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Author: Vincent Rouvreau Introduced in: GUDHI 1.3.0 Copyright: GPL v3 Requires: \ref cgal ≥ 4.7.0 and \ref eigen3 |
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 |
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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 |
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Author: Clément Maria, Pawel Dlotko, Vincent Rouvreau Introduced in: GUDHI 2.0.0 Copyright: GPL v3 |
Rips_complex is a simplicial complex constructed from a one skeleton graph. The filtration value of each edge is computed from a user-given distance function and is inserted until a user-given threshold value. This complex can be built from a point cloud and a distance function, or from a distance matrix. User manual: \ref rips_complex - Reference manual: Gudhi::rips_complex::Rips_complex |
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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 |
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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 |
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Author: Clément Jamin Introduced in: GUDHI 2.0.0 Copyright: GPL v3 Requires: \ref cgal ≥ 4.8.1 and \ref eigen3 |
A Tangential Delaunay complex is a simplicial complex
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$. User manual: \ref tangential_complex - Reference manual: Gudhi::tangential_complex::Tangential_complex |
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Author: Siargey Kachanovich Introduced in: GUDHI 1.3.0 Copyright: GPL v3 Euclidean version requires: \ref cgal ≥ 4.6.0 and \ref eigen3 |
Witness complex \f$ 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 |
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Author: François Godi Introduced in: GUDHI 2.0.0 Copyright: GPL v3 Requires: \ref cgal ≥ 4.8.1 and \ref eigen3 |
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.
User manual: \ref bottleneck_distance |
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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 |
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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 |
cmake . && make CGAL (or even
* cmake -DCGAL_HEADER_ONLY=ON . for CGAL version ≥ 4.8.0), thereafter you will be able to compile
* GUDHI by calling cmake -DCGAL_DIR=/your/path/to/CGAL-X.Y .. && make
*
* Having CGAL version 4.4.0 or higher installed is recommended. The procedure to install this library according to
* your operating system is detailed here http://doc.cgal.org/latest/Manual/installation.html
*
* The following examples require the Computational Geometry Algorithms
* Library (CGAL \cite cgal:eb-15b) and will not be built if CGAL is not installed:
* \li
* Persistent_cohomology/alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/exact_alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/weighted_alpha_complex_3d_persistence.cpp
* \li
* Simplex_tree/example_alpha_shapes_3_simplex_tree_from_off_file.cpp
*
* The following example requires CGAL version ≥ 4.6.0:
* \li
* Witness_complex/witness_complex_sphere.cpp
*
* The following example requires CGAL version ≥ 4.7.0:
* \li
* Alpha_complex/Alpha_complex_from_off.cpp
* \li
* Alpha_complex/Alpha_complex_from_points.cpp
* \li
* Persistent_cohomology/alpha_complex_persistence.cpp
* \li
* Persistent_cohomology/periodic_alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/custom_persistence_sort.cpp
*
* The following example requires CGAL version ≥ 4.8.1:
* \li
* Bottleneck_distance/alpha_rips_persistence_bottleneck_distance.cpp.cpp
* \li
* Bottleneck_distance/bottleneck_basic_example.cpp
* \li
* Bottleneck_distance/bottleneck_read_file_example.cpp
* \li
* Spatial_searching/example_spatial_searching.cpp
* \li
* Subsampling/example_choose_n_farthest_points.cpp
* \li
* Subsampling/example_custom_kernel.cpp
* \li
* Subsampling/example_pick_n_random_points.cpp
* \li
* Subsampling/example_sparsify_point_set.cpp
* \li
* Tangential_complex/example_basic.cpp
* \li
* Tangential_complex/example_with_perturb.cpp
*
* \subsection eigen3 Eigen3
* The \ref alpha_complex data structure and few examples requires
* Eigen3 is a C++ template library for linear algebra:
* matrices, vectors, numerical solvers, and related algorithms.
*
* The following example requires the Eigen3 and will not be
* built if Eigen3 is not installed:
* \li
* Alpha_complex/Alpha_complex_from_off.cpp
* \li
* Alpha_complex/Alpha_complex_from_points.cpp
* \li
* Persistent_cohomology/alpha_complex_persistence.cpp
* \li
* Persistent_cohomology/periodic_alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/custom_persistence_sort.cpp
*
* \subsection tbb Threading Building Blocks
* Intel® TBB lets you easily write parallel
* C++ programs that take full advantage of multicore performance, that are portable and composable, and that have
* future-proof scalability.
*
* Having Intel® TBB installed is recommended to parallelize and accelerate some GUDHI computations.
*
* The following examples are using Intel® TBB if installed:
* \li
* Alpha_complex/Alpha_complex_from_off.cpp
* \li
* Alpha_complex/Alpha_complex_from_points.cpp
* \li
* Bitmap_cubical_complex/Bitmap_cubical_complex.cpp
* \li
* Bitmap_cubical_complex/Bitmap_cubical_complex_periodic_boundary_conditions.cpp
* \li
* Bitmap_cubical_complex/Random_bitmap_cubical_complex.cpp
* \li
* Persistent_cohomology/alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/alpha_complex_persistence.cpp
* \li
* Simplex_tree/simple_simplex_tree.cpp
* \li
* Simplex_tree/example_alpha_shapes_3_simplex_tree_from_off_file.cpp
* \li
* Simplex_tree/simplex_tree_from_cliques_of_graph.cpp
* \li
* Persistent_cohomology/alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/alpha_complex_persistence.cpp
* \li
* Persistent_cohomology/rips_persistence_via_boundary_matrix.cpp
* \li
* Persistent_cohomology/persistence_from_file.cpp
* \li
* Persistent_cohomology/persistence_from_simple_simplex_tree.cpp
* \li
* Persistent_cohomology/plain_homology.cpp
* \li
* Persistent_cohomology/rips_distance_matrix_persistence.cpp
* \li
* Persistent_cohomology/rips_multifield_persistence.cpp
* \li
* Persistent_cohomology/rips_persistence.cpp
* \li
* Persistent_cohomology/rips_persistence_step_by_step.cpp
* \li
* Persistent_cohomology/exact_alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/weighted_alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/periodic_alpha_complex_3d_persistence.cpp
* \li
* Persistent_cohomology/custom_persistence_sort.cpp
* \li
* Rips_complex/example_one_skeleton_rips_from_points.cpp
* \li
* Rips_complex/example_rips_complex_from_off_file.cpp
*
* \section Contributions Bug reports and contributions
* Please help us improving the quality of the GUDHI library. You may report bugs or suggestions to:
* \verbatim Contact: gudhi-users@lists.gforge.inria.fr \endverbatim
*
* GUDHI is open to external contributions. If you want to join our development team, please contact us.
*
*/
/*! \page Citation Acknowledging the GUDHI library
* We kindly ask users to cite the GUDHI library as appropriately as possible in their papers, and to mention the use
* of the GUDHI library on the web pages of their projects using GUDHI and provide us with links to these web pages.
* Feel free to contact us in case you have any question or remark on this topic.
*
* We provide \ref GudhiBibtex entries for the modules of the User and Reference Manual, as well as for publications
* directly related to the GUDHI library.
* \section GudhiBibtex GUDHI bibtex
* \verbinclude biblio/how_to_cite_gudhi.bib
*/
// List of GUDHI examples - Doxygen needs at least a file tag to analyse comments
/*! @file Examples
* @example Alpha_complex/Alpha_complex_from_off.cpp
* @example Alpha_complex/Alpha_complex_from_points.cpp
* @example Bottleneck_distance/alpha_rips_persistence_bottleneck_distance.cpp
* @example Bottleneck_distance/bottleneck_basic_example.cpp
* @example Bottleneck_distance/bottleneck_read_file_example.cpp
* @example Bitmap_cubical_complex/Bitmap_cubical_complex.cpp
* @example Bitmap_cubical_complex/Bitmap_cubical_complex_periodic_boundary_conditions.cpp
* @example Bitmap_cubical_complex/Random_bitmap_cubical_complex.cpp
* @example common/example_CGAL_3D_points_off_reader.cpp
* @example common/example_CGAL_points_off_reader.cpp
* @example Contraction/Garland_heckbert.cpp
* @example Contraction/Rips_contraction.cpp
* @example Persistent_cohomology/alpha_complex_3d_persistence.cpp
* @example Persistent_cohomology/alpha_complex_persistence.cpp
* @example Persistent_cohomology/rips_persistence_via_boundary_matrix.cpp
* @example Persistent_cohomology/exact_alpha_complex_3d_persistence.cpp
* @example Persistent_cohomology/weighted_alpha_complex_3d_persistence.cpp
* @example Persistent_cohomology/periodic_alpha_complex_3d_persistence.cpp
* @example Persistent_cohomology/persistence_from_file.cpp
* @example Persistent_cohomology/persistence_from_simple_simplex_tree.cpp
* @example Persistent_cohomology/plain_homology.cpp
* @example Persistent_cohomology/rips_multifield_persistence.cpp
* @example Persistent_cohomology/rips_distance_matrix_persistence.cpp
* @example Persistent_cohomology/rips_persistence.cpp
* @example Persistent_cohomology/custom_persistence_sort.cpp
* @example Persistent_cohomology/rips_persistence_step_by_step.cpp
* @example Rips_complex/example_one_skeleton_rips_from_points.cpp
* @example Rips_complex/example_rips_complex_from_off_file.cpp
* @example Simplex_tree/mini_simplex_tree.cpp
* @example Simplex_tree/simple_simplex_tree.cpp
* @example Simplex_tree/example_alpha_shapes_3_simplex_tree_from_off_file.cpp
* @example Simplex_tree/simplex_tree_from_cliques_of_graph.cpp
* @example Skeleton_blocker/Skeleton_blocker_from_simplices.cpp
* @example Skeleton_blocker/Skeleton_blocker_iteration.cpp
* @example Skeleton_blocker/Skeleton_blocker_link.cpp
* @example Spatial_searching/example_spatial_searching.cpp
* @example Subsampling/example_choose_n_farthest_points.cpp
* @example Subsampling/example_custom_kernel.cpp
* @example Subsampling/example_pick_n_random_points.cpp
* @example Subsampling/example_sparsify_point_set.cpp
* @example Tangential_complex/example_basic.cpp
* @example Tangential_complex/example_with_perturb.cpp
* @example Witness_complex/example_nearest_landmark_table.cpp
* @example Witness_complex/example_strong_witness_complex_off.cpp
* @example Witness_complex/example_strong_witness_persistence.cpp
* @example Witness_complex/example_witness_complex_off.cpp
* @example Witness_complex/example_witness_complex_persistence.cpp
* @example Witness_complex/example_witness_complex_sphere.cpp
*/