/*! \mainpage * \tableofcontents * \image html "Gudhi_banner.png" "" width=20cm * * \section Introduction Introduction * The Gudhi library (Geometry Understanding in Higher Dimensions) is a generic open source C++ library for * Computational Topology and Topological Data Analysis * (TDA). * The GUDHI library intends to help the development of new algorithmic solutions in TDA and their transfer to * applications. It provides robust, efficient, flexible and easy to use implementations of state-of-the-art * algorithms and data structures. * * The current release of the GUDHI library includes: * * \li Data structures to represent, construct and manipulate simplicial complexes. * \li Algorithms to compute persistent homology and multi-field persistent homology. * \li Simplication of simplicial complexes by edge contraction. * * All data-structures are generic and several of their aspects can be parameterized via template classes. * We refer to \cite gudhilibrary_ICMS14 for a detailed description of the design of the library. * \section DataStructures Data structures \subsection AlphaComplexDataStructure Alpha complex \image html "alpha_complex_representation.png" "Alpha complex representation"
Author: Vincent Rouvreau Introduced in: GUDHI 1.3.0 Copyright: GPL v3 |
Alpha_complex is a simplicial complex constructed from the finite cells of a Delaunay Triangulation. 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. User manual: \ref alpha_complex - Reference manual: Gudhi::alpha_complex::Alpha_complex |
Author: Pawel Dlotko Introduced in: GUDHI 1.3.0 Copyright: GPL v3 |
The cubical complex is an example of a structured complex useful in computational mathematics (specially
rigorous numerics) and image analysis. User manual: \ref cubical_complex - Reference manual: Gudhi::cubical_complex::Bitmap_cubical_complex |
Author: Clément Maria Introduced in: GUDHI 1.0.0 Copyright: GPL v3 |
The simplex tree is an efficient and flexible
data structure for representing general (filtered) simplicial complexes. The data structure
is described in \cite boissonnatmariasimplextreealgorithmica . User manual: \ref simplex_tree - Reference manual: Gudhi::Simplex_tree |
Author: David Salinas Introduced in: GUDHI 1.1.0 Copyright: GPL v3 |
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. User manual: \ref skbl - Reference manual: Gudhi::skeleton_blocker::Skeleton_blocker_complex |
Author: Siargey Kachanovich Introduced in: GUDHI 1.3.0 Copyright: GPL v3 |
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 . User manual: \ref witness_complex - Reference manual: Gudhi::witness_complex::SimplicialComplexForWitness |
Author: David Salinas Introduced in: GUDHI 1.1.0 Copyright: GPL v3 |
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. User manual: \ref contr |
Author: Clément Maria Introduced in: GUDHI 1.0.0 Copyright: GPL v3 |
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 . User manual: \ref persistent_cohomology - Reference manual: Gudhi::persistent_cohomology::Persistent_cohomology |