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Cubical complex

## Classes

class | Gudhi::cubical_complex::Bitmap_cubical_complex< T >
| Cubical complex represented as a bitmap. More...
class | Gudhi::cubical_complex::Bitmap_cubical_complex_base< T >
| Cubical complex represented as a bitmap, class with basic implementation. More...
class | Gudhi::cubical_complex::Bitmap_cubical_complex_periodic_boundary_conditions_base< T >
| Cubical complex with periodic boundary conditions represented as a bitmap. More...

## Detailed Description

Author
  Pawel Dlotko

Bitmap_cubical_complex is an example of a structured complex useful in computational mathematics (specially rigorous numerics) and image analysis. The presented implementation of cubical complexes is based on the following definition.

An elementary interval is an interval of a form \( [n,n+1] \), or \([n,n]\), for \( n \in \mathcal{Z} \). The first one is called non-degenerate, while the second one is degenerate interval. A boundary of a elementary interval is a chain \(\partial [n,n+1] = [n+1,n+1]-[n,n] \) in case of non-degenerated elementary interval and \(\partial [n,n] = 0 \) in case of degenerate elementary interval. An elementary cube \( C \) is a product of elementary intervals, \(C=I_1 \times \ldots \times I_n\). Embedding dimension of a cube is n, the number of elementary intervals (degenerate or not) in the product. A dimension of a cube \(C=I_1 \times ... \times I_n\) is the number of non degenerate elementary intervals in the product. A boundary of a cube \(C=I_1 \times \ldots \times I_n\) is a chain obtained in the following way:

\[\partial C = (\partial I_1 \times \ldots \times I_n) + (I_1 \times \partial I_2 \times \ldots \times I_n) + \ldots + (I_1 \times I_2 \times \ldots \times \partial I_n)\]

(when working with a field of characteristic other than 2, the non-zero terms of this sum come with alternating signs ±1).

A cubical complex \(\mathcal{K}\) is a collection of cubes closed under operation of taking boundary (i.e. boundary of every cube from the collection is in the collection). A cube \(C\) in cubical complex \(\mathcal{K}\) is maximal if it is not in a boundary of any other cube in \(\mathcal{K}\). A support of a cube \(C\) is the set in \(\mathbb{R}^n\) occupied by \(C\) ( \(n\) is the embedding dimension of \(C\)).

Cubes may be equipped with a filtration values in which case we have filtered cubical complex. All the cubical complexes considered in this implementation are filtered cubical complexes (although, the range of a filtration may be a set of two elements).

For further details and theory of cubical complexes, please consult [kaczynski2004computational] as well as the following paper [peikert2012topological] .

# Data structure

The implementation of Cubical complex provides a representation of complexes that occupy a rectangular region in \(\mathbb{R}^n\). This extra assumption allows for a memory efficient way of storing cubical complexes in a form of so called bitmaps. Let \(R = [b_1,e_1] \times \ldots \times [b_n,e_n]\), for \(b_1,...b_n,e_1,...,e_n \in \mathbb{Z}\), \(b_i \leq d_i\) be the considered rectangular region and let \(\mathcal{K}\) be a filtered cubical complex having the rectangle \(R\) as its support. Note that the structure of the coordinate system gives a way a lexicographical ordering of cells of \(\mathcal{K}\). This ordering is a base of the presented bitmap-based implementation. In this implementation, the whole cubical complex is stored as a vector of the values of filtration. This, together with dimension of \(\mathcal{K}\) and the sizes of \(\mathcal{K}\) in all directions, allows to determine, dimension, neighborhood, boundary and coboundary of every cube \(C \in \mathcal{K}\).

Cubical complex.

Note that the cubical complex in the figure above is, in a natural way, a product of one dimensional cubical complexes in \(\mathbb{R}\). The number of all cubes in each direction is equal \(2n+1\), where \(n\) is the number of maximal cubes in the considered direction. Let us consider a cube at the position \(k\) in the bitmap. Knowing the sizes of the bitmap, by a series of modulo operation, we can determine which elementary intervals are present in the product that gives the cube \(C\). In a similar way, we can compute boundary and the coboundary of each cube. Further details can be found in the literature.

# Input Format

In the current implementation, filtration is given at the maximal cubes, and it is then extended by the lower star filtration to all cubes. Alternatively, it is also possible to give the filtration at the vertices and extend it to all cubes. There are a number of constructors that can be used to construct cubical complex by users who want to use the code directly. They can be found in the Bitmap_cubical_complex class. Currently one input from a text file is used. It uses a format inspired from the Perseus software (http://www.sas.upenn.edu/~vnanda/perseus/) by Vidit Nanda.

Note
  While Perseus assume the filtration of all maximal cubes to be non-negative, over here we do not enforce this and we allow any filtration values. As a consequence one cannot use -1's to indicate missing cubes. If you have missing cubes in your complex, please set their filtration to \(+\infty\) (aka. inf in the file).

The file format is described in details in Perseus file format section.

# Periodic boundary conditions

Often one would like to impose periodic boundary conditions to the cubical complex. Let \( I_1\times ... \times I_n \) be a box that is decomposed with a cubical complex \( \mathcal{K} \). Imposing periodic boundary conditions in the direction i, means that the left and the right side of a complex \( \mathcal{K} \) are considered the same. In particular, if for a bitmap \( \mathcal{K} \) periodic boundary conditions are imposed in all directions, then complex \( \mathcal{K} \) became n-dimensional torus. One can use various constructors from the file Bitmap_cubical_complex_periodic_boundary_conditions_base.h to construct cubical complex with periodic boundary conditions. One can also use Perseus style input files (see Perseus).

# Examples

End user programs are available in example/Bitmap_cubical_complex and utilities/Bitmap_cubical_complex folders.