This library can be used for computing a discrete vector field preserving the multiparameter persistent homology of a multiparameter filtration.
The expected input is an .off file representing a simplicial complex (up to dimension three) with multiple scalar values defined at its vertices.
Example mesh.off (if you use this remove the commented lines form the input file)
OFF #keyword
8 12 0 #number of vertices - number of top simplices
0 1 0 1 2 #0 1 0 are the (x,y,z) coordinates, 1 2 are the vertex scalar values
1 1 -1 4 5
1 1 1 3 5
2 0 0 3 1
2 2 0 2 1
3 1 -1 6 0
3 1 1 9 9
4 1 0 1 2
3 0 1 3 #list of triangles 3 is the number of vertices 0 1 3 are the indexes of the correspinding vertices composing the triangle
3 0 2 3
3 0 1 4
3 0 2 4
3 1 3 5
3 2 3 6
3 1 4 5
3 2 4 6
3 3 5 7
3 3 6 7
3 4 5 7
3 4 6 7If no scalar value is defined of the vertices the program will ask it should use the z coordinates or the x and y coordinates combined. Since the program can be used with an arbitrary number of scalar functions (even 1) it can be used for computing the Forman gradient of a filtration.
The output will be two txt files encoding the boundary matrices of the original simplicial complex and of the computed gradient.
Example boundaryForman.txt
2 #number of filtrations
1 2 3 6 9 #scalar values (grades) for the first filtration
0 1 2 5 9 #scalar values (grades) for the second filtration
0 : 0 2 #0 dimension of the simplex, 0 2 indices of the filtration values in the list of grades
0 : 0 2
0 : 1 1
0 : 2 1
0 : 3 0
1 0 2 : 1 2 #1 dimension of the simplex, 0 2 indices of the boundary simplices, 1 2 indices of the filtration values in the list of grades
1 1 2 : 1 2
1 0 3 : 2 2
1 1 3 : 2 2
1 2 4 : 3 1
1 3 4 : 3 1
2 6 8 9 10 : 3 2
2 5 7 9 10 : 3 3
2 5 6 7 8 : 4 4Boost is the only required library
From the main folder type the following command to compile the library
mkdir build
cd build
cmake ..
makeAt this point, given an input file mesh.off you can run the program as follows
./NM_Forman mesh.off