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diff --git a/src/common/doc/main_page.md b/src/common/doc/main_page.md new file mode 100644 index 00000000..d8cbf97f --- /dev/null +++ b/src/common/doc/main_page.md @@ -0,0 +1,391 @@ +[TOC] + +# The C++ library {#main_page} +\image html "Gudhi_banner.png" +<br><br><br><br> + +## Complexes {#Complexes} +### Cubical complex + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "Cubical_complex_representation.png" + </td> + <td width="50%"> + The cubical complex is an example of a structured complex useful in computational mathematics (specially + rigorous numerics) and image analysis.<br> + </td> + <td width="15%"> + <b>Author:</b> Pawel Dlotko<br> + <b>Introduced in:</b> GUDHI 1.3.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref cubical_complex + </td> + </tr> +</table> + +### Simplicial complex + +#### Alpha complex + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "alpha_complex_representation.png" + </td> + <td width="50%"> + Alpha complex is a simplicial complex constructed from the finite cells of a Delaunay Triangulation.<br> + The filtration value of each simplex is computed as the square of the circumradius of the simplex if the + circumsphere is empty (the simplex is then said to be Gabriel), and as the minimum of the filtration + values of the codimension 1 cofaces that make it not Gabriel otherwise. + All simplices that have a filtration value strictly greater than a given alpha squared value are not inserted into + the complex.<br> + </td> + <td width="15%"> + <b>Author:</b> Vincent Rouvreau<br> + <b>Introduced in:</b> GUDHI 1.3.0<br> + <b>Copyright:</b> MIT [(GPL v3)](../../licensing/)<br> + <b>Requires:</b> \ref eigen ≥ 3.1.0 and \ref cgal ≥ 4.11.0 + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref alpha_complex + </td> + </tr> +</table> + +#### Čech complex + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "cech_complex_representation.png" + </td> + <td width="50%"> + The Čech complex is a simplicial complex constructed from a proximity graph. + The set of all simplices is filtered by the radius of their minimal enclosing ball. + </td> + <td width="15%"> + <b>Author:</b> Vincent Rouvreau<br> + <b>Introduced in:</b> GUDHI 2.2.0<br> + <b>Copyright:</b> MIT [(GPL v3)](../../licensing/)<br> + <b>Includes:</b> [Miniball](https://people.inf.ethz.ch/gaertner/subdir/software/miniball.html)<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref cech_complex + </td> + </tr> +</table> + +#### Rips complex + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "rips_complex_representation.png" + </td> + <td width="50%"> + Rips complex is a simplicial complex constructed from a one skeleton graph.<br> + The filtration value of each edge is computed from a user-given distance function and is inserted until a + user-given threshold value.<br> + This complex can be built from a point cloud and a distance function, or from a distance matrix. + </td> + <td width="15%"> + <b>Author:</b> Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse<br> + <b>Introduced in:</b> GUDHI 2.0.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref rips_complex + </td> + </tr> +</table> + +#### Witness complex + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "Witness_complex_representation.png" + </td> + <td width="50%"> + Witness complex \f$ Wit(W,L) \f$ is a simplicial complex defined on two sets of points in \f$\mathbb{R}^D\f$. + The data structure is described in \cite boissonnatmariasimplextreealgorithmica . + </td> + <td width="15%"> + <b>Author:</b> Siargey Kachanovich<br> + <b>Introduced in:</b> GUDHI 1.3.0<br> + <b>Copyright:</b> MIT ([GPL v3](../../licensing/) for Euclidean version)<br> + <b>Euclidean version requires:</b> \ref eigen ≥ 3.1.0 and \ref cgal ≥ 4.11.0 + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref witness_complex + </td> + </tr> +</table> + +### Cover Complexes +<table> + <tr> + <td width="35%" rowspan=2> + \image html "gicvisu.jpg" + </td> + <td width="50%"> + Nerves and Graph Induced Complexes are cover complexes, i.e. simplicial complexes that provably contain + topological information about the input data. They can be computed with a cover of the + data, that comes i.e. from the preimage of a family of intervals covering the image + of a scalar-valued function defined on the data. <br> + </td> + <td width="15%"> + <b>Author:</b> Mathieu Carrière<br> + <b>Introduced in:</b> GUDHI 2.1.0<br> + <b>Copyright:</b> MIT [(GPL v3)](../../licensing/)<br> + <b>Requires:</b> \ref cgal ≥ 4.11.0 + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref cover_complex + </td> + </tr> +</table> + +## Data structures and basic operations {#DataStructuresAndBasicOperations} + +### Data structures + +#### Simplex tree +<table> + <tr> + <td width="35%" rowspan=2> + \image html "Simplex_tree_representation.png" + </td> + <td width="50%"> + The simplex tree is an efficient and flexible + data structure for representing general (filtered) simplicial complexes. The data structure + is described in \cite boissonnatmariasimplextreealgorithmica . + </td> + <td width="15%"> + <b>Author:</b> Clément Maria<br> + <b>Introduced in:</b> GUDHI 1.0.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref simplex_tree + </td> + </tr> +</table> + +#### Skeleton blocker + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "ds_representation.png" + </td> + <td width="50%"> + The Skeleton-Blocker data-structure proposes a light encoding for simplicial complexes by storing only an *implicit* + representation of its simplices \cite socg_blockers_2011,\cite blockers2012. Intuitively, it just stores the + 1-skeleton of a simplicial complex with a graph and the set of its "missing faces" that is very small in practice. + This data-structure handles all simplicial complexes operations such as simplex enumeration or simplex removal but + operations that are particularly efficient are operations that do not require simplex enumeration such as edge + iteration, link computation or simplex contraction. + </td> + <td width="15%"> + <b>Author:</b> David Salinas<br> + <b>Introduced in:</b> GUDHI 1.1.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref skbl + </td> + </tr> +</table> + +#### Toplex Map + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "map.png" + </td> + <td width="50%"> + The Toplex map data structure is composed firstly of a raw storage of toplices (the maximal simplices) + and secondly of a map which associate any vertex to a set of pointers toward all toplices + containing this vertex. + </td> + <td width="15%"> + <b>Author:</b> François Godi<br> + <b>Introduced in:</b> GUDHI 2.1.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref toplex_map + </td> + </tr> +</table> + +### Basic operations + +#### Contraction + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "sphere_contraction_representation.png" + </td> + <td width="50%"> + The purpose of this package is to offer a user-friendly interface for edge contraction simplification of huge + simplicial complexes. It uses the \ref skbl data-structure whose size remains small during simplification of most + used geometrical complexes of topological data analysis such as the Rips or the Delaunay complexes. In practice, + the size of this data-structure is even much lower than the total number of simplices. + </td> + <td width="15%"> + <b>Author:</b> David Salinas<br> + <b>Introduced in:</b> GUDHI 1.1.0<br> + <b>Copyright:</b> MIT [(LGPL v3)](../../licensing/)<br> + <b>Requires:</b> \ref cgal ≥ 4.11.0 + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref contr + </td> + </tr> +</table> + +## Topological descriptors computation {#TopologicalDescriptorsComputation} + +### Persistent Cohomology + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "3DTorus_poch.png" + </td> + <td width="50%"> + 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. + 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 . + </td> + <td width="15%"> + <b>Author:</b> Clément Maria<br> + <b>Introduced in:</b> GUDHI 1.0.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref persistent_cohomology + </td> + </tr> +</table> + +## Manifold reconstruction {#ManifoldReconstruction} + +### Tangential complex + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "tc_examples.png" + </td> + <td width="50%"> + A Tangential Delaunay complex is a <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a> + designed to reconstruct a \f$ k \f$-dimensional manifold embedded in \f$ d \f$-dimensional Euclidean space. + The input is a point sample coming from an unknown manifold. + The running time depends only linearly on the extrinsic dimension \f$ d \f$ + and exponentially on the intrinsic dimension \f$ k \f$. + </td> + <td width="15%"> + <b>Author:</b> Clément Jamin<br> + <b>Introduced in:</b> GUDHI 2.0.0<br> + <b>Copyright:</b> MIT [(GPL v3)](../../licensing/)<br> + <b>Requires:</b> \ref eigen ≥ 3.1.0 and \ref cgal ≥ 4.11.0 + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref tangential_complex + </td> + </tr> +</table> + +## Topological descriptors tools {#TopologicalDescriptorsTools} + +### Bottleneck distance + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "perturb_pd.png" + </td> + <td width="50%"> + Bottleneck distance measures the similarity between two persistence diagrams. + It's the shortest distance b for which there exists a perfect matching between + the points of the two diagrams (+ all the diagonal points) such that + any couple of matched points are at distance at most b, + where the distance between points is the sup norm in \f$\mathbb{R}^2\f$ + (not the Euclidean distance). + </td> + <td width="15%"> + <b>Author:</b> François Godi<br> + <b>Introduced in:</b> GUDHI 2.0.0<br> + <b>Copyright:</b> MIT [(GPL v3)](../../licensing/)<br> + <b>Requires:</b> \ref cgal ≥ 4.11.0 + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref bottleneck_distance + </td> + </tr> +</table> + +### Persistence representations + +<table> + <tr> + <td width="35%" rowspan=2> + \image html "average_landscape.png" + </td> + <td width="50%"> + It contains implementation of various representations of persistence diagrams; diagrams themselves, persistence + landscapes (rigorous and grid version), persistence heat maps, vectors and others. It implements basic + functionalities which are necessary to use persistence in statistics and machine learning. + </td> + <td width="15%"> + <b>Author:</b> Pawel Dlotko<br> + <b>Introduced in:</b> GUDHI 2.1.0<br> + <b>Copyright:</b> MIT<br> + </td> + </tr> + <tr> + <td colspan=2 height="25"> + <b>User manual:</b> \ref Persistence_representations + </td> + </tr> +</table> |