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diff --git a/doc/Bitmap_cubical_complex/Gudhi_Cubical_Complex_doc.h b/doc/Bitmap_cubical_complex/Gudhi_Cubical_Complex_doc.h new file mode 100644 index 00000000..4c9c04d9 --- /dev/null +++ b/doc/Bitmap_cubical_complex/Gudhi_Cubical_Complex_doc.h @@ -0,0 +1,159 @@ +/* This file is part of the Gudhi Library. The Gudhi library + * (Geometric Understanding in Higher Dimensions) is a generic C++ + * library for computational topology. + * + * Author(s): Pawel Dlotko + * + * Copyright (C) 2015 INRIA Sophia-Saclay (France) + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <http://www.gnu.org/licenses/>. + */ + + +#ifndef DOC_GUDHI_CUBICAL_COMPLEX_COMPLEX_H_ +#define DOC_GUDHI_CUBICAL_COMPLEX_COMPLEX_H_ + +namespace Gudhi { + +namespace cubical_complex { + +/** \defgroup cubical_complex Cubical complex + * + * \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 <em>elementary interval</em> is an interval of a form \f$ [n,n+1] \f$, or \f$[n,n]\f$, for \f$ n \in \mathcal{Z} + * \f$. The first one is called <em>non-degenerate</em>, while the second one is \a degenerate interval. A + * <em>boundary of a elementary interval</em> is a chain \f$\partial [n,n+1] = [n+1,n+1]-[n,n] \f$ in case of + * non-degenerated elementary interval and \f$\partial [n,n] = 0 \f$ in case of degenerate elementary interval. An + * <em>elementary cube</em> \f$ C \f$ is a product of elementary intervals, \f$C=I_1 \times \ldots \times I_n\f$. + * <em>Embedding dimension</em> of a cube is n, the number of elementary intervals (degenerate or not) in the product. + * A <em>dimension of a cube</em> \f$C=I_1 \times ... \times I_n\f$ is the number of non degenerate elementary + * intervals in the product. A <em>boundary of a cube</em> \f$C=I_1 \times \ldots \times I_n\f$ is a chain obtained + * in the following way: + * \f[\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).\f] + * A <em>cubical complex</em> \f$\mathcal{K}\f$ 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 \f$C\f$ in cubical complex + * \f$\mathcal{K}\f$ is <em>maximal</em> if it is not in a boundary of any other cube in \f$\mathcal{K}\f$. A \a + * support of a cube \f$C\f$ is the set in \f$\mathbb{R}^n\f$ occupied by \f$C\f$ (\f$n\f$ is the embedding dimension + * of \f$C\f$). + * + * 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 \cite kaczynski2004computational as well as the + * following paper \cite peikert2012topological . + * + * \section datastructure Data structure. + * + * The implementation of Cubical complex provides a representation of complexes that occupy a rectangular region in + * \f$\mathbb{R}^n\f$. This extra assumption allows for a memory efficient way of storing cubical complexes in a form + * of so called bitmaps. Let \f$R = [b_1,e_1] \times \ldots \times [b_n,e_n]\f$, for \f$b_1,...b_n,e_1,...,e_n \in + * \mathbb{Z}\f$, \f$b_i \leq d_i\f$ be the considered rectangular region and let \f$\mathcal{K}\f$ be a filtered + * cubical complex having the rectangle \f$R\f$ as its support. Note that the structure of the coordinate system gives + * a way a lexicographical ordering of cells of \f$\mathcal{K}\f$. 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 \f$\mathcal{K}\f$ and the sizes of \f$\mathcal{K}\f$ in all + * directions, allows to determine, dimension, neighborhood, boundary and coboundary of every cube \f$C \in + * \mathcal{K}\f$. + * + * \image html "Cubical_complex_representation.png" Cubical complex. + * + * Note that the cubical complex in the figure above is, in a natural way, a product of one dimensional cubical + * complexes in \f$\mathbb{R}\f$. The number of all cubes in each direction is equal \f$2n+1\f$, where \f$n\f$ is the + * number of maximal cubes in the considered direction. Let us consider a cube at the position \f$k\f$ 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 \f$C\f$. In a similar way, we can compute boundary and the coboundary of + * each cube. Further details can be found in the literature. + * + * \section inputformat Input Format. + * + * In the current implantation, filtration is given at the maximal cubes, and it is then extended by the lower star + * filtration 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 \a Bitmap_cubical_complex class. + * Currently one input from a text file is used. It uses a format used already in Perseus software + * (http://www.sas.upenn.edu/~vnanda/perseus/) by Vidit Nanda. + * Below we are providing a description of the format. The first line contains a number d begin the dimension of the + * bitmap (2 in the example below). Next d lines are the numbers of top dimensional cubes in each dimensions (3 and 3 + * in the example below). Next, in lexicographical order, the filtration of top dimensional cubes is given (1 4 6 8 + * 20 4 7 6 5 in the example below). + * + * + * \image html "exampleBitmap.png" "Example of a input data." + * + * The input file for the following complex is: + * \verbatim +2 +3 +3 +1 +4 +6 +8 +20 +4 +7 +6 +5 +\endverbatim + + * \section PeriodicBoundaryConditions Periodic boundary conditions + * Often one would like to impose periodic boundary conditions to the cubical complex. Let \f$ I_1\times ... \times + * I_n \f$ be a box that is decomposed with a cubical complex \f$ \mathcal{K} \f$. Imposing periodic boundary + * conditions in the direction i, means that the left and the right side of a complex \f$ \mathcal{K} \f$ are + * considered the same. In particular, if for a bitmap \f$ \mathcal{K} \f$ periodic boundary conditions are imposed + * in all directions, then complex \f$ \mathcal{K} \f$ 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. To indicate periodic boundary conditions in a + * given direction, then number of top dimensional cells in this direction have to be multiplied by -1. For instance: + + *\verbatim +2 +-3 +3 +1 +4 +6 +8 +20 +4 +7 +6 +5 +\endverbatim + + * Indicate that we have imposed periodic boundary conditions in the direction x, but not in the direction y. + + * \section BitmapExamples Examples + * End user programs are available in example/Bitmap_cubical_complex folder. + * + * \copyright GNU General Public License v3. + */ +/** @} */ // end defgroup cubical_complex + +} // namespace cubical_complex + +namespace Cubical_complex = cubical_complex; + +} // namespace Gudhi + +#endif // DOC_GUDHI_CUBICAL_COMPLEX_COMPLEX_H_ |