The literature on watersheds is itself divided by a divide line: on one side the watersheds on node weighted graphs, on the other side the watersheds on edge weighted graphs. Flooding graphs introduced below show that this division makes no sense as ooding graphs oer a perfect coupling between edge and node weights: one set of weights may be deduced from the other, and the catch- ment basins associated to either one or the other are the same.
15.1 De nition and basic properties
De nition 7. An edge and node weighted graph G = [N, E] is a ooding graph i its weight distribution (n, e) veri es the relations:
- δenn = e
- εnee = n
Corollary 1. For a ooding graph, the weight distribution (n, e)verifies : - n ∈ Inv(ϕn)
- e ∈ Inv(γe)
Proof. e = δenn = δenεnee = γeeand n = εnee = εneδenn = ϕnn
Properties of the ooding graph As G is invariant by γe,all its edges are
the lowest edge of one of their extremities.
As G is invariant by ϕn,it has no isolated regional minimum.
Lemma 6. In a ooding graph, a node has no adjacent edges with a lower weight but each node has at least one adjacent edge with the same weight.
Proof. In a ooding graph, as εnee = n, all edges adjacent to a node i have
weights which are higher or equal than this node and at least one of them, say (i, j) has the same weight : eij= ni.
15.2 Constructing ooding graphs.
Flooding graphs oer a perfect coupling between the edge and the node weights. As a matter of fact, constructing the watershed for node weighted graphs or constructing it for edge weighted is strictly equivalent, since, as we will show below, any node weighted graph without edge weights, or any edge weighted graph without node weights may be completed to become a ooding graph, on which the watershed will be ultimately constructed.
Transforming an arbitrary edge weighted graph into a ooding graph Lemma 7. If G = (e, ) ∈ Inv(γe),then (e, εnee)is a ooding graph
Proof. a) e = γee = δenεnee = δenn); b) n = εneeby construction.
Lemma 8. If G = (e, ) is an arbitrary edge weighted graph, then (↓ e, εne↓ e)
is a ooding graph
Proof. If G = (e, ) is an arbitrary edge weighted graph, then ↓ (e, ) ∈ Inv(γe),
as the edges lowered by γe, have been suppressed, the other edges keep their
weights. Furthermore we have proved above that εnee = εne ↓ e.Applying the
previous lemma shows that (↓ e, εne↓ e)is a ooding graph.
Transforming an arbitrary node weighted graph into ooding graph Lemma 9. If G = (−, n) ∈ Inv(ϕn), then (δenn, n) is a ooding graph
Proof. a) e = δenn)by construction ; b) n = ϕnn = εneδenn = εnee.
Lemma 10. If G = (−, n) is an arbitrary node weighted graph, then (δen (
n, ( n) is a ooding graph
Proof. If G = (e, ) is an arbitrary node weighted graph, then (−, ( n) ∈ Inv(ϕn),as each isolated regional minimum has been duplicated by the operator
( n. Applying the preceding lemma shows that (δen ( n, ( n) is a ooding
graph.
Partial graph of a ooding graph Suppressing edges in a ooding graph, but leaving at least one lower neighboring edge for each node (like that, no isolated regional minima are created) produces a partial graph which also is a ooding graph, keeping an identical distribution of weights on the nodes and on the remaining edges.
15.3 Regional minima of ooding graphs We proved earlier these theorems:
If G ∈ Inv(γe)is an edge weighted graph, and mi is an edge regional mini-
mum of G, then the nodes spanned by mi form a node regional minimum of
the node weighted graph εneG = (−, εnee)
If G ∈ Inv(ϕn)is a node weighted graph, and miis a node regional minimum
of G, then the edges spanning miform an edge regional minimum of the edge
weighted graph δenG = (δenn, )
Theorem 4. If G is a ooding graph with the weight distribution (e, n), then the node weighted graph (−, n) and the edge weighted graph (e, ) have the same regional minima subgraph.
Fig.17 presents the same ooding graph, on the left with its edge weights and on the right with its node weights : they have exactly the same regional minima. The minima of a ooding graph being identical for the edge weigths and for the node weights may be assigned the same labels. The labels may be hold by the edges of the minima or by their extremities, or by both, as illustrated by g.18.
15.4 Perfect equivalence between node and edge weights on ooding graphs
We now understand why the watershed of a ooding graph may be constructed on the basis of the node weights or on the basis of the edge weights : they produce the same result :
the node weights and the edge weights may be derived from each other the minima are the same
each node has no lower adjacent edges but at least one with the same weight. A node and the adjacent edge with the same weight is called ooding pair. A series of non increasing ooding pairs is both a non increasing path of nodes and a non increasing chain of edges.
We de ne the catchment basin of a regional minimum as the set of nodes which are linked by a non increasing path for node weighted graphs and a non increasing chain for edge weighted graphs with this minimum. As the minima are the same and the non increasing paths or chains also are equivalent, this shows that indeed the watershed constructed on the node weights or on the edge weights of a ooding graph are the same.
Fig. 17. Whether one consider the egde weights or the node weights produces the same regional minima.
labeling the regional minima on the nodes and/or the edges
2 2 2 1 1 1 1 1 1 1 4 4 4 4 4 7 8 6
Fig. 18. As the minima are identical on the nodes or the edges of a ooding graph, it is possible to assign the same labels to nodes or to edges.