From b16152118574b9f2127df3b7f4495f75f3b079c1 Mon Sep 17 00:00:00 2001 From: vrouvrea Date: Thu, 19 May 2016 15:27:00 +0000 Subject: Seperate doc from code in Persistent_cohomology module. git-svn-id: svn+ssh://scm.gforge.inria.fr/svnroot/gudhi/branches/get_persistence@1184 636b058d-ea47-450e-bf9e-a15bfbe3eedb Former-commit-id: 90b406b11cfa6356243bb4e881c8c7db06f14d6e --- .../include/gudhi/Persistent_cohomology.h | 136 +-------------------- 1 file changed, 4 insertions(+), 132 deletions(-) (limited to 'src/Persistent_cohomology/include') diff --git a/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h b/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h index 1b86f1f9..cd1bd438 100644 --- a/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h +++ b/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h @@ -46,136 +46,10 @@ namespace Gudhi { namespace persistent_cohomology { -/** \defgroup persistent_cohomology Persistent Cohomology - * - \author Clément Maria - - Computation of persistent cohomology using the algorithm of - \cite DBLP:journals/dcg/SilvaMV11 and \cite DBLP:journals/corr/abs-1208-5018 - and the Compressed Annotation Matrix - implementation of \cite DBLP:conf/esa/BoissonnatDM13 - - The theory of homology consists in attaching to a topological space a sequence of - (homology) groups, - capturing global topological features - like connected components, holes, cavities, etc. Persistent homology studies the evolution - -- birth, life and death -- of - these features when the topological space is changing. Consequently, the theory is essentially - composed of three elements: - topological spaces, their homology groups and an evolution scheme. - -
Topological Spaces:
- Topological spaces are represented by simplicial complexes. - Let \f$V = \{1, \cdots ,|V|\}\f$ be a set of vertices. - A simplex \f$\sigma\f$ is a subset of vertices - \f$\sigma \subseteq V\f$. A simplicial complex \f$\mathbf{K}\f$ - on \f$V\f$ is a collection of simplices \f$\{\sigma\}\f$, - \f$\sigma \subseteq V\f$, such that \f$\tau \subseteq \sigma \in \mathbf{K} - \Rightarrow \tau \in \mathbf{K}\f$. The dimension \f$n=|\sigma|-1\f$ of \f$\sigma\f$ - is its number of elements minus 1. A filtration of a simplicial complex is - a function \f$f:\mathbf{K} \rightarrow \mathbb{R}\f$ satisfying \f$f(\tau)\leq - f(\sigma)\f$ whenever \f$\tau \subseteq \sigma\f$. - - We define the concept FilteredComplex which enumerates the requirements for a class - to represent a filtered complex from which persistent homology may be computed. - We use the vocabulary of simplicial complexes, but the concept - is valid for any type of cell complex. The main requirements - are the definition of: - \li type Indexing_tag, which is a model of the concept - IndexingTag, - describing the nature of the indexing scheme, - \li type Simplex_handle to manipulate simplices, - \li method int dimension(Simplex_handle) returning - the dimension of a simplex, - \li type and method Boundary_simplex_range - boundary_simplex_range(Simplex_handle) that returns - a range giving access to the codimension 1 subsimplices of the - input simplex, as-well-as the coefficients \f$(-1)^i\f$ in the - definition of the operator \f$\partial\f$. The iterators have - value type Simplex_handle, - \li type and method - Filtration_simplex_range filtration_simplex_range () - that returns a range giving - access to all the simplices of the complex read in the order - assigned by the indexing scheme, - \li type and method - Filtration_value filtration (Simplex_handle) that returns the value of - the filtration on the simplex represented by the handle. - -
Homology:
- For a ring \f$\mathcal{R}\f$, the group of n-chains, - denoted \f$\mathbf{C}_n(\mathbf{K},\mathcal{R})\f$, of \f$\mathbf{K}\f$ is the - group of formal sums of - n-simplices with \f$\mathcal{R}\f$ coefficients. The boundary operator is a - linear operator - \f$\partial_n: \mathbf{C}_n(\mathbf{K},\mathcal{R}) \rightarrow \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R})\f$ - such that \f$\partial_n \sigma = \partial_n [v_0, \cdots , v_n] = - \sum_{i=0}^n (-1)^{i}[v_0,\cdots ,\widehat{v_i}, \cdots,v_n]\f$, - where \f$\widehat{v_i}\f$ means \f$v_i\f$ is omitted from the list. The chain - groups form a sequence: - - \f[\cdots \ \ \mathbf{C}_n(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_n\ } \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R}) - \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\ \partial_2 \ } - \mathbf{C}_1(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_1 \ } \mathbf{C}_0(\mathbf{K},\mathcal{R}) \f] - - of finitely many groups \f$\mathbf{C}_n(\mathbf{K},\mathcal{R})\f$ and homomorphisms - \f$\partial_n\f$, indexed by the dimension \f$n \geq 0\f$. - The boundary operators satisfy the property \f$\partial_n \circ \partial_{n+1}=0\f$ - for every \f$n > 0\f$ - and we define the homology groups: - - \f[\mathbf{H}_n(\mathbf{K},\mathcal{R}) = \ker \partial_n / \mathrm{im} \ \partial_{n+1}\f] - - We refer to \cite Munkres-elementsalgtop1984 for an introduction to homology - theory and to \cite DBLP:books/daglib/0025666 for an introduction to persistent homology. - -
Indexing Scheme:
- "Changing" a simplicial complex consists in applying a simplicial map. - An indexing scheme is a directed graph together with a traversal - order, such that two - consecutive nodes in the graph are connected by an arrow (either forward or backward). - The nodes represent simplicial complexes and the directed edges simplicial maps. - - From the computational point of view, there are two types of indexing schemes of - interest - in persistent homology: linear ones - \f$\bullet \longrightarrow \bullet \longrightarrow \cdots \longrightarrow \bullet - \longrightarrow \bullet\f$ - in persistent homology \cite DBLP:journals/dcg/ZomorodianC05 , - and zigzag ones - \f$\bullet \longrightarrow \bullet \longleftarrow \cdots - \longrightarrow \bullet - \longleftarrow \bullet \f$ in zigzag persistent - homology \cite DBLP:journals/focm/CarlssonS10. - These indexing schemes have a natural left-to-right traversal order, and we - describe them with ranges and iterators. - In the current release of the Gudhi library, only the linear case is implemented. - - In the following, we consider the case where the indexing scheme is induced - by a filtration. - Ordering the simplices - by increasing filtration values (breaking ties so as a simplex appears after - its subsimplices of same filtration value) provides an indexing scheme. - -\section Examples - We provide several example files: run these examples with -h for details on their use, and read the README file. - -\li rips_persistence.cpp computes the Rips complex of a point cloud and its persistence diagram. - -\li rips_multifield_persistence.cpp computes the Rips complex of a point cloud and its persistence diagram -with a family of field coefficients. - -\li performance_rips_persistence.cpp provides timings for the construction of the Rips complex on a set of -points sampling a Klein bottle in \f$\mathbb{R}^5\f$ with a simplex tree, its conversion to a -Hasse diagram and the computation of persistent homology and multi-field persistent homology for the -different representations. - - \copyright GNU General Public License v3. - @{ - */ - /** \brief Computes the persistent cohomology of a filtered complex. * + * \ingroup persistent_cohomology + * * The computation is implemented with a Compressed Annotation Matrix * (CAM)\cite DBLP:conf/esa/BoissonnatDM13, * and is adapted to the computation of Multi-Field Persistent Homology (MF) @@ -184,8 +58,8 @@ different representations. * \implements PersistentHomology * */ -// Memory allocation policy: classic, use a mempool, etc.*/ -template // to do mem allocation policy: classic, mempool, etc. +// TODO(CM): Memory allocation policy: classic, use a mempool, etc. +template class Persistent_cohomology { public: typedef FilteredComplex Complex_ds; @@ -766,8 +640,6 @@ class Persistent_cohomology { Simple_object_pool cell_pool_; }; -/** @} */ // end defgroup persistent_cohomology - } // namespace persistent_cohomology } // namespace Gudhi -- cgit v1.2.3