Digital topological method for computing genus and the Betti numbers

Abstract

This paper concerns with computation of topological invariants such as genus and the Betti numbers. We design a linear time algorithm that determines such invariants for digital spaces in 3D. These computations could have applications in medical imaging as they can be used to identify patterns in 3D image. Our method is based on cubical images with direct adjacency, also called ( 6 , 26 ) -connectivity images in discrete geometry. There are only six types of local surface points in such a digital surface. Two mathematical ingredients are used. First, we use the Gauss–Bonnett Theorem in differential geometry to determine the genus of 2-dimensional digital surfaces. This is done by counting the contribution for each of the six types of local surface points. The new formula derived in this paper that calculates genus is g = 1 + ( | M 5 | + 2 ⋅ | M 6 | − | M 3 | ) / 8 where M i indicates the set of surface-points each of which has i adjacent points on the surface. Second, we apply the Alexander duality to express the homology groups of a 3D manifold in the usual 3D space in terms of the homology groups of its boundary surface. While our result is stated for digital spaces, the same idea can be applied to simplicial complexes in 3D or more general cell complexes.