Added a big chunk of documentation. +small fixes

git-svn-id: svn+ssh://scm.gforge.inria.fr/svnroot/gudhi/branches/relaxed-witness@1679 636b058d-ea47-450e-bf9e-a15bfbe3eedb

Former-commit-id: d3166034bf662121bc21583bb027c67f736e904c

1 files changed, 34 insertions, 12 deletions

diff --git a/src/Witness_complex/doc/Witness_complex_doc.h b/src/Witness_complex/doc/Witness_complex_doc.h
index 60dfd27b..1d6e9da2 100644
--- a/src/Witness_complex/doc/Witness_complex_doc.h
+++ b/src/Witness_complex/doc/Witness_complex_doc.h
@@ -6,33 +6,55 @@
 
    \author Siargey Kachanovich
 
-   \image html "Witness_complex_representation.png" "Witness complex representation"
+   \image html "Witness_complex_representation.png" "Witness complex representation in a Simplex tree (from \cite boissonnatmariasimplextreealgorithmica)"
 
    \section Definitions
 
-   Witness complex \f$ Wit(W,L) \f$  is a simplicial complex defined on two sets of points in \f$\mathbb{R}^D\f$:
+   Witness complex is a simplicial complex defined on two sets of points in \f$\mathbb{R}^D\f$:
 
    \li \f$W\f$ set of **witnesses** and 
-   \li \f$L \subseteq W\f$ set of **landmarks**.
+   \li \f$L\f$ set of **landmarks**.
+
+   Even though often the set of landmarks \f$L\f$ is a subset of the set of witnesses \f$ W\f$, it is not a requirement for the current implementation.
 
    The simplices are based on landmarks
-   and a simplex belongs to the witness complex if and only if it is witnessed, that is:
+   and witnesses help to decide on which simplices are inserted via a predicate "is witnessed".
+
+   De Silva and Carlsson in their paper \cite de2004topological differentiate **weak witnessing** and **strong witnessing**:
+
+   - *weak*:   \f$ \sigma \subset L \f$ is witnessed by \f$ w \in W\f$ if \f$ \forall l \in \sigma,\ \forall l' \in L \setminus \sigma,\ d(w,l) \leq d(w,l') \f$
+   - *strong*: \f$ \sigma \subset L \f$ is witnessed by \f$ w \in W\f$ if \f$ \forall l \in \sigma,\ \forall l' \in L,\ d(w,l) \leq d(w,l') \f$
+
+   where \f$ d(.,.) \f$ is a distance function.
+
+   Both definitions can be relaxed by a real value \f$\alpha\f$:
+
+   - *weak*:   \f$ \sigma \subset L \f$ is \f$\alpha\f$-witnessed by \f$ w \in W\f$ if \f$ \forall l \in \sigma,\ \forall l' \in L \setminus \sigma,\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2 \f$
+   - *strong*: \f$ \sigma \subset L \f$ is \f$\alpha\f$-witnessed by \f$ w \in W\f$ if \f$ \forall l \in \sigma,\ \forall l' \in L,\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2 \f$
 
-   \f$ \sigma \subset L \f$ is witnessed if there exists a point \f$w \in W\f$ such that 
-   w is closer to the vertices of \f$ \sigma \f$ than other points in \f$ L \f$ and all of its faces are witnessed as well. 
- 
-   The data structure is described in \cite boissonnatmariasimplextreealgorithmica .
+   which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and **strong relaxed witness complex** respectively.
 
    \section Implementation
+   
+   The two complexes described above are implemented in the corresponding classes 
+   - Gudhi::witness_complex::Witness_complex
+   - Gudhi::witness_complex::Strong_witness_complex
+
+   The construction of both of them follow the same scheme:
+   1. Construct a search tree on landmarks (for that Gudhi::spatial_searching::Kd_tree_search is used internally).
+   2. Construct lists of nearest landmarks for each witness (special internal structure Gudhi::spatial_searching::Active_witness is used internally).
+   3. Construct the witness complex for nearest landmark lists.
+
+   The constructors take on the step 1, while the function 'create_complex' executes the steps 2 and 3.
 
-   The principal class of this module is Gudhi::Witness_complex.
+   \section Examples
 
-   In both cases, the constructor for this class takes a {witness}x{closest_landmarks} table, where each row represents a witness and consists of landmarks sorted by distance to this witness.
-   This table can be constructed by two additional classes Landmark_choice_by_furthest_point and Landmark_choice_by_random_point also included in the module.
+   Here is an example of constructing a strong witness complex filtration and computing persistence on it:
+
+   \include Witness_complex/example_strong_witness_persistence.cpp
 
    *\image html "bench_Cy8.png" "Running time as function on number of landmarks" width=10cm
    *\image html "bench_sphere.png" "Running time as function on number of witnesses for |L|=300" width=10cm
-   
 
    \copyright GNU General Public License v3.