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

1 files changed, 4 insertions, 132 deletions

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.
-
- <DT>Topological Spaces:</DT>
- Topological spaces are represented by simplicial complexes.
- Let \f$V = \{1, \cdots ,|V|\}\f$ be a set of <EM>vertices</EM>.
- A <EM>simplex</EM> \f$\sigma\f$ is a subset of vertices
- \f$\sigma \subseteq V\f$. A <EM>simplicial complex</EM> \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 <EM>filtration</EM> 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 <CODE>Indexing_tag</CODE>, which is a model of the concept
- <CODE>IndexingTag</CODE>,
- describing the nature of the indexing scheme,
- \li type Simplex_handle to manipulate simplices,
- \li method <CODE>int dimension(Simplex_handle)</CODE> returning
- the dimension of a simplex,
- \li type and method <CODE>Boundary_simplex_range
- boundary_simplex_range(Simplex_handle)</CODE> 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 <CODE>Simplex_handle</CODE>,
- \li type and method
- <CODE>Filtration_simplex_range filtration_simplex_range ()</CODE>
- 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
- <CODE>Filtration_value filtration (Simplex_handle)</CODE> that returns the value of
- the filtration on the simplex represented by the handle.
-
- <DT>Homology:</DT>
- For a ring \f$\mathcal{R}\f$, the group of <EM>n-chains</EM>,
- 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 <EM>boundary operator</EM> 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.
-
- <DT>Indexing Scheme:</DT>
- "Changing" a simplicial complex consists in applying a simplicial map.
- An <EM>indexing scheme</EM> 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: <EM>linear</EM> ones
- \f$\bullet \longrightarrow \bullet \longrightarrow \cdots \longrightarrow \bullet
- \longrightarrow \bullet\f$
- in persistent homology \cite DBLP:journals/dcg/ZomorodianC05 ,
- and <EM>zigzag</EM> 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 <CODE>rips_persistence.cpp</CODE> computes the Rips complex of a point cloud and its persistence diagram.
-
-\li <CODE>rips_multifield_persistence.cpp</CODE> computes the Rips complex of a point cloud and its persistence diagram 
-with a family of field coefficients.
-
-\li <CODE>performance_rips_persistence.cpp</CODE> 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<class FilteredComplex, class CoefficientField>  // to do mem allocation policy: classic, mempool, etc.
+// TODO(CM): Memory allocation policy: classic, use a mempool, etc.
+template<class FilteredComplex, class CoefficientField>
 class Persistent_cohomology {
  public:
   typedef FilteredComplex Complex_ds;
@@ -766,8 +640,6 @@ class Persistent_cohomology {
   Simple_object_pool<Cell> cell_pool_;
 };
 
-/** @} */  // end defgroup persistent_cohomology
-
 }  // namespace persistent_cohomology
 
 }  // namespace Gudhi