Constrained connectivity hierarchy¶
Alphaomega constrained connectivity hierarchy based on the given vertex weighted graph. 

Strongly constrained connectivity hierarchy based on the given edge weighted graph. 

constrained_connectivity_hierarchy_alpha_omega
(graph, vertex_weights)[source]¶ Alphaomega constrained connectivity hierarchy based on the given vertex weighted graph.
For \((i,j)\) be an edge of the graph, we define \(w(i,j)=w(i)  w(j)\), the weight of this edge. Let \(X\) be a set of vertices, the range of \(X\) is the maximal absolute difference between the weights of any two vertices in \(X\): \(R(X) = \max\{w(i)  w(j), (i,j)\in X^2\}\)
Let \(\alpha\) be a positive real number, a set of vertices \(X\) is \(\alpha\)connected, if for any two vertices \(i\) and \(j\) in \(X\), there exists a path from \(i\) to \(j\) in \(X\) composed of edges of weights lower than or equal to \(\alpha\).
Let \(\alpha\) and \(\omega\) be a two positive real numbers, the \(\alpha\omega\)connected components of the graph are the maximal \(\alpha'\)connected sets of vertices with a range lower than or equal to \(\omega\), with \(\alpha'\leq\alpha\).
Finally, the alphaomega constrained connectivity hierarchy is defined as the hierarchy composed of all the \(kk\)connected components for all positive \(k\).
The definition used follows the one given in:
P. Soille, “Constrained connectivity for hierarchical image partitioning and simplification,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 7, pp. 11321145, July 2008. doi: 10.1109/TPAMI.2007.70817
The algorithm runs in time \(\mathcal{O}(n\log(n))\) and proceeds by filtering a quasiflat zone hierarchy (see
quasi_flat_zones_hierarchy()
) Parameters
graph – input graph
vertex_weights – edge_weights: edge weights of the input graph
 Returns
a tree (Concept
CptHierarchy
) and its node altitudes

constrained_connectivity_hierarchy_strong_connection
(graph, edge_weights)[source]¶ Strongly constrained connectivity hierarchy based on the given edge weighted graph.
Let \(X\) be a set of vertices, the range of \(X\) is the maximal weight of the edges linking two vertices inside \(X\).
Let \(\alpha\) be a positive real number, a set of vertices \(X\) is \(\alpha\)connected, if for any two vertices \(i\) and \(j\) in \(X\), there exists a path from \(i\) to \(j\) in \(X\) composed of edges of weights lower than or equal to \(\alpha\).
Let \(\alpha\) be a positive real numbers, the \(\alpha\)strongly connected components of the graph are the maximal \(\alpha'\)connected sets of vertices with a range lower than or equal to \(\alpha\) with \(\alpha'\leq\alpha\).
Finally, the strongly constrained connectivity hierarchy is defined as the hierarchy composed of all the \(\alpha\) strongly connected components for all positive \(\alpha\).
The definition used follows the one given in:
P. Soille, “Constrained connectivity for hierarchical image partitioning and simplification,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 7, pp. 11321145, July 2008. doi: 10.1109/TPAMI.2007.70817
The algorithm runs in time \(\mathcal{O}(n\log(n))\) and proceeds by filtering a quasiflat zone hierarchy (see
quasi_flat_zones_hierarchy()
) Parameters
graph – input graph
edge_weights – edge_weights: edge weights of the input graph
 Returns
a tree (Concept
CptHierarchy
) and its node altitudes