Digital connectedness: An algebraic approach

Authors:
Ludvik Janos;Azriel Rosenfeld
Affiliations:
Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD 20742, U.S.A.;Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD 20742, U.S.A.
Venue:
Pattern Recognition Letters
Year:
1983

Citing 5
Cited 0

Connectivity in Digital Pictures

Journal of the ACM (JACM)
The Boundary Count of Digital Pictures

Journal of the ACM (JACM)
On the Topological Properties of Quantized Spaces, I. The Notion of Dimension

Journal of the ACM (JACM)
Some Results in Computational Topology

Journal of the ACM (JACM)
Local Properties of Binary Images in Two Dimensions

IEEE Transactions on Computers

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Abstract

Let E be the incidence matrix of a graph G having m nodes; then the number of connected components of G is equal to m - r, where r is the rank of E. In particular, if G represents an adjacency relation between points in a digital picture (or higher-dimensional array), this shows that the connected components of points can be counted by computing the rank of E. Two proofs of this result are given, one based on results from algebraic topology and the other based on a self-contained graph-theoretic argument. The former proof can be generalized to yield a method of counting holes.