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diff --git a/doc/Witness_complex/Witness_complex_doc.h b/doc/Witness_complex/Witness_complex_doc.h deleted file mode 100644 index 62203054..00000000 --- a/doc/Witness_complex/Witness_complex_doc.h +++ /dev/null @@ -1,122 +0,0 @@ -#ifndef WITNESS_COMPLEX_DOC_H_ -#define WITNESS_COMPLEX_DOC_H_ - -/** - \defgroup witness_complex Witness complex - - \author Siargey Kachanovich - - \image html "Witness_complex_representation.png" "Witness complex representation" - - \section witnessdefinitions Definitions - - 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$L\f$ set of **landmarks**. - - Even though often the set of landmarks \f$L\f$ is a subset of the set of witnesses \f$ W\f$, it is not a requirement for the current implementation. - - Landmarks are the vertices of the simplicial complex - and witnesses help to decide on which simplices are inserted via a predicate "is witnessed". - - De Silva and Carlsson in their paper \cite de2004topological differentiate **weak witnessing** and **strong witnessing**: - - - *weak*: \f$ \sigma \subset L \f$ is witnessed by \f$ w \in W\f$ if \f$ \forall l \in \sigma,\ \forall l' \in \mathbf{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 \mathbf{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 \mathbf{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 \mathbf{L},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2 \f$ - - which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and **strong relaxed witness complex** respectively. - - \image html "swit.svg" "Strongly witnessed simplex" - - In particular case of 0-relaxation, weak complex corresponds to **witness complex** introduced in \cite de2004topological, whereas 0-relaxed strong witness complex consists of just vertices and is not very interesting. - Hence for small relaxation weak version is preferable. - However, to capture the homotopy type (for example using Gudhi::persistent_cohomology::Persistent_cohomology) it is often necessary to work with higher filtration values. In this case strong relaxed witness complex is faster to compute and offers similar results. - - \section witnessimplementation Implementation - - The two complexes described above are implemented in the corresponding classes - - Gudhi::witness_complex::Witness_complex - - Gudhi::witness_complex::Euclidean_witness_complex - - Gudhi::witness_complex::Strong_witness_complex - - Gudhi::witness_complex::Euclidean_strong_witness_complex - - The construction of the Euclidean versions of complexes 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 structure Gudhi::witness_complex::Active_witness is used internally). - 3. Construct the witness complex for nearest landmark lists. - - In the non-Euclidean classes, the lists of nearest landmarks are supposed to be given as input. - - The constructors take on the steps 1 and 2, while the function 'create_complex' executes the step 3. - - \section witnessexample1 Example 1: Constructing weak relaxed witness complex from an off file - - Let's start with a simple example, which reads an off point file and computes a weak witness complex. - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{.cpp} - -#include <gudhi/Simplex_tree.h> -#include <gudhi/Euclidean_witness_complex.h> -#include <gudhi/pick_n_random_points.h> -#include <gudhi/Points_off_io.h> - -#include <CGAL/Epick_d.h> - -#include <string> -#include <vector> - -typedef CGAL::Epick_d<CGAL::Dynamic_dimension_tag> K; -typedef typename K::Point_d Point_d; -typedef typename Gudhi::witness_complex::Euclidean_witness_complex<K> Witness_complex; -typedef std::vector< Vertex_handle > typeVectorVertex; -typedef std::vector< Point_d > Point_vector; - -int main(int argc, char * const argv[]) { - std::string file_name = argv[1]; - int nbL = atoi(argv[2]), lim_dim = atoi(argv[4]); - double alpha2 = atof(argv[3]); - Gudhi::Simplex_tree<> simplex_tree; - - // Read the point file - Point_vector point_vector, landmarks; - Gudhi::Points_off_reader<Point_d> off_reader(file_name); - point_vector = Point_vector(off_reader.get_point_cloud()); - - // Choose landmarks (one can choose either of the two methods below) - // Gudhi::subsampling::pick_n_random_points(point_vector, nbL, std::back_inserter(landmarks)); - Gudhi::subsampling::choose_n_farthest_points(K(), point_vector, nbL, Gudhi::subsampling::random_starting_point, std::back_inserter(landmarks)); - - // Compute witness complex - Witness_complex witness_complex(landmarks, - point_vector); - - witness_complex.create_complex(simplex_tree, alpha2, lim_dim); -} - - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - \section witnessexample2 Example2: Computing persistence using strong relaxed witness complex - - Here is an example of constructing a strong witness complex filtration and computing persistence on it: - - \include Witness_complex/strong_witness_persistence.cpp - - \section witnessexample3 Example3: Computing relaxed witness complex persistence from a distance matrix - - In this example we compute the relaxed witness complex persistence from a given matrix of closest landmarks to each witness. - Each landmark is given as the couple (index, distance). - - \include Witness_complex/example_nearest_landmark_table.cpp - - */ - -#endif // WITNESS_COMPLEX_DOC_H_ |