CubicalComplex: R6 Class for Cubical Complex
In rgudhi: An Interface to the GUDHI Library for Topological Data Analysis
CubicalComplex R Documentation
R6 Class for Cubical Complex
Description
The CubicalComplex is an example of a structured complex useful
in computational mathematics (specially rigorous numerics) and image
analysis.
Super class
rgudhi::PythonClass -> CubicalComplex
Methods
Public methods
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Inherited methods
Method new()
Constructor from either top_dimensional_cells (and
possibly dimensions) or from a Perseus-style file name.
Usage
CubicalComplex$new(
perseus_file,
top_dimensional_cells,
dimensions = NULL,
py_class = NULL
)
Arguments
perseus_fileA character string specifying the path to a
Perseus-style file name.
top_dimensional_cellsEither a numeric vector (in which case,
dimensions should be provided as well) or a multidimensional array
specifying cell filtration values.
dimensionsAn integer vector specifying the number of top
dimensional cells. Defaults to NULL.
py_classAn existing CubicalComplex Python class. Defaults to
NULL which uses the Python class constructor instead.
Returns
A new CubicalComplex object.
Method betti_numbers()
This function returns the Betti numbers of the complex.
Usage
CubicalComplex$betti_numbers()
Details
The $betti_numbers() method always returns [1, 0, 0, ...] as
infinity filtration cubes are not removed from the complex.
Returns
An integer vector storing the Betti numbers.
Method cofaces_of_persistence_pairs()
A persistence interval is described by a pair of cells, one
that creates the feature and one that kills it. The filtration values
of those 2 cells give coordinates for a point in a persistence diagram,
or a bar in a barcode. Structurally, in the cubical complexes provided
here, the filtration value of any cell is the minimum of the filtration
values of the maximal cells that contain it. Connecting persistence
diagram coordinates to the corresponding value in the input (i.e. the
filtration values of the top-dimensional cells) is useful for
differentiation purposes.
Usage
CubicalComplex$cofaces_of_persistence_pairs()
Details
This function returns a list of pairs of top-dimensional cells
corresponding to the persistence birth and death cells of the
filtration. The cells are represented by their indices in the input
list of top-dimensional cells (and not their indices in the internal
data structure that includes non-maximal cells). Note that when two
adjacent top-dimensional cells have the same filtration value, we
arbitrarily return one of the two when calling the function on one of
their common faces.
Returns
The top-dimensional cells/cofaces of the positive and negative
cells, together with the corresponding homological dimension, in two
lists of integer arrays. The first list contains the regular
persistence pairs, grouped by dimension. It contains numpy arrays of
shape [number_of_persistence_points, 2]. The indices of the arrays in
the list correspond to the homological dimensions, and the integers of
each row in each array correspond to: (index of positive top-dimensional cell, index of negative top-dimensional cell). The
second list contains the essential features, grouped by dimension. It
contains numpy arrays of shape [number_of_persistence_points, 1]. The
indices of the arrays in the list correspond to the homological
dimensions, and the integers of each row in each array correspond to:
(index of positive top-dimensional cell).
Method compute_persistence()
This method computes the persistence of the complex, so it
can be accessed through $persistent_betti_numbers(),
$persistence_intervals_in_dimension(), etc. It is equivalent to the
$persistence() method when you do not want the list $persistence()
returns.
Usage
CubicalComplex$compute_persistence(
homology_coeff_field = 11,
min_persistence = 0
)
Arguments
homology_coeff_fieldAn integer value specifying the homology
coefficient field. Must be a prime number. Defaults to 11L. Maximum
is 46337L.
min_persistenceA numeric value specifying the minimum persistence
value to take into account (strictly greater than min_persistence).
Defaults to 0.0. Set min_persistence = -1.0 to see all values.
Returns
The updated CubicalComplex class itself invisibly.
Method dimension()
This function returns the dimension of the complex.
Usage
CubicalComplex$dimension()
Returns
An integer value giving the complex dimension.
Method num_simplices()
This function returns the number of all cubes in the
complex.
Usage
CubicalComplex$num_simplices()
Returns
An integer value giving the number of all cubes in the complex.
Method persistence()
This function computes and returns the persistence of the
complex.
Usage
CubicalComplex$persistence(homology_coeff_field = 11, min_persistence = 0)
Arguments
homology_coeff_fieldAn integer value specifying the homology
coefficient field. Must be a prime number. Defaults to 11L. Maximum
is 46337L.
min_persistenceA numeric value specifying the minimum persistence
value to take into account (strictly greater than min_persistence).
Defaults to 0.0. Set min_persistence = -1.0 to see all values.
Returns
A tibble listing all persistence feature
summarised by 3 variables: dimension, birth and death.
Method persistence_intervals_in_dimension()
This function returns the persistence intervals of the
complex in a specific dimension.
Usage
CubicalComplex$persistence_intervals_in_dimension(dimension)
Arguments
dimensionAn integer value specifying the desired dimension.
Returns
A tibble storing the persistence intervals
by row.
Method persistent_betti_numbers()
This function returns the persistent Betti numbers of the
complex.
Usage
CubicalComplex$persistent_betti_numbers(from_value, to_value)
Arguments
from_valueA numeric value specifying the persistence birth limit
to be added in the numbers (persistent birth <= from_value).
to_valueA numeric value specifying the persistence death limit to
be added in the numbers (persistent death > to_value).
Returns
An integer vector storing the persistent Betti numbers.
Method clone()
The objects of this class are cloneable with this method.
Usage
CubicalComplex$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Author(s)
Pawel Dlotko
See Also
Other data structures for cell complexes:
PeriodicCubicalComplex,
SimplexTree
Examples
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$betti_numbers()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$cofaces_of_persistence_pairs()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$dimension()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$num_simplices()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$persistence()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$persistence_intervals_in_dimension(0)
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$persistent_betti_numbers(0, 1)
rgudhi documentation built on March 31, 2023, 11:38 p.m.
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| CubicalComplex | R Documentation |
R6 Class for Cubical Complex
Description
The CubicalComplex is an example of a structured complex useful in computational mathematics (specially rigorous numerics) and image analysis.
Super class
rgudhi::PythonClass -> CubicalComplex
Methods
Public methods
Inherited methods
Method new()
Constructor from either top_dimensional_cells (and
possibly dimensions) or from a Perseus-style file name.
Usage
CubicalComplex$new( perseus_file, top_dimensional_cells, dimensions = NULL, py_class = NULL )
Arguments
perseus_fileA character string specifying the path to a Perseus-style file name.
top_dimensional_cellsEither a numeric vector (in which case,
dimensionsshould be provided as well) or a multidimensional array specifying cell filtration values.dimensionsAn integer vector specifying the number of top dimensional cells. Defaults to
NULL.py_classAn existing
CubicalComplexPython class. Defaults toNULLwhich uses the Python class constructor instead.
Returns
A new CubicalComplex object.
Method betti_numbers()
This function returns the Betti numbers of the complex.
Usage
CubicalComplex$betti_numbers()
Details
The $betti_numbers() method always returns [1, 0, 0, ...] as
infinity filtration cubes are not removed from the complex.
Returns
An integer vector storing the Betti numbers.
Method cofaces_of_persistence_pairs()
A persistence interval is described by a pair of cells, one that creates the feature and one that kills it. The filtration values of those 2 cells give coordinates for a point in a persistence diagram, or a bar in a barcode. Structurally, in the cubical complexes provided here, the filtration value of any cell is the minimum of the filtration values of the maximal cells that contain it. Connecting persistence diagram coordinates to the corresponding value in the input (i.e. the filtration values of the top-dimensional cells) is useful for differentiation purposes.
Usage
CubicalComplex$cofaces_of_persistence_pairs()
Details
This function returns a list of pairs of top-dimensional cells corresponding to the persistence birth and death cells of the filtration. The cells are represented by their indices in the input list of top-dimensional cells (and not their indices in the internal data structure that includes non-maximal cells). Note that when two adjacent top-dimensional cells have the same filtration value, we arbitrarily return one of the two when calling the function on one of their common faces.
Returns
The top-dimensional cells/cofaces of the positive and negative
cells, together with the corresponding homological dimension, in two
lists of integer arrays. The first list contains the regular
persistence pairs, grouped by dimension. It contains numpy arrays of
shape [number_of_persistence_points, 2]. The indices of the arrays in
the list correspond to the homological dimensions, and the integers of
each row in each array correspond to: (index of positive top-dimensional cell, index of negative top-dimensional cell). The
second list contains the essential features, grouped by dimension. It
contains numpy arrays of shape [number_of_persistence_points, 1]. The
indices of the arrays in the list correspond to the homological
dimensions, and the integers of each row in each array correspond to:
(index of positive top-dimensional cell).
Method compute_persistence()
This method computes the persistence of the complex, so it
can be accessed through $persistent_betti_numbers(),
$persistence_intervals_in_dimension(), etc. It is equivalent to the
$persistence() method when you do not want the list $persistence()
returns.
Usage
CubicalComplex$compute_persistence( homology_coeff_field = 11, min_persistence = 0 )
Arguments
homology_coeff_fieldAn integer value specifying the homology coefficient field. Must be a prime number. Defaults to
11L. Maximum is46337L.min_persistenceA numeric value specifying the minimum persistence value to take into account (strictly greater than
min_persistence). Defaults to0.0. Setmin_persistence = -1.0to see all values.
Returns
The updated CubicalComplex class itself invisibly.
Method dimension()
This function returns the dimension of the complex.
Usage
CubicalComplex$dimension()
Returns
An integer value giving the complex dimension.
Method num_simplices()
This function returns the number of all cubes in the complex.
Usage
CubicalComplex$num_simplices()
Returns
An integer value giving the number of all cubes in the complex.
Method persistence()
This function computes and returns the persistence of the complex.
Usage
CubicalComplex$persistence(homology_coeff_field = 11, min_persistence = 0)
Arguments
homology_coeff_fieldAn integer value specifying the homology coefficient field. Must be a prime number. Defaults to
11L. Maximum is46337L.min_persistenceA numeric value specifying the minimum persistence value to take into account (strictly greater than
min_persistence). Defaults to0.0. Setmin_persistence = -1.0to see all values.
Returns
A tibble listing all persistence feature
summarised by 3 variables: dimension, birth and death.
Method persistence_intervals_in_dimension()
This function returns the persistence intervals of the complex in a specific dimension.
Usage
CubicalComplex$persistence_intervals_in_dimension(dimension)
Arguments
dimensionAn integer value specifying the desired dimension.
Returns
A tibble storing the persistence intervals
by row.
Method persistent_betti_numbers()
This function returns the persistent Betti numbers of the complex.
Usage
CubicalComplex$persistent_betti_numbers(from_value, to_value)
Arguments
from_valueA numeric value specifying the persistence birth limit to be added in the numbers (
persistent birth <= from_value).to_valueA numeric value specifying the persistence death limit to be added in the numbers (
persistent death > to_value).
Returns
An integer vector storing the persistent Betti numbers.
Method clone()
The objects of this class are cloneable with this method.
Usage
CubicalComplex$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Author(s)
Pawel Dlotko
See Also
Other data structures for cell complexes:
PeriodicCubicalComplex,
SimplexTree
Examples
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$betti_numbers()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$cofaces_of_persistence_pairs()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$dimension()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$num_simplices()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$persistence()
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$persistence_intervals_in_dimension(0)
n <- 10
X <- cbind(seq(0, 1, len = n), seq(0, 1, len = n))
cc <- CubicalComplex$new(top_dimensional_cells = X)
cc$compute_persistence()$persistent_betti_numbers(0, 1)
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