Edge contraction

Classes

class  Gudhi::contraction::Cost_policy< EdgeProfile >
 Policy to specify the cost of contracting an edge. More...
 
class  Gudhi::contraction::Link_condition_valid_contraction< EdgeProfile >
 Policy that only accept edges verifying the link condition (and therefore whose contraction preserving homotopy type). More...
 
class  Gudhi::contraction::Placement_policy< EdgeProfile >
 Policy to specify where the merged point had to be placed after an edge contraction. More...
 
class  Gudhi::contraction::Valid_contraction_policy< EdgeProfile >
 Policy to specify if an edge contraction is valid or not. More...
 
class  Gudhi::contraction::Contraction_visitor< EdgeProfile >
 Interface for a visitor of the edge contraction process. More...
 
class  Gudhi::contraction::Skeleton_blocker_contractor< GeometricSimplifiableComplex, EdgeProfile >
 Class that allows to contract iteratively edges of a simplicial complex. More...
 

Detailed Description

Author
David Salinas

Introduction

The purpose of this package is to offer a user-friendly interface for edge contraction simplification of huge simplicial complexes. It uses the Skeleton-Blocker 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.

The edge contraction operation consists in identifying two vertices of a simplicial complex. A lot of algorithms have been developed in computer graphics that allows to reduce efficiently the size of 2-dimensional simplicial complexes while preserving its geometry (for instance see [12] [15]). These approaches can be extended to higher-dimensional simplicial complexes. The main advantage of using the Skeleton-Blocker data structure for edge contraction is that when the number of blockers is small, the operations needed for edge contraction algorithms have polynomial complexity regarding the size the graph. Therefore, the simplification can be done without enumerating the set of simplices that is often non tracktable in high-dimension and is then very efficient (sub-linear with regards to the number of simplices in practice).

A typical application of this package is homology group computation. It is illustrated in the next figure where a Rips complex is built uppon a set of high-dimensional points and simplified with edge contractions. It has initially a big number of simplices (around 20 millions) but simplifying it to a much reduced form with only 15 vertices (and 714 simplices) takes only few seconds on a desktop machine (see the example bellow). One can then compute homology group with a simplicial complex having very few simplices instead of running the homology algorithm on the much bigger initial set of simplices which would take much more time and memory.

so3.png
Edge contraction illustration

Design

This class design is policy based and heavily inspired from the similar edge collapse package of CGAL http://doc.cgal.org/latest/Surface_mesh_simplification/index.html (which is however restricted to 2D triangulations).

Policies

Four policies can be customized in this package:

Visitor

A visitor which implements the class Contraction_visitor gets called at several key moments during the simplification:

This allows to implements most of edge contraction based algorithm with this package without having to change the main simplification source code.

Performance

The next figure shows the computation time to reduce a random 2-sphere to a single tetrahedron with this package and with the CGAL equivalent package. Despite this package is able to deal with arbitrary simplicial complexes (any dimensions, manifold or non manifold), it is still 65% times faster than the CGAL package which is focused on 2-manifold. The main reason is that few blockers appears during the simplification and hence, the algorithm only have to deal with the graph and not higher-dimensional simplices (in this case triangles). However, we recall that higher-dimensional simplices are implicitely stored in the Skeleton-Blocker data-structure. Hence, one has to store simplices in an external map if some information needs to be associated with them (information that could be a filtration value or an orientation for instance).

sphere_contraction.png
Time in seconds to simplify random 2-spheres to a tetrahedron

Example

This example loads points from an OFF file and builds the Rips complex with an user provided parameter. It then simplifies the built Rips complex while ensuring its homotopy type is preserved during the contraction (edge are contracted only when the link condition is valid).

#include <boost/timer/timer.hpp>
#include <iostream>
#include <gudhi/Edge_contraction.h>
#include <gudhi/Skeleton_blocker.h>
#include <gudhi/Off_reader.h>
using namespace std;
using namespace Gudhi;
using namespace skeleton_blocker;
using namespace contraction;
struct Geometry_trait{
typedef std::vector<double> Point;
};
typedef Geometry_trait::Point Point;
typedef Skeleton_blocker_simple_geometric_traits<Geometry_trait> Complex_geometric_traits;
typedef Skeleton_blocker_geometric_complex< Complex_geometric_traits > Complex;
typedef Edge_profile<Complex> Profile;
typedef Skeleton_blocker_contractor<Complex> Complex_contractor;
template<typename Point>
double eucl_distance(const Point& a,const Point& b){
double res = 0;
auto a_coord = a.begin();
auto b_coord = b.begin();
for(; a_coord != a.end(); a_coord++, b_coord++){
res += (*a_coord - *b_coord) * (*a_coord - *b_coord);
}
return sqrt(res);
}
template<typename ComplexType>
void build_rips(ComplexType& complex, double offset){
if (offset<=0) return;
auto vertices = complex.vertex_range();
for (auto p = vertices.begin(); p != vertices.end(); ++p)
for (auto q = p; ++q != vertices.end(); )
if (eucl_distance(complex.point(*p), complex.point(*q)) < 2 * offset)
complex.add_edge_without_blockers(*p,*q);
}
int main (int argc, char *argv[])
{
if (argc!=3){
std::cerr << "Usage "<<argv[0]<<" ../../data/SO3_10000.off 0.3 to load the file ../../data/SO3_10000.off and contract the Rips complex built with paremeter 0.3.\n";
return -1;
}
Complex complex;
// load the points
Skeleton_blocker_off_reader<Complex> off_reader(argv[1],complex,true);
if(!off_reader.is_valid()){
std::cerr << "Unable to read file:"<<argv[1]<<std::endl;
return EXIT_FAILURE;
}
std::cout << "Build the Rips complex"<<std::endl;
build_rips(complex,atof(argv[2]));
boost::timer::auto_cpu_timer t;
std::cout << "Initial complex has "<<
complex.num_vertices()<<" vertices and "<<
complex.num_edges()<<" edges"<<std::endl;
Complex_contractor contractor(complex,
new Edge_length_cost<Profile>,
contraction::make_first_vertex_placement<Profile>(),
contraction::make_link_valid_contraction<Profile>(),
contraction::make_remove_popable_blockers_visitor<Profile>());
contractor.contract_edges();
std::cout << "Counting final number of simplices \n";
unsigned num_simplices = std::distance(complex.star_simplex_range().begin(),complex.star_simplex_range().end());
std::cout << "Final complex has "<<
complex.num_vertices()<<" vertices, "<<
complex.num_edges()<<" edges, "<<
complex.num_blockers()<<" blockers and "<<
num_simplices<<" simplices"<<std::endl;
std::cout << "Time to simplify and enumerate simplices:\n";
return EXIT_SUCCESS;
}
}
./example/Contraction/RipsContraction ../../data/SO3_10000.off 0.3
[ 50%] [100%] Built target SkeletonBlockerIteration
Built target RipsContraction
Build the Rips complex
Initial complex has 10000 vertices and 195664 edges
Counting final number of simplices 
Final complex has 15 vertices, 101 edges, 165 blockers and 714 simplices
Time to simplify and enumerate simplices:
 3.166621s wall, 3.150000s user + 0.010000s system = 3.160000s CPU (99.8%)
GUDHI  Version 2.0.0  - C++ library for Topological Data Analysis (TDA) and Higher Dimensional Geometry Understanding. Generated on Wed Apr 19 2017 22:26:16 for GUDHI by doxygen 1.8.11