An approximation to Čech complex using median of triangles for computing Betti numbers of some point cloud data

Abstract

An approach has been developed to create an approximated simplicial complex in between the Vietoris-Rips complex and the Čech complex using median of triangles for computing Betti numbers of some point cloud data. The Vietoris-Rips complex has been built first for this. Then the sample points have been classified into three classes based on three conditions of the median (l) of any triangle’s maximum edge (2r) for any three points in〖 R〗^n. Then the values of the filtration (ε) have been chosen in such a way that ε=r for l<r, ε=r for l=r, and ε=r+((l-r))/3 for l>r. The approach has been extended for higher dimensional triangles calculating l by the distance of the centroid from the opposite vertex of the maximum face and considering r as the filtration value of the maximum edge. Then an algorithm has been introduced to calculate the simplicial complex after building simplices for each filtration value. Finally, to validate the study results of the approximated simplicial complex have been compared with the Vietoris-Rips complex and the Čech complex. The proposed approximated simplicial complex has been found computationally effective than the Čech complex and its filtration values are lying between filtration values of the Vietoris-Rips complex and the Čech complex without any loss of persistent data.