CGAL::HDVF::Filtration_lower_star< CoefficientType, ComplexType, DegreeType > Class Template Reference

#include <CGAL/HDVF/Filtration_lower_star.h>

Inherits from

CGAL::HDVF::Filtration_core< CoefficientType, ComplexType, DegreeType >.

Definition

template<typename CoefficientType, typename ComplexType, typename DegreeType>
class CGAL::HDVF::Filtration_lower_star< CoefficientType, ComplexType, DegreeType >

The class Filtration_lower_star implements the lower star filtration on a given complex implementing the concept AbstractChainComplex.

A filtration associated to a chain complex K associates to each cell of K a scalar value (called degree) such that the degree of a cell is larger than the degrees of its faces.

Let DegreeType be a scalar type. The lower star filtration is a filtration obtained from a map \(\mathrm{deg}\,:\, K_0 \to \mathrm{DegreeType}\) (where \(K_0\) denotes the set of vertices of \(K\)) associating a degree to each vertex of the complex \(K\).

The map is extended to cells of any dimension by setting, for a cell \(\sigma\):

\[\mathrm{deg}(\sigma) = \max_{\substack{v\in K_0\\v\text{ face of }\sigma}} \mathrm{deg}(v) \]

For geometric complexes, standard lower star filtrations are obtained by taking as a degree function the \(x\), \(y\) or \(z\) coordinate of vertices. The image below illustrates such a filtration ((left) lower star filtration with a \(z\) degree map on vertices, (right) lower star filtration with a \(y\) degree map on vertices).

The Filtration_lower_star class provides constructors taking as input:

• either the vector of vertex degrees
• or a function mapping each vertex to its degree.

Is model of

Filtration

Template Parameters

CoefficientType	a model of the Ring concept (ring used for homology computation).
ComplexType	a model of the AbstractChainComplex concept (type of the underlying chain complex).
DegreeType	the scalar type of degrees.

Examples

HDVF/main_per_hdvf.cpp.

Public Member Functions
	Filtration_lower_star (const Filtration_lower_star &f)
	Copy constructor.
	Filtration_lower_star (const ComplexType &K, const std::vector< DegreeType > &deg)
	Constructor from vertex degrees.
	Filtration_lower_star (const ComplexType &K, std::function< DegreeType(size_t)> &deg_fun)
	Constructor from a function mapping vertices to degrees.
Public Member Functions inherited from CGAL::HDVF::Filtration_core< CoefficientType, ComplexType, DegreeType >
	Filtration_core (const ComplexType &K)
	Filtration_core default constructor.
	Filtration_core (const ComplexType &K, const std::vector< CellDim > &filtration, const std::vector< DegreeType > &deg)
	Constructor from a vector of cells (ordering of cells) and an associated vector of degrees.
	Filtration_core (const Filtration_core &f)
	Constructor by copy.
iterator	begin ()
	Iterator to the beginning of the filtration.
iterator	end ()
	Iterator to the ending of the filtration.
std::size_t	get_filtration_size () const
	Gets the filtration size.
CellDim	get_cell_dim (std::size_t i) const
	Gets the cell (that is cell index and dimension) at the index i of the filtration.
DegreeType	get_degree (std::size_t i) const
	Gets the degree of the ith element of the filtration.
vector< vector< std::size_t > >	export_filtration () const
	Exports the filtration time indices.
bool	is_valid_filtration () const
	Checks if a filtration is valid.

Protected Member Functions
void	star_filtration (const std::vector< DegreeType > &deg)
	Function building the filtration from the vector of vertices degrees.
void	star_filtration (std::function< DegreeType(size_t)> &deg_fun)
	Function building the filtration from a function mapping vertices to their degree.
Protected Member Functions inherited from CGAL::HDVF::Filtration_core< CoefficientType, ComplexType, DegreeType >
void	build_filtration_structure ()

Additional Inherited Members
Public Types inherited from CGAL::HDVF::Filtration_core< CoefficientType, ComplexType, DegreeType >
typedef std::pair< std::size_t, int >	CellDim
	Type for indexing uniquely a cell.
Protected Types inherited from CGAL::HDVF::Filtration_core< CoefficientType, ComplexType, DegreeType >
typedef OSM::Sparse_matrix< CoefficientType, OSM::COLUMN >	CMatrix
	Type of column-major sparse matrices.
typedef OSM::Sparse_matrix< CoefficientType, OSM::ROW >	RMatrix
	Type of row-major sparse matrices.
typedef OSM::Sparse_chain< CoefficientType, OSM::COLUMN >	CChain
	Type of column-major chains.
typedef OSM::Sparse_chain< CoefficientType, OSM::ROW >	RChain
	Type of row-major chains.
Protected Attributes inherited from CGAL::HDVF::Filtration_core< CoefficientType, ComplexType, DegreeType >
const ComplexType &	_K
	Constant reference to the underlying chain complex.
std::vector< CellDim >	_filtration
	Vector of cells of the filtration (full ordering of cells).
std::vector< DegreeType >	_deg
	Vector of degrees of cells along the filtration.
std::map< CellDim, std::size_t >	_cell_to_t
	Map from cells to their index in the filtration.

Constructor & Destructor Documentation

Filtration_lower_star() [1/3]

template<typename CoefficientType , typename ComplexType , typename DegreeType >

CGAL::HDVF::Filtration_lower_star< CoefficientType, ComplexType, DegreeType >::Filtration_lower_star

(

const Filtration_lower_star< CoefficientType, ComplexType, DegreeType > &

f

)

Copy constructor.

Builds a filtration by copy from another.

Parameters

[in]

f

An initial lower star filtration.

Filtration_lower_star() [2/3]

template<typename CoefficientType , typename ComplexType , typename DegreeType >

CGAL::HDVF::Filtration_lower_star< CoefficientType, ComplexType, DegreeType >::Filtration_lower_star	(	const ComplexType &	K,
		const std::vector< DegreeType > &	deg
	)

Constructor from vertex degrees.

The constructor computes all cells degrees as the minimum of the degrees of their vertices and sorts all the cells of the complex to fulfill the filtration ordering constraints.

Parameters

[in]	K	Constant reference to the underlying complex.
[in]	deg	Vector of vertex degrees.

Filtration_lower_star() [3/3]

template<typename CoefficientType , typename ComplexType , typename DegreeType >

CGAL::HDVF::Filtration_lower_star< CoefficientType, ComplexType, DegreeType >::Filtration_lower_star	(	const ComplexType &	K,
		std::function< DegreeType(size_t)> &	deg_fun
	)

Constructor from a function mapping vertices to degrees.

The constructor computes all cells degrees as the minimum of the degrees of their vertices (obtained through deg_fun) and sorts all the cells of the complex to fulfill the filtration ordering constraints.

Parameters

[in]	K	Constant reference to the underlying complex.
[in]	deg_fun	Function mapping vertices of K to their degree.