Connected Components Matrix

For example there are 3 sccs in the following graph.

Connected components matrix.

First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph s laplacian matrix is k. A mark is initialized and incremented for every detected object in the image. Implementation of connected components in three dimensions using a 26 18 or 6 connected neighborhood in 3d or 4 and 8 connected in 2d. A strongly connected component scc of a directed graph is a maximal strongly connected subgraph.

Tarjan s algorithm to find strongly connected components finding connected components for an undirected graph is an easier task. This package uses a 3d variant of the two pass method by rosenfeld and pflatz augmented with union find and a decision tree based on the 2d 8 connected work of wu otoo and suzuki. Connected components of an image or matrix image components is a fortran77 library which seeks to count and label the connected nonzero nonblack components in an image or integer vector matrix or 3d block. A counter is initialized to count the number of objects.

For a graph with multiple connected components l is a block diagonal matrix where each block is the respective laplacian matrix for each component possibly after reordering the vertices i e. As discussed in section 3 the ranking algorithm requires a suitable connectivity matrix to be identified. Connected component matrix is initialized to size of image matrix. L is permutation similar to a block diagonal matrix.

For example there are 3 sccs in the following graph. Below are steps based on dfs. A row major scan is started for the entire image. We simple need to do either bfs or dfs starting from every unvisited vertex and we get all strongly connected components.