From fd41fbbb321b36c143129b5a311e015bf7c44a6f Mon Sep 17 00:00:00 2001 From: vrouvrea Date: Fri, 11 Dec 2015 13:04:06 +0000 Subject: Removed pdf to use png Biblio issue git-svn-id: svn+ssh://scm.gforge.inria.fr/svnroot/gudhi/branches/bitmap@944 636b058d-ea47-450e-bf9e-a15bfbe3eedb Former-commit-id: 68e93aa53ebadcde4b44eec881b72080e6ceaa70 --- .../doc/Gudhi_Cubical_Complex_doc.h | 89 ++++++++++------------ 1 file changed, 42 insertions(+), 47 deletions(-) (limited to 'src/Bitmap_cubical_complex/doc/Gudhi_Cubical_Complex_doc.h') diff --git a/src/Bitmap_cubical_complex/doc/Gudhi_Cubical_Complex_doc.h b/src/Bitmap_cubical_complex/doc/Gudhi_Cubical_Complex_doc.h index 6e821ce4..1a6310fb 100644 --- a/src/Bitmap_cubical_complex/doc/Gudhi_Cubical_Complex_doc.h +++ b/src/Bitmap_cubical_complex/doc/Gudhi_Cubical_Complex_doc.h @@ -38,70 +38,65 @@ namespace Cubical_complex *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 \emph{elementary interval} is an interval of a form $[n,n+1]$, or $[n,n]$, for $n \in \mathcal{Z}$. The first one is called \emph{non-degenerated}, while the second one is \emph{degenerated} interval. A \emph{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 degenerated elementary interval. An \emph{elementary cube} $C$ is a -*product of elementary intervals, $C=I_1 \times \ldots \times I_n$. \emph{Embedding dimension} of a cube is n, the number of elementary intervals (degenerated or not) in the product. A \emph{dimension of a cube} $C=I_1 \times ... \times I_n$ is the -*number of non degenerated elementary intervals in the product. A \emph{boundary of a cube} $C=I_1 \times \ldots \times I_n$ is a chain obtained in the following way: -*\[\partial C = (\partial I_1 \times \ldots \times I_n) + (I_1 \times \partial I_2 \times \ldots \times I_n) + \ldots + (I_1 \times I_2 \times \ldots \times \partial I_n).\] -*A \emph{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 \emph{maximal} if it is not in -*a boundary of any other cube in $\mathcal{K}$. A \emph{support} of a cube $C$ is the set in $\mathbb{R}^n$ occupied by $C$ ($n$ is the embedding dimension of $C$). +* An elementary interval is an interval of a form \f$ [n,n+1] \f$, or \f$[n,n]\f$, for \f$ n \in \mathcal{Z} \f$. The first one is called non-degenerated, while the second one is \a degenerated interval. A boundary of a elementary +*interval is a chain \f$\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 degenerated elementary interval. An elementary cube \f$ C \f$ is a + +*product of elementary intervals, \f$C=I_1 \times \ldots \times I_n\f$. Embedding dimension of a cube is n, the number of elementary intervals (degenerated or not) in the product. A dimension of a cube \f$C=I_1 \times ... \times I_n\f$ is the +*number of non degenerated elementary intervals in the product. A boundary of a cube \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 cubical complex \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 maximal 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 a book:\\ -*Computational homology, by Tomasz Kaczynski, Konstantin Mischaikow, and Marion Mrozek, Appl. Math. Sci., vol. 157, Springer-Verlag, New York, 2004 +*For further details and theory of cubical complexes, please consult \cite kaczynski2004computational . * -*as well as the paper: -*Efficient computation of persistent homology for cubical data by Hubert Wagner, Chao Chen, Erald Vuçini (published in the proceedings of Workshop on Topology-based Methods in Data -*Analysis and Visualization) +*as well as the following paper \cite peikert2012topological . * -*\section{Data structure.} +*\section datastructure 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 $C \in \mathcal{K}$. +*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 "bitmapAllCubes.pdf" "Cubical complex in $\mathbb{R}^2". +*\image html "bitmapAllCubes.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 $\mathbb{R}$. 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 +*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{Input Format.} +*\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 \emph{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. +*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. * -*\begin{enumerate} -*\item The first line of the file is $d$, the embedding dimension of a complex. -*\item The next $d$ lines consist of positive numbers being the numbers of top dimensional cubes in the given direction. Let us call those numbers $n_1,\ldots,n_d$. -*\item Later there is a sequence of $n_1 \dot \ldots \dot n_d$ numbers in a lexicographical ordering. Those numbers are filtrations of top dimensional cubes. -*\end{enumerate} * -*\image html "exampleBitmap.pdf" "Example of a input data." +*\image html "exampleBitmap.png" "Example of a input data." * *The input file for the following complex is: -*\begin{verbatim} -*2 -*3 -*3 -*1 -*2 -*3 -*8 -*20 -*4 -*7 -*6 -*5 -*\end{verbatim} -* -* +*\verbatim +2 +3 +3 +1 +2 +3 +8 +20 +4 +7 +6 +5 +\endverbatim + +*/ +/** @} */ // end defgroup cubical_complex + *@}//end of the group } } -- cgit v1.2.3