From 0ef4de65595354c0bad3a9f19d314a5de7d29d92 Mon Sep 17 00:00:00 2001 From: salinasd Date: Wed, 17 Dec 2014 16:37:27 +0000 Subject: merge doc git-svn-id: svn+ssh://scm.gforge.inria.fr/svnroot/gudhi/trunk@377 636b058d-ea47-450e-bf9e-a15bfbe3eedb Former-commit-id: f50f6eda2f484864be2bf27960a58dbbbaba9793 --- src/Skeleton_blocker/include/gudhi/Skeleton_blocker.h | 17 ++++++++--------- 1 file changed, 8 insertions(+), 9 deletions(-) (limited to 'src/Skeleton_blocker/include/gudhi/Skeleton_blocker.h') diff --git a/src/Skeleton_blocker/include/gudhi/Skeleton_blocker.h b/src/Skeleton_blocker/include/gudhi/Skeleton_blocker.h index e67323e5..77f59e35 100644 --- a/src/Skeleton_blocker/include/gudhi/Skeleton_blocker.h +++ b/src/Skeleton_blocker/include/gudhi/Skeleton_blocker.h @@ -41,10 +41,9 @@ namespace skbl{ \author David Salinas \section Introduction -The Skeleton-Blocker data-structure had been introduced in the two papers -\cite socg_blockers_2011,\cite blockers2012. -It proposes a light encoding for simplicial complexes by storing only an *implicit* representation of its -simplices. +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 (see next section for a formal definition). This data-structure handles all simplicial complexes operations such as @@ -60,7 +59,7 @@ An abstract simplex is a finite non-empty set and its dimension is its number of Whenever \f$\tau \subset \sigma\f$ and \f$\tau \neq \emptyset \f$, \f$ \tau \f$ is called a face of \f$ \sigma\f$ and \f$ \sigma\f$ is called a coface of \f$ \tau \f$ . Furthermore, when \f$ \tau \neq \sigma\f$ we say that \f$ \tau\f$ is a proper-face of \f$ \sigma\f$. -An abstract simplicial complex is a set of simplices that contains all the faces of their simplices. +An abstract simplicial complex is a set of simplices that contains all the faces of its simplices. The 1-skeleton of a simplicial complex (or its graph) consists of its elements of dimension lower than 2. *\image html "ds_representation.png" "Skeleton-blocker representation" width=20cm @@ -81,8 +80,8 @@ In practice, the set of blockers of a simplicial complex remains also small when simplifying a Rips complex with edge contractions but also for most of the simplicial complexes used in topological data-analysis such as Delaunay, Cech or Witness complexes. For instance, the numbers of blockers is depicted for random 3-dimensional spheres embedded into \f$R^4\f$ -in next figure. Storing the graph and blockers of such simplicial complex is much compact in this case than storing -its simplices. +in next figure. Storing the graph and blockers of such simplicial complexes is much compact in this case than storing +their simplices. *\image html "blockers_curve.png" "Number of blockers of random triangulations of 3-spheres" width=10cm @@ -94,7 +93,7 @@ its simplices. \subsection Overview -Four classes are implemented for simplicial complex in this representation namely (most user will just need to use Skeleton_blocker_geometric_complex) +Four classes are implemented for simplicial complex in this representation namely (most user will just need to use Skeleton_blocker_geometric_complex): \li Skeleton_blocker_complex : a simplicial complex with basic operations such as vertex/edge/simplex enumeration and construction \li Skeleton_blocker_link_complex : the link of a simplex in a parent complex. It is represented as a sub complex @@ -181,7 +180,7 @@ The Euler Characteristic is 1 \subsection Acknowledgements The author wishes to thank Dominique Attali and André Lieutier for -their collaboration to write the two initial papers about this data-structure +their collaboration to write the two initial papers (cite socg_blockers_2011,\cite blockers2012) about this data-structure and also Dominique for leaving him use a prototype. -- cgit v1.2.3