Rewrite some summaries

7 files changed, 24 insertions, 34 deletions

diff --git a/src/python/doc/alpha_complex_sum.inc b/src/python/doc/alpha_complex_sum.inc
index 3aba0d71..aeab493f 100644
--- a/src/python/doc/alpha_complex_sum.inc
+++ b/src/python/doc/alpha_complex_sum.inc
@@ -3,15 +3,13 @@
 
    +----------------------------------------------------------------+-------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
    | .. figure::                                                    | Alpha complex is a simplicial complex constructed from the finite       | :Author: Vincent Rouvreau                                                                                                 |
-   |      ../../doc/Alpha_complex/alpha_complex_representation.png  | cells of a Delaunay Triangulation.                                      |                                                                                                                           |
-   |      :alt: Alpha complex representation                        |                                                                         | :Since: GUDHI 2.0.0                                                                                                       |
-   |      :figclass: align-center                                   | The filtration value of each simplex is computed as the **square** of   |                                                                                                                           |
-   |                                                                | the circumradius of the simplex if the circumsphere is empty (the       | :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
-   |                                                                | simplex is then said to be Gabriel), and as the minimum of the          |                                                                                                                           |
-   |                                                                | filtration values of the codimension 1 cofaces that make it not         | :Requires: `Eigen <installation.html#eigen>`_ :math:`\geq` 3.1.0 and `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0 |
-   |                                                                | Gabriel otherwise.                                                      |                                                                                                                           |
+   |      ../../doc/Alpha_complex/alpha_complex_representation.png  | cells of a Delaunay Triangulation. It has the same persistent homology  |                                                                                                                           |
+   |      :alt: Alpha complex representation                        | as the Čech complex and is significantly smaller.                       | :Since: GUDHI 2.0.0                                                                                                       |
+   |      :figclass: align-center                                   |                                                                         |                                                                                                                           |
+   |                                                                |                                                                         | :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
+   |                                                                |                                                                         |                                                                                                                           |
+   |                                                                |                                                                         | :Requires: `Eigen <installation.html#eigen>`_ :math:`\geq` 3.1.0 and `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0 |
    |                                                                |                                                                         |                                                                                                                           |
-   |                                                                | For performances reasons, it is advised to use CGAL :math:`\geq` 5.0.0. |                                                                                                                           |
    +----------------------------------------------------------------+-------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
    | * :doc:`alpha_complex_user`                                    | * :doc:`alpha_complex_ref`                                                                                                                                                                          |
    +----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
diff --git a/src/python/doc/alpha_complex_user.rst b/src/python/doc/alpha_complex_user.rst
index d49f45b4..e31194a7 100644
--- a/src/python/doc/alpha_complex_user.rst
+++ b/src/python/doc/alpha_complex_user.rst
@@ -9,7 +9,7 @@ Definition
 
 .. include:: alpha_complex_sum.inc
 
-`AlphaComplex` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using
+:doc:`AlphaComplex <alpha_complex_ref>` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using
 `Delaunay Triangulation  <http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations>`_
 :cite:`cgal:hdj-t-19b` from the `Computational Geometry Algorithms Library <http://www.cgal.org/>`_
 :cite:`cgal:eb-19b`.
@@ -19,6 +19,8 @@ Remarks
 When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`,
 the complex is a Delaunay complex (with special filtration values).
 
+For performances reasons, it is advised to use CGAL :math:`\geq` 5.0.0.
+
 Example from points
 -------------------
 
diff --git a/src/python/doc/cubical_complex_sum.inc b/src/python/doc/cubical_complex_sum.inc
index 28bf8e94..87db184d 100644
--- a/src/python/doc/cubical_complex_sum.inc
+++ b/src/python/doc/cubical_complex_sum.inc
@@ -2,9 +2,9 @@
    :widths: 30 40 30
 
    +--------------------------------------------------------------------------+----------------------------------------------------------------------+-----------------------------+
-   | .. figure::                                                              | The cubical complex is an example of a structured complex useful in  | :Author: Pawel Dlotko       |
-   |      ../../doc/Bitmap_cubical_complex/Cubical_complex_representation.png | computational mathematics (specially rigorous numerics) and image    |                             |
-   |      :alt: Cubical complex representation                                | analysis.                                                            | :Since: GUDHI 2.0.0         |
+   | .. figure::                                                              | The cubical complex represents a grid as a cell complex with         | :Author: Pawel Dlotko       |
+   |      ../../doc/Bitmap_cubical_complex/Cubical_complex_representation.png | cells of all dimensions.                                             |                             |
+   |      :alt: Cubical complex representation                                |                                                                      | :Since: GUDHI 2.0.0         |
    |      :figclass: align-center                                             |                                                                      |                             |
    |                                                                          |                                                                      | :License: MIT               |
    |                                                                          |                                                                      |                             |
diff --git a/src/python/doc/index.rst b/src/python/doc/index.rst
index 13e51047..05bc18b4 100644
--- a/src/python/doc/index.rst
+++ b/src/python/doc/index.rst
@@ -53,8 +53,8 @@ Tangential complex
 Topological descriptors computation
 ***********************************
 
-Persistence cohomology
-======================
+Persistent cohomology
+=====================
 
 .. include:: persistent_cohomology_sum.inc
 
diff --git a/src/python/doc/persistent_cohomology_sum.inc b/src/python/doc/persistent_cohomology_sum.inc
index a1ff2eee..58e44b8a 100644
--- a/src/python/doc/persistent_cohomology_sum.inc
+++ b/src/python/doc/persistent_cohomology_sum.inc
@@ -6,9 +6,7 @@
    |      ../../doc/Persistent_cohomology/3DTorus_poch.png           | a sequence of (homology) groups, capturing global topological         |                                               |
    |      :figclass: align-center                                    | features like connected components, holes, cavities, etc. Persistent  | :Since: GUDHI 2.0.0                           |
    |                                                                 | homology studies the evolution -- birth, life and death -- of these   |                                               |
-   |      Rips Persistent Cohomology on a 3D                         | features when the topological space is changing. Consequently, the    | :License: MIT                                 |
-   |      Torus                                                      | theory is essentially composed of three elements: topological spaces, |                                               |
-   |                                                                 | their homology groups and an evolution scheme.                        |                                               |
+   |      Rips Persistent Cohomology on a 3D Torus                   | features when the topological space is changing.                      | :License: MIT                                 |
    |                                                                 |                                                                       |                                               |
    |                                                                 | Computation of persistent cohomology using the algorithm of           |                                               |
    |                                                                 | :cite:`DBLP:journals/dcg/SilvaMV11` and                               |                                               |
diff --git a/src/python/doc/rips_complex_sum.inc b/src/python/doc/rips_complex_sum.inc
index 9cd8074b..c123ea2a 100644
--- a/src/python/doc/rips_complex_sum.inc
+++ b/src/python/doc/rips_complex_sum.inc
@@ -2,21 +2,13 @@
    :widths: 30 40 30
 
    +----------------------------------------------------------------+------------------------------------------------------------------------+----------------------------------------------------------------------+
-   | .. figure::                                                    | Rips complex is a simplicial complex constructed from a one skeleton   | :Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse |
-   |      ../../doc/Rips_complex/rips_complex_representation.png    | graph.                                                                 |                                                                      |
+   | .. figure::                                                    | The Vietoris-Rips complex is a simplicial complex built as the         | :Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse |
+   |      ../../doc/Rips_complex/rips_complex_representation.png    | clique-complex of a proximity graph.                                   |                                                                      |
    |      :figclass: align-center                                   |                                                                        | :Since: GUDHI 2.0.0                                                  |
-   |                                                                | The filtration value of each edge is computed from a user-given        |                                                                      |
-   |                                                                | distance function and is inserted until a user-given threshold         | :License: MIT                                                        |
-   |                                                                | value.                                                                 |                                                                      |
+   |                                                                | We also provide sparse approximations, to speed-up the computation     |                                                                      |
+   |                                                                | of persistent homology, and weighted versions, which are more robust   | :License: MIT                                                        |
+   |                                                                | to outliers.                                                           |                                                                      |
    |                                                                |                                                                        |                                                                      |
-   |                                                                | This complex can be built from a point cloud and a distance function,  |                                                                      |
-   |                                                                | or from a distance matrix.                                             |                                                                      |
-   |                                                                |                                                                        |                                                                      |
-   |                                                                | Weighted Rips complex constructs a simplicial complex from a distance  |                                                                      |
-   |                                                                | matrix and weights on vertices.                                        |                                                                      |
-   |                                                                |                                                                        |                                                                      |
-   |                                                                | DTM Rips complex builds a simplicial complex from a point set or       |                                                                      |
-   |                                                                | a distance matrix.                                                     |                                                                      |
    +----------------------------------------------------------------+------------------------------------------------------------------------+----------------------------------------------------------------------+
    | * :doc:`rips_complex_user`                                     | * :doc:`rips_complex_ref`                                                                                                                     |
    +----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
diff --git a/src/python/doc/tangential_complex_sum.inc b/src/python/doc/tangential_complex_sum.inc
index 22314a2d..2f330a07 100644
--- a/src/python/doc/tangential_complex_sum.inc
+++ b/src/python/doc/tangential_complex_sum.inc
@@ -3,10 +3,10 @@
 
    +----------------------------------------------------------------+------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
    | .. figure::                                                    | A Tangential Delaunay complex is a simplicial complex designed to      | :Author:    Clément Jamin                                                                                                 |
-   |      ../../doc/Tangential_complex/tc_examples.png              | reconstruct a :math:`k`-dimensional manifold embedded in :math:`d`-    |                                                                                                                           |
-   |      :figclass: align-center                                   | dimensional Euclidean space. The input is a point sample coming from   | :Since:    GUDHI 2.0.0                                                                                                    |
-   |                                                                | an unknown manifold. The running time depends only linearly on the     |                                                                                                                           |
-   |                                                                | extrinsic dimension :math:`d` and exponentially on the intrinsic       | :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
+   |      ../../doc/Tangential_complex/tc_examples.png              | reconstruct a :math:`k`-dimensional manifold embedded in               |                                                                                                                           |
+   |      :figclass: align-center                                   | :math:`d`-dimensional Euclidean space. The input is a point sample     | :Since:    GUDHI 2.0.0                                                                                                    |
+   |                                                                | coming from an unknown manifold. The running time depends only linearly|                                                                                                                           |
+   |                                                                | on the extrinsic dimension :math:`d` and exponentially on the intrinsic| :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
    |                                                                | dimension :math:`k`.                                                   |                                                                                                                           |
    |                                                                |                                                                        | :Requires: `Eigen <installation.html#eigen>`_ :math:`\geq` 3.1.0 and `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0 |
    +----------------------------------------------------------------+------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+