Aim: a geometric kernel to compute the Delaunay triangulation of digital points with integer-only arithmetic. It casts lattice point into a higher dimensional space and computes its convex hull. Facets pointing toward the bottom form the simplices of the Delaunay triangulation. More...

#include <DGtal/geometry/tools/QuickHullKernels.h>

Inheritance diagram for DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >:

Public Types
typedef ConvexHullCommonKernel< dim+1, TCoordinateInteger, TInternalInteger >	Base

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinatePoint

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinateVector

typedef CoordinateInteger	CoordinateScalar

typedef DGtal::PointVector< dim, InternalInteger >	InternalPoint

typedef DGtal::PointVector< dim, InternalInteger >	InternalVector

typedef InternalInteger	InternalScalar

typedef std::size_t	Size

typedef Size	Index

typedef std::vector< Index >	IndexRange

typedef std::array< Index, dim >	CombinatorialPlaneSimplex

Public Types inherited from DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >
typedef TCoordinateInteger	CoordinateInteger

typedef TInternalInteger	InternalInteger

typedef CoordinateInteger	CoordinateScalar

typedef InternalInteger	InternalScalar

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinatePoint

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinateVector

typedef DGtal::PointVector< dim, InternalInteger >	InternalPoint

typedef DGtal::PointVector< dim, InternalInteger >	InternalVector

typedef std::size_t	Size

typedef Size	Index

typedef std::vector< Index >	IndexRange

typedef std::array< Index, dim >	CombinatorialPlaneSimplex

typedef IntegerConverter< dim, CoordinateInteger >	Outer
	Converter to outer coordinate integers or lattice points / vector. More...

typedef IntegerConverter< dim, InternalInteger >	Inner
	Converter to inner internal integers or lattice points / vector. More...

Public Member Functions
	DelaunayIntegralKernel ()=default
	Default constructor. More...

bool	hasInfiniteFacets () const

bool	isHalfSpaceFacetInfinite (const HalfSpace &hs) const

template<typename InputPoint >
void	makeInput (std::vector< CoordinatePoint > &processed_points, IndexRange &input2comp, IndexRange &comp2input, const std::vector< InputPoint > &input_points, bool remove_duplicates)

template<typename OutputPoint >
void	convertPointTo (const CoordinatePoint &p, OutputPoint &out_p) const

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex, Index idx_below)

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex)

CoordinateVector	normal (const HalfSpace &H) const

CoordinateScalar	intercept (const HalfSpace &H) const

InternalScalar	dot (const HalfSpace &H1, const HalfSpace &H2) const

bool	equal (const HalfSpace &H1, const HalfSpace &H2) const

InternalScalar	height (const HalfSpace &H, const CoordinatePoint &p) const

InternalScalar	volume (const HalfSpace &H, const CoordinatePoint &p) const

bool	above (const HalfSpace &H, const CoordinatePoint &p) const

bool	aboveOrOn (const HalfSpace &H, const CoordinatePoint &p) const

bool	on (const HalfSpace &H, const CoordinatePoint &p) const

Public Member Functions inherited from DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >
	BOOST_CONCEPT_ASSERT ((concepts::CInteger< TCoordinateInteger >))

	BOOST_CONCEPT_ASSERT ((concepts::CInteger< TInternalInteger >))

	ConvexHullCommonKernel ()=default
	Default constructor. More...

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex, Index idx_below)

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex)

CoordinateVector	normal (const HalfSpace &H) const

CoordinateScalar	intercept (const HalfSpace &H) const

InternalScalar	dot (const HalfSpace &H1, const HalfSpace &H2) const

bool	equal (const HalfSpace &H1, const HalfSpace &H2) const

InternalScalar	height (const HalfSpace &H, const CoordinatePoint &p) const

InternalScalar	volume (const HalfSpace &H, const CoordinatePoint &p) const

bool	above (const HalfSpace &H, const CoordinatePoint &p) const

bool	aboveOrOn (const HalfSpace &H, const CoordinatePoint &p) const

bool	on (const HalfSpace &H, const CoordinatePoint &p) const

Static Public Attributes
static const Dimension	dimension

Static Public Attributes inherited from DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >
static const Dimension	dimension = dim

Detailed Description

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>
struct DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >

Aim: a geometric kernel to compute the Delaunay triangulation of digital points with integer-only arithmetic. It casts lattice point into a higher dimensional space and computes its convex hull. Facets pointing toward the bottom form the simplices of the Delaunay triangulation.

Description of template class 'DelaunayIntegralKernel'

See also: QuickHull algorithm in arbitrary dimension for convex hull and Delaunay cell complex computation

Template Parameters

dim	the dimension of the space of processed points.
TCoordinateInteger	the integer type that represents coordinates of lattice points, a model of concepts::CInteger.
TInternalInteger	the integer type that is used for internal computations of above/below plane tests, a model of concepts::CInteger. Must be at least as precise as TCoordinateInteger.

Examples: geometry/tools/exampleLatticeBallDelaunay2D.cpp.

Definition at line 508 of file QuickHullKernels.h.

Member Typedef Documentation

◆ Base

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef ConvexHullCommonKernel< dim+1, TCoordinateInteger, TInternalInteger > DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::Base

Definition at line 511 of file QuickHullKernels.h.

◆ CombinatorialPlaneSimplex

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef std::array< Index, dim > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CombinatorialPlaneSimplex

Definition at line 195 of file QuickHullKernels.h.

◆ CoordinatePoint

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, CoordinateInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CoordinatePoint

Definition at line 188 of file QuickHullKernels.h.

◆ CoordinateScalar

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef CoordinateInteger DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CoordinateScalar

Definition at line 184 of file QuickHullKernels.h.

◆ CoordinateVector

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, CoordinateInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CoordinateVector

Definition at line 189 of file QuickHullKernels.h.

◆ Index

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef Size DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::Index

Definition at line 193 of file QuickHullKernels.h.

◆ IndexRange

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef std::vector< Index > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::IndexRange

Definition at line 194 of file QuickHullKernels.h.

◆ InternalPoint

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, InternalInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::InternalPoint

Definition at line 190 of file QuickHullKernels.h.

◆ InternalScalar

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef InternalInteger DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::InternalScalar

Definition at line 185 of file QuickHullKernels.h.

◆ InternalVector

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, InternalInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::InternalVector

Definition at line 191 of file QuickHullKernels.h.

◆ Size

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef std::size_t DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::Size

Definition at line 192 of file QuickHullKernels.h.

Constructor & Destructor Documentation

◆ DelaunayIntegralKernel()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::DelaunayIntegralKernel ( )

default

Default constructor.

Member Function Documentation

◆ above()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::above	(	const HalfSpace &	H,
		const CoordinatePoint &	p
	)		const

inline

Parameters

H	the half-space
p	any point

Returns: 'true' iff p is strictly above this plane (so in direction N ).

Definition at line 334 of file QuickHullKernels.h.

335 { return height( H, p ) > 0; }

DGtal::ProbingMode::H

@ H

◆ aboveOrOn()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::aboveOrOn	(	const HalfSpace &	H,
		const CoordinatePoint &	p
	)		const

inline

Parameters

H	the half-space
p	any point

Returns: 'true' iff p is above or lies on this plane (so in direction N ).

Definition at line 340 of file QuickHullKernels.h.

341 { return height( H, p ) >= 0; }

◆ compute() [1/2]

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

HalfSpace DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::compute	(	const std::vector< CoordinatePoint > &	vpoints,
		const CombinatorialPlaneSimplex &	simplex
	)

inline

Computes an halfspace from dimension points specified by simplex with vertices in a range vpoints of Point. Orientation is induced by the order of the points. If the simplex is degenrated, the half-space is invalid and has null normal.

Parameters

[in]	vpoints	a range of points over which the simplex is defined.
[in]	simplex	a range of dimension indices of points defining an hyperplane.

Returns: the corresponding halfspace (has null normal vector if simplex was not full dimensional)

Definition at line 257 of file QuickHullKernels.h.

    {
      typedef DGtal::SimpleMatrix< InternalScalar, dimension, dimension > Matrix;
      Matrix A;
      InternalVector N;
      InternalScalar c;
      for ( Dimension i = 1; i < dimension; i++ )
        for ( Dimension j = 0; j < dimension; j++ )
          A.setComponent( i-1, j,
                          Inner::cast( vpoints[ simplex[ i ] ][ j ]
                                       - vpoints[ simplex[ 0 ] ][ j ] ) );
      for ( Dimension j = 0; j < dimension; j++ )
        N[ j ] = A.cofactor( dimension-1, j );
      const InternalPoint ip = Inner::cast( vpoints[ simplex[ 0 ] ] );
      // c = N.dot( vpoints[ simplex[ 0 ] ] );
      return HalfSpace { N, N.dot( ip ) };
    }

◆ compute() [2/2]

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

HalfSpace DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::compute	(	const std::vector< CoordinatePoint > &	vpoints,
		const CombinatorialPlaneSimplex &	simplex,
		Index	idx_below
	)

inline

Computes an halfspace from dimension points specified by simplex with vertices in a range vpoints of Point. It is oriented such that the point of index idx_below is included in the half-space (i.e. below).

Parameters

[in]	vpoints	a range of points over which the simplex is defined.
[in]	simplex	a range of dimension indices of points defining an hyperplane.
[in]	idx_below	the index of a p-oint that is below the hyperplane.

Returns: the corresponding halfspace (has null normal vector if simplex was not full dimensional)

Definition at line 230 of file QuickHullKernels.h.

    {
      HalfSpace hs = compute( vpoints, simplex );
      if ( hs.N != InternalVector::zero ) 
        {
          const InternalPoint  ip = Inner::cast( vpoints[ idx_below ] );
          const InternalScalar nu = hs.N.dot( ip );
          //const Scalar nu = hs.N.dot( vpoints[ idx_below ] );
          if ( nu > hs.c ) { hs.N = -hs.N; hs.c = -hs.c; }
        }
      return hs;
    }

◆ convertPointTo()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

template<typename OutputPoint >

void DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::convertPointTo	(	const CoordinatePoint &	p,
		OutputPoint &	out_p
	)		const

inline

Template Parameters

OutputPoint a model of point such that processing type Point is convertible to it.

Definition at line 614 of file QuickHullKernels.h.

    {
      for ( Dimension k = 0; k < dimension-1; k++ )
        out_p[ k ] = p[ k ];
    }

References DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::dimension.

◆ dot()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

InternalScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::dot	(	const HalfSpace &	H1,
		const HalfSpace &	H2
	)		const

inline

Equivalent of the dot product of the normals of the half-spaces.

Parameters

H1	an half-space
H2	an half-space

Returns: a positive scalar if both half-spaces points to to the same hemisphere, a negative scalar if they point to opposite hemispheres, and zero if they are orthogonal.

Definition at line 298 of file QuickHullKernels.h.

    {
      return H1.N.dot( H2.N );
    }

◆ equal()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::equal	(	const HalfSpace &	H1,
		const HalfSpace &	H2
	)		const

inline

Parameters

H1	an half-space
H2	an half-space

Returns: 'true' if the half-spaces have the smae members.

Note: two half-spaces may be not equal but may represent the same set of points. For instance H1={{1,0},3} and H1={{2,0},6}.

Definition at line 311 of file QuickHullKernels.h.

    {
      return H1.c == H2.c && H1.N == H2.N;
    }

◆ hasInfiniteFacets()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::hasInfiniteFacets ( ) const

inline

Returns: 'true' if a kernel may induce infinite facets. This is typically the case of a Delaunay computation kernel, which casts points in higher dimension.

Definition at line 547 of file QuickHullKernels.h.

548 { return true; }

◆ height()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

InternalScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::height	(	const HalfSpace &	H,
		const CoordinatePoint &	p
	)		const

inline

Parameters

H	the half-space
p	any point

Returns: the (signed) height of p wrt this plane.

Definition at line 319 of file QuickHullKernels.h.

320 { return H.N.dot( Inner::cast( p ) ) - H.c; }

◆ intercept()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

CoordinateScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::intercept ( const HalfSpace & H ) const

inline

Parameters

H	the half-space

Returns: the intercept of this facet.

Definition at line 285 of file QuickHullKernels.h.

    {
      return Outer::cast( H.c );
    }

◆ isHalfSpaceFacetInfinite()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::isHalfSpaceFacetInfinite ( const HalfSpace & hs ) const

inline

Parameters

[in] hs an half-space corresponding to a facet.

Returns: 'true' if the facet associated to this half-space correspond to an infinite facet.

Definition at line 554 of file QuickHullKernels.h.

    {
      return hs.internalNormal()[ dimension - 1 ] >= InternalScalar( 0 );
    }

References DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::dimension, and DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::HalfSpace::internalNormal().

◆ makeInput()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

template<typename InputPoint >

void DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::makeInput	(	std::vector< CoordinatePoint > &	processed_points,
		IndexRange &	input2comp,
		IndexRange &	comp2input,
		const std::vector< InputPoint > &	input_points,
		bool	remove_duplicates
	)

inline

Transforms a range input_points of input points to a range processed_points of points adapted to a processing by QuickHull convex hull algorithm. Keep the mapping information between input and processed points. This kernel transforms dim-dimensional into dim+1-dimensional points, where the last coordinate lies on a paraboloid. This is the classical way for computing a Delaunay triangulation as the convex hull of the points onto a paraboloid of higher dimension.

Template Parameters

InputPoint any model of point whose components are convertible to Scalar.

Parameters

[out]	processed_points	the range of points prepared for a process by QuickHull.
[out]	input2comp	the surjective mapping between the input_points range and the processed_points range used for computation.
[out]	comp2input	the injective mapping between the processed_points range used for computation and the input_points range.
[in]	input_points	the range of input points.
[in]	remove_duplicates	when 'true', this method removes possible duplicates in input_points and processed_points may thus be of smaller size, otherwise, when 'false', it means that there are no duplicates in input_points.

Definition at line 589 of file QuickHullKernels.h.

    {
      const auto F = [&] ( InputPoint input ) -> CoordinatePoint
        {
          CoordinatePoint p;
          CoordinateScalar z = 0;
          for ( Dimension i = 0; i < dimension-1; i++ ) {
            const CoordinateScalar x = CoordinateScalar( input[ i ] );
            p[ i ]   = x;
            z       += x*x;
          }
          p[ dimension-1 ] = z;
          return p;
        };
      DGtal::detail::transform( processed_points, input2comp, comp2input,
                                input_points.cbegin(), input_points.cend(),
                                F, remove_duplicates );
    }

References DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::dimension, and DGtal::detail::transform().

◆ normal()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

CoordinateVector DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::normal ( const HalfSpace & H ) const

inline

Parameters

H	the half-space

Returns: the normal to this facet.

Definition at line 278 of file QuickHullKernels.h.

    {
      return Outer::cast( H.N );
    }

◆ on()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::on	(	const HalfSpace &	H,
		const CoordinatePoint &	p
	)		const

inline

Parameters

H	the half-space
p	any point

Returns: 'true' iff p lies on this plane.

Definition at line 346 of file QuickHullKernels.h.

347 { return height( H, p ) == 0; }

◆ volume()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

InternalScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::volume	(	const HalfSpace &	H,
		const CoordinatePoint &	p
	)		const

inline

Parameters

H	the half-space
p	any point

Returns: the volume of the vectors spanned by the simplex and this point.

Definition at line 325 of file QuickHullKernels.h.

    {
      InternalScalar v = height( H, p );
      return v < InternalScalar( 0 ) ? -v : v;
    }

Field Documentation

◆ dimension

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

const Dimension DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::dimension

static

Definition at line 196 of file QuickHullKernels.h.

Referenced by DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::convertPointTo(), DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::isHalfSpaceFacetInfinite(), and DGtal::DelaunayIntegralKernel< dim, TCoordinateInteger, TInternalInteger >::makeInput().

The documentation for this struct was generated from the following file:

QuickHullKernels.h

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Member Typedef Documentation

◆ Base

◆ CombinatorialPlaneSimplex

◆ CoordinatePoint

◆ CoordinateScalar

◆ CoordinateVector

◆ Index

◆ IndexRange

◆ InternalPoint

◆ InternalScalar

◆ InternalVector

◆ Size

Constructor & Destructor Documentation

◆ DelaunayIntegralKernel()

Member Function Documentation

◆ above()

◆ aboveOrOn()

◆ compute() [1/2]

◆ compute() [2/2]

◆ convertPointTo()

◆ dot()

◆ equal()

◆ hasInfiniteFacets()

◆ height()

◆ intercept()

◆ isHalfSpaceFacetInfinite()

◆ makeInput()

◆ normal()

◆ on()

◆ volume()

Field Documentation

◆ dimension