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diff --git a/src/Witness_complex/doc/Witness_complex_doc.h b/src/Witness_complex/doc/Witness_complex_doc.h index 60dfd27b..171a185f 100644 --- a/src/Witness_complex/doc/Witness_complex_doc.h +++ b/src/Witness_complex/doc/Witness_complex_doc.h @@ -8,31 +8,106 @@ \image html "Witness_complex_representation.png" "Witness complex representation" - \section Definitions + \section witnessdefinitions Definitions - Witness complex \f$ Wit(W,L) \f$ is a simplicial complex defined on two sets of points in \f$\mathbb{R}^D\f$: + 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$L \subseteq W\f$ set of **landmarks**. + \li \f$L\f$ set of **landmarks**. - The simplices are based on landmarks - and a simplex belongs to the witness complex if and only if it is witnessed, that is: + Even though often the set of landmarks \f$L\f$ is a subset of the set of witnesses \f$ W\f$, it is not a requirement for the current implementation. - \f$ \sigma \subset L \f$ is witnessed if there exists a point \f$w \in W\f$ such that - w is closer to the vertices of \f$ \sigma \f$ than other points in \f$ L \f$ and all of its faces are witnessed as well. - - The data structure is described in \cite boissonnatmariasimplextreealgorithmica . + Landmarks are the vertices of the simplicial complex + and witnesses help to decide on which simplices are inserted via a predicate "is witnessed". - \section Implementation + De Silva and Carlsson in their paper \cite de2004topological differentiate **weak witnessing** and **strong witnessing**: - The principal class of this module is Gudhi::Witness_complex. + - *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$ - In both cases, the constructor for this class takes a {witness}x{closest_landmarks} table, where each row represents a witness and consists of landmarks sorted by distance to this witness. - This table can be constructed by two additional classes Landmark_choice_by_furthest_point and Landmark_choice_by_random_point also included in the module. + where \f$ d(.,.) \f$ is a distance function. - *\image html "bench_Cy8.png" "Running time as function on number of landmarks" width=10cm - *\image html "bench_sphere.png" "Running time as function on number of witnesses for |L|=300" width=10cm + 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 + Gudhi::subsampling::pick_n_random_points(point_vector, nbL, 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/example_strong_witness_persistence.cpp \copyright GNU General Public License v3. |