to store output normals.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of ForwardIterator is convertible to Point_3<Kernel>.
NormalPMapis a model of WritablePropertyMap with value type Vector_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
SvdTraitstemplate parameter for the class Monge_via_jet_fitting that can be ommited under conditions described in the documentation of Monge_via_jet_fitting.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of ForwardIterator -> Point_3.
normal_pmapproperty map: value_type of ForwardIterator -> Vector_3.
knumber of neighbors.
degree_fittingfitting degree

#include <CGAL/jet_estimate_normals.h>

template<typename Concurrency_tag , typename InputIterator , typename PointPMap , typename Kernel , typename SvdTraits >
void CGAL::jet_smooth_point_set ( InputIterator  first,
InputIterator  beyond,
PointPMap  point_pmap,
unsigned int  k,
const Kernel ,
unsigned int  degree_fitting = 2,
unsigned int  degree_monge = 2 
)

Smoothes the [first, beyond) range of points using jet fitting on the k nearest neighbors and reprojection onto the jet.

As this method relocates the points, it should not be called on containers sorted w.r.t. point locations.

Precondition
k >= 2
Template Parameters
Concurrency_tagenables sequential versus parallel algorithm. Possible values are Sequential_tag and Parallel_tag.
InputIteratoriterator over input points.
PointPMapis a model of ReadWritePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of InputIterator is convertible to Point_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
SvdTraitstemplate parameter for the class Monge_via_jet_fitting that can be ommited under conditions described in the documentation of Monge_via_jet_fitting.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of InputIterator -> Point_3.
knumber of neighbors.
degree_fittingfitting degree
degree_mongeMonge degree

#include <CGAL/jet_smooth_point_set.h>

template<typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
ForwardIterator CGAL::mst_orient_normals ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
unsigned int  k,
const Kernel kernel 
)

Orients the normals of the [first, beyond) range of points using the propagation of a seed orientation through a minimum spanning tree of the Riemannian graph [Hoppe92].

This method modifies the order of input points so as to pack all sucessfully oriented points first, and returns an iterator over the first point with an unoriented normal (see erase-remove idiom). For this reason it should not be called on sorted containers.

Warning
This function may fail when Boost version 1.54 is used, because of the following bug: https://svn.boost.org/trac/boost/ticket/9012
Precondition
Normals must be unit vectors
k >= 2
Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of ForwardIterator is convertible to Point_3<Kernel>.
NormalPMapis a model of ReadWritePropertyMap with value type Vector_3<Kernel> .
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Returns
iterator over the first point with an unoriented normal.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of ForwardIterator -> Point_3.
normal_pmapproperty map: value_type of ForwardIterator -> Vector_3.
knumber of neighbors
kernelgeometric traits.

#include <CGAL/mst_orient_normals.h>

Examples:
Point_set_processing_3/normals_example.cpp.
template<typename Concurrency_tag , typename ForwardIterator , typename PointPMap , typename NormalPMap , typename Kernel >
void CGAL::pca_estimate_normals ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
unsigned int  k,
const Kernel  
)

Estimates normal directions of the [first, beyond) range of points by linear least squares fitting of a plane over the k nearest neighbors.

The output normals are randomly oriented.

Precondition
k >= 2
Template Parameters
Concurrency_tagenables sequential versus parallel algorithm. Possible values are Sequential_tag and Parallel_tag.
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of ForwardIterator is convertible to Point_3<Kernel>.
NormalPMapis a model of WritablePropertyMap with value type Vector_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of ForwardIterator -> Point_3.
normal_pmapproperty map: value_type of ForwardIterator -> Vector_3.
knumber of neighbors.

#include <CGAL/pca_estimate_normals.h>

template<typename ForwardIterator , typename PointPMap , typename Kernel >
ForwardIterator CGAL::random_simplify_point_set ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  ,
double  removed_percentage,
const Kernel  
)

Randomly deletes a user-specified fraction of the input points.

This method modifies the order of input points so as to pack all remaining points first, and returns an iterator over the first point to remove (see erase-remove idiom). For this reason it should not be called on sorted containers.

Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted if the value type of ForwardIterator is convertible to Point_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Returns
iterator over the first point to remove.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
removed_percentagepercentage of points to remove.

#include <CGAL/random_simplify_point_set.h>

template<typename InputIterator , typename PointPMap , typename Kernel >
InputIterator CGAL::remove_outliers ( InputIterator  first,
InputIterator  beyond,
PointPMap  point_pmap,
unsigned int  k,
double  threshold_percent,
double  threshold_distance,
const Kernel  
)

Removes outliers:

  • computes average squared distance to the K nearest neighbors,
  • and sorts the points in increasing order of average distance.

This method modifies the order of input points so as to pack all remaining points first, and returns an iterator over the first point to remove (see erase-remove idiom). For this reason it should not be called on sorted containers.

Precondition
k >= 2
Template Parameters
InputIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with value type Point_3<Kernel>. It can be omitted ifthe value type of InputIterator is convertible to Point_3<Kernel>.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap.
Returns
iterator over the first point to remove.
Note
There are two thresholds that can be used: threshold_percent and threshold_distance. This function returns the smallest number of outliers such that at least one of these threshold is fullfilled. This means that if threshold_percent=100, only threshold_distance is taken into account; if threshold_distance=0 only threshold_percent is taken into account.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of InputIterator -> Point_3
knumber of neighbors.
threshold_percentmaximum percentage of points to remove.
threshold_distanceminimum distance for a point to be considered as outlier (distance here is the square root of the average squared distance to K nearest neighbors)

#include <CGAL/remove_outliers.h>

Examples:
Point_set_processing_3/remove_outliers_example.cpp.
template<typename Traits , typename OutputIterator >
OutputIterator CGAL::structure_point_set ( typename Traits::Input_range::iterator  first,
typename Traits::Input_range::iterator  beyond,
typename Traits::Point_map  point_map,
typename Traits::Normal_map  normal_map,
OutputIterator  output,
Shape_detection_3::Efficient_RANSAC< Traits > &  shape_detection,
double  epsilon,
double  attraction_factor = 3. 
)

This is an implementation of the Point Set Structuring algorithm.

This algorithm takes advantage of a set of detected planes: it detects adjacency relationships between planes and resamples the detected planes, edges and corners to produce a structured point set.

The size parameter epsilon is used both for detecting adjacencies and for setting the sampling density of the structured point set.

For more details, please refer to [6].

Template Parameters
Traitsa model of EfficientRANSACTraits that must provide in addition a function Intersect_3 intersection_3_object() const and a functor Intersect_3 with:
  • boost::optional< boost::variant< Traits::Plane_3, Traits::Line_3 > > operator()(typename Traits::Plane_3, typename Traits::Plane_3)
  • boost::optional< boost::variant< Traits::Line_3, Traits::Point_3 > > operator()(typename Traits::Line_3, typename Traits::Plane_3)
OutputIteratorType of the output iterator. The type of the objects put in it is std::pair<Traits::Point_3, Traits::Vector_3>. Note that the user may use a function_output_iterator to match specific needs.
Note
If no plane is found in the shape detection object, the algorithm does nothing and the output points are the unaltered input points.
Both property maps can be omitted if the default constructors of these property maps can be safely used.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_mapproperty map: value_type of InputIterator -> Point_3.
normal_mapproperty map: value_type of InputIterator -> Vector_3.
outputoutput iterator where output points are written
shape_detectionshape detection object
epsilonsize parameter
attraction_factorattraction factor

#include <CGAL/structure_point_set.h>

Examples:
Point_set_processing_3/structuring_example.cpp.
template<typename ForwardIterator , typename PointPMap , typename NormalPMap , typename VCMTraits >
void CGAL::vcm_estimate_normals ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
double  offset_radius,
double  convolution_radius,
VCMTraits   
)

Estimates normal directions of the points in the range [first, beyond) using the Voronoi Covariance Measure with a radius for the convolution.

The output normals are randomly oriented.

See compute_vcm() for a detailed description of the parameters offset_radius and convolution_radius and of the Voronoi Covariance Measure.

Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with a value_type = Kernel::Point_3.
NormalPMapis a model of WritablePropertyMap with a value_type = Kernel::Vector_3.
VCMTraitsis a model of DiagonalizeTraits. It can be omitted: if Eigen 3 (or greater) is available and CGAL_EIGEN3_ENABLED is defined then an overload using Eigen_diagonalize_traits is provided. Otherwise, the internal implementation Diagonalize_traits is used.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of ForwardIterator -> Point_3.
normal_pmapproperty map: value_type of ForwardIterator -> Vector_3.
offset_radiusoffset radius.
convolution_radiusconvolution radius.

#include <CGAL/vcm_estimate_normals.h>

template<typename ForwardIterator , typename PointPMap , typename NormalPMap , typename VCMTraits >
void CGAL::vcm_estimate_normals ( ForwardIterator  first,
ForwardIterator  beyond,
PointPMap  point_pmap,
NormalPMap  normal_pmap,
double  offset_radius,
unsigned int  k,
VCMTraits   
)

Estimates normal directions of the points in the range [first, beyond) using the Voronoi Covariance Measure with a number of neighbors for the convolution.

The output normals are randomly oriented.

See compute_vcm() for a detailed description of the parameter offset_radius and of the Voronoi Covariance Measure.

Template Parameters
ForwardIteratoriterator over input points.
PointPMapis a model of ReadablePropertyMap with a value_type = Kernel::Point_3.
NormalPMapis a model of WritablePropertyMap with a value_type = Kernel::Vector_3.
VCMTraitsis a model of DiagonalizeTraits. It can be omitted: if Eigen 3 (or greater) is available and CGAL_EIGEN3_ENABLED is defined then an overload using Eigen_diagonalize_traits is provided. Otherwise, the internal implementation Diagonalize_traits is used.
Parameters
firstiterator over the first input point.
beyondpast-the-end iterator over the input points.
point_pmapproperty map: value_type of ForwardIterator -> Point_3.
normal_pmapproperty map: value_type of ForwardIterator -> Vector_3.
offset_radiusoffset radius.
knumber of neighbor points used for the convolution.

#include <CGAL/vcm_estimate_normals.h>

template<class FT , class VCMTraits >
bool CGAL::vcm_is_on_feature_edge ( cpp11::array< FT, 6 > &  cov,
double  threshold,
VCMTraits   
)

determines if a point is on a sharp feature edge from a point set for which the Voronoi covariance Measures have been computed.

The sharpness of the edge, specified by parameter threshold, is used to filtered points according to the external angle around a sharp feature.

A point is considered to be on a sharp feature if the external angle alpha at the edge is such that alpha >= 2 / sqrt(3) * sqrt(threshold). In particular this means that if the input contains sharp features with different external angles, the one with the smallest external angle should be considered, which however would result in selecting more points on sharper regions. More details are provided in [7].

Template Parameters
VCMTraitsis a model of DiagonalizeTraits. It can be omitted: if Eigen 3 (or greater) is available and CGAL_EIGEN3_ENABLED is defined then an overload using Eigen_diagonalize_traits is provided. Otherwise, the internal implementation Diagonalize_traits is used.
See Also
CGAL::compute_vcm()`

#include <CGAL/vcm_estimate_edges.h>

Examples:
Point_set_processing_3/edges_example.cpp.
template<typename Concurrency_tag , typename OutputIterator , typename RandomAccessIterator , typename PointPMap , typename Kernel >
OutputIterator CGAL::wlop_simplify_and_regularize_point_set ( RandomAccessIterator  first,
RandomAccessIterator  beyond,
OutputIterator  output,
PointPMap  point_pmap,
double  select_percentage,
double  radius,
unsigned int  iter_number,
bool  require_uniform_sampling,
const Kernel  
)

This is an implementation of the Weighted Locally Optimal Projection (WLOP) simplification algorithm.

The WLOP simplification algorithm can produce a set of denoised, outlier-free and evenly distributed particles over the original dense point cloud. The core of the algorithm is a Weighted Locally Optimal Projection operator with a density uniformization term. For more details, please refer to [4].

A parallel version of WLOP is provided and requires the executable to be linked against the Intel TBB library. To control the number of threads used, the user may use the tbb::task_scheduler_init class. See the TBB documentation for more details.

Template Parameters
Concurrency_tagenables sequential versus parallel algorithm. Possible values are Sequential_tag and Parallel_tag.
OutputIteratorType of the output iterator. It must accept objects of type Kernel::Point_3.
RandomAccessIteratorIterator over input points.
PointPMapis a model of ReadablePropertyMap with the value type of ForwardIterator as key type and Kernel::Point_3 as value type. It can be omitted if the value type of RandomAccessIterator is convertible to Kernel::Point_3.
KernelGeometric traits class. It can be omitted and deduced automatically from the value type of PointPMap using Kernel_traits.
Parameters
firstrandom-access iterator to the first input point.
beyondpast-the-end iterator.
outputoutput iterator where output points are put.
point_pmappoint property map.
select_percentagepercentage of points to retain. The default value is set to 5 (%).
radiusspherical neighborhood radius. This is a key parameter that needs to be finely tuned. The result will be irregular if too small, but a larger value will impact the runtime. In practice, choosing a radius such that the neighborhood of each sample point includes at least two rings of neighboring sample points gives satisfactory result. The default value is set to 8 times the average spacing of the point set.
iter_numbernumber of iterations to solve the optimsation problem. The default value is 35. More iterations give a more regular result but increase the runtime.
require_uniform_samplingan optional preprocessing, which will give better result if the distribution of the input points is highly non-uniform. The default value is false.

#include <CGAL/wlop_simplify_and_regularize_point_set.h>

Examples:
Point_set_processing_3/wlop_simplify_and_regularize_point_set_example.cpp.