Filtration Concept Reference

Definition

The concept Filtration describes the requirements for persistent filtrations associated to persistent homology computation.

A filtration is associated to an AbstractChainComplex. Each cell is equiped with a scalar value (called its degree) and the filtration is an enumeration of cells in any dimension by increasing degrees.

A filtration class provides:

• an iterator to visit all cells by increasing degrees.
• getters to get the degree of a cell, and the cell of a given index along the filtration.
• an overload of operator<<() to output filtrations.

Cells are indexed along each dimension and thus identified by their index together with their dimension.

Has models

CGAL::Homological_discrete_vector_field::Filtration_core<ChainComplex,Degree>

CGAL::Homological_discrete_vector_field::Filtration_lower_star<ChainComplex,Degree>

Examples

HDVF/main_per_hdvf.cpp.

Concepts
concept	Filtration_iter_value
	Value type of the filtration iterator. More...
concept	iterator
	The iterator over filtration. More...

Public Types
typedef unspecified_type	Complex
	Type of the underlying complex.
typedef unspecified_type	Degree
	(Scalar) type of degrees (model of RealEmbeddable).
typedef Complex::Coefficient_ring	Coefficient_ring
	Type of coefficients used to compute homology.
typedef std::pair< size_t, int >	Cell
	Type for indexing uniquely a cell.

Public Member Functions
iterator	begin ()
	Iterator to the beginning of the filtration.
iterator	end ()
	Returns a past-the-end iterator.
size_t	size () const
	Gets the filtration size.
Cell	cell_index_dimension (size_t i) const
	Gets the cell (that is cell index and dimension) at the index i of the filtration.
Degree	degree (size_t i) const
	Gets the degree of the ith element of the filtration.
bool	is_valid () const
	Checks that a filtration is valid.

Protected Types
typedef SparseMatrix< Coefficient_ring, CGAL::OSM::COLUMN >	Column_matrix
	Type of column-major sparse matrices.
typedef SparseMatrix< Coefficient_ring, CGAL::OSM::ROW >	Row_matrix
	Type of row-major sparse matrices.
typedef SparseChain< Coefficient_ring, CGAL::OSM::COLUMN >	Column_chain
	Type of column-major chains.
typedef SparseChain< Coefficient_ring, CGAL::OSM::ROW >	Row_chain
	Type of row-major chains.

Related Functions
(Note that these are not member functions.)
istream &	operator>> (istream &in, Filtration &f)
	extracts a filtration from a stream.
ostream &	operator<< (ostream &out, const Filtration &f)
	Inserts a filtration into a stream.

Member Typedef Documentation

Cell

typedef std::pair<size_t, int> Filtration::Cell

Type for indexing uniquely a cell.

As stated in AbstractChainComplex, cells are identified by their dimension, together with their index along this dimension. The type Cell stores this pair:

• First element of the pair: index of the cell.
• Second element of the pair: dimension of the cell.

Member Function Documentation

is_valid()

bool Filtration::is_valid

(

)

const

Checks that a filtration is valid.

Checks that cells are ordered in increasing degrees and all cells have indices larger than their faces.