Pruning digraphs for reducing catchment zones

Transcription

1 Pruning digraphs for reducing catchment zones Fernand Meyer Centre de Morphologie Mathématique 19 March 2017 ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

2 Why do catchment zones overlap? A node p linked by owing paths with 2 distinct regional minima m 1 and m 2 belongs to the catchment basins of both of them. If the rst edge of the paths leading to m 2 is cut, then p does not belong anymore to the catchment zone of m 2. In this section we show how to compare the owing paths with a common origin according to some steepness measure. We then introduce a pruning operator able to cut the rst edge of the paths which are not steep enough. Pruning permits to reduce the extension of the catchment zones and of their overlappings. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

3 Pruning node weighted graphs. The pruning of the owing paths will be done on a node weighted digraph associated to the initial graphs: G (ν, nil) : Each edge is a owing edge for one of its extremities. It is replaced by an arrow in the digraph! G (ν, nil). In a owing path of G (ν, δ en ν) each node is followed by an edge with the same weight except the ultimate node, when this node is an isolated regional minimum. G (nil, η) : In G (nil, η) not each edge is a owing edge of one of its extremities. These edges are cut in # G (nil, η). Each remaining edge is a owing edge of one of its extremities. It is replaced by an arrow in the graph #! G (ε ne η, nil). In a owing path of # G (ε ne η, nil) each node is followed by an edge with the same weight. In each case, each node of a owing path is followed by an edge with the same weight. The edge weights are redundant, and only the node weights are used. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

4 The owing digraph associated to a node weighted graph From G (ν, nil) in gure A to! G (ν, nil) in gure B: the same owing paths and regional minima. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

5 The owing digraph associated to an edge weighted graph From G (nil, η) to! # G (ε ne η, nil): the same owing paths and regional minima. A: an edge weighted graph G (nil, η) B: the edge weighted digraph! # G (nil, η) in which each arrow represents a owing edge of # G (nil, η). C: the node weighted digraph! # G (ε ne η, nil). Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

6 A lexicographic order relation ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

7 The steepness order relation between owing paths An order relation between the owing paths with a common origin permits to compare them. To each owing path is associated an in nite list of weights, whose rst elements are the weights of its nodes and the last elements an in nite repetition of the weight of the regional minimum. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

8 The steepness order relation between owing paths Consider two such lists λ = (λ 1, λ 2,...λ k, λ k+1,...λ n,...) and µ = (µ 1, µ 2,...µ k, µ k+1,...µ n,...). We de ne a lexicographic order relation between these lists: * λ µ if λ 1 < µ 1 or there exists k such that 8l < k : λ i = µ i λ k < µ k * If {λ µ} or λ = µ, we write λ µ. The relation is symmetrical and transitive. It is a preorder relation and not an order relation, as it is not antisymmetric: λ µ and µ λ does not imply that λ = µ. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

9 The steepness order relation between owing paths Each node is the rst node of one or several owing paths.and is linked with at least one regional minimum. It is the rst node of one or several owing paths. Among all in nitely prolonged owing paths with a same origin, there is at least one (maybe several) which has the lowest lexicographic weight. Such paths with a minimal lexicographic weight are said to be steep. A path µ starting at node p is said k steep if its k rst nodes have the same weights as an steep with the same origin. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

10 k-steep owing graphs A owing graph G is k steep (resp. steep) if each of its owing paths is at least k steep (resp. steep). Remark: In a k steep graph, all (prolongated) owing paths with the same origin have identical weights on their k rst nodes. This is more and more unlikely when k increases. Similarly, in an steep graph, all (prolongated) owing paths with the same origin have identical weights, from the rst node until the last node belonging to a regional minimum. Corollary: If the regional minima have distinct weights, then each node of an steep graph is linked with one and only one regional minimum: the catchment zones do not overlap. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

11 The steepness measure associated to a node Among all owing paths with the same origin p, there is at least one which has a smaller lexicographic weight than all others. There may be several, the weights of their successive nodes being θ 1 (p) = ν p, θ 2 (p),..., θ k (p),... The function θ i is a steepness measure associated to each node. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

12 The steepness of a subpath let π p = p B π q be a directed path obtained by appending to the node p the path π q of origin q. Lemma π p = k steep ) π q = (k 1) steep Proof. If σ q is path of origin q which is steeper than the path π q, then p B σ q is steeper than p B π q. Corollary If (p! q) is the rst arc of a k θ k (p) = θ k 1 (q) By contraposition: if θ k (p) < θ k of a k steep path of origin p steep path of origin p, then 1 (q), then (p! q) is not the rst arc ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

13 k-steep graphs : a criterion Theorem If 8l < k : θ l+1 (p) = θ l (q) for any owing arc (p! q), then any owing path of length k is k steep and the graph G is k steep. Proof. Consider an arbitrary path π k 1 = (p 1, p 2,...p k ) of length k. The weights of the nodes are (ν p1, ν p2,...ν pk ). As θ 1 (p i ) is the weight of the rst node of any path starting with p i, we have ν pi = θ 1 (p i ). Since 8l < k : θ l+1 (p) = θ l (q), we may perform the following substitutions: ν pi = θ 1 (p i ) = θ 2 (p i 1 ) = θ 3 (p i 2 ) = = θ i (p 1 ). Hence the list of weights of π k 1 = (ν p 1, ν p2,...ν pk ) = (θ 1 (p 1 ), θ 2 (p 1 ),...θ k (p 1 )) = θ k 1 (p 1 ), showing that π k 1 is indeed k steep. ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

14 Pruning owing digraphs ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

15 The pruning operators (1) The erosion! ε assigns to each node p the minimal weight of its downstream neighbors:! ε ν p = V ν q. qjp!q Remark: we suppose that p! p, which permits to attribute a weight to nodes which are not the origin of an arrow. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

16 The pruning operators (2) The operator # 2! G (ν, nil) keeps all arcs (p! q) of origin p such that ν q is minimal and suppresses all others. Suppose that the arrow p! q is suppressed and the arrow p! u preserved, implying ν p ν q > ν u, which in turn implies u 9 p. These properties characterize "authorized prunings", preserving the regional minima of the digraph. Only the catchment zones are reduced, as some owing paths have been cut. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

17 The pruning operators (3) Applying the operator! ε to # 2! G (ν, nil) lets the node weights move upstream along the remaining owing paths. Applying again the operator # 2 on! ε # 2! G (ν, nil) produces a second authorized pruning. We obtain like that the pruning operator # 3 = # 2! ε # 2 Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

18 The pruning operators (4) Applying n times the operator # 2! ε to # 2! G (ν, nil) produces the authorized pruning: # n+2 = # 2! n ε # 2 At each iteration, new arcs are cut, the catchment zones reduced but the regional minima preserved (as the black holes of a gravitational graph after a series of authorized prunings). Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

19 The pruning operator creates k-steep graphs Theorem Thus the operator # 2! k 1 ε # 2 leaves only the owing paths of origin p which are at least (k + 1) steep. Consider two paths with the same origin p. The rst,! π = p! q! u!..., is steep and the second,! σ = p! s! t!..., is k steep : θ i+1 (p) = θ i (q) = θ i (s) for i < k but σ is not (k + 1) steep as θ k (s) > θ k (q). The operator! ε # 2! k 2 ε # 2! G uses k times the operator! ε, letting the weights of the nodes glide upstream along the owing paths, producing: ν p = θ k (p), ν q = θ k (q) = θ k+1 (p) and ν s = θ k (s) > θ k (q). The subsequent operator # 2 cuts the arc between p and s. ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

20 Illustration : lexicographic distances The follownig 3 images show respectively the shortest lexicographic distances of depth 1, 2 and 3. If a path is the shortest path for a lexicographic distance of depth k, it also is a shortest path for a lexicographic distance of smaller depth. The following three gures present the lexicographic distances of depth 1, 2 and 3. The partition of catchment basins for the distance 3 is also solution for the distance 2 and 1. Similarly the partition for distance 2 is also solution for distance 1. The number of solutions decreases with the lexicographic depth. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

21 Illustration : lexicographic distance of depth 1 Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

22 Illustration : lexicographic distance of depth 2 Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

23 Illustration : lexicographic distance of depth 3 Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

24 Illustration of the pruning operator A: G (ν, nil) ; B:! G (ν, nil) C: 2! G (ν, nil): a rst catchment zone separated from the rest of the graph. D:! ε 2! G (ν, nil) ; E: 2! ε 2! G (ν, nil) F: 2! ε 2! ε 2! G (ν, nil) : all three catchment zones are separated. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

25 Illustration of the pruning operator A: an edge weighted graph G ; B: the owing digraph! G ; C: the owing digraph with labeled regional minima Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

26 Illustration of the pruning operator A: a owing digraph! G ; B: 2! G (ν, nil) ; C:! ε 2! G (ν, nil) ; D: 2! ε 2! G (ν, nil) Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

27 The e ect of the pruning operator ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

28 Increasing pruning intensity The owing paths (in violet) and the catchment zone of the dark dot are compared for increasing degrees of pruning (1, 2, 3, 4, 5, 7, ). Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

29 Increasing pruning intensity Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

30 Downstream trajectories Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

31 Downstream, upstream, upstream of downstream Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

32 Downstream, upstream, upstream of downstream Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

33 Locality of the pruning operator ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

34 Locality of the pruning operator The operators # 2 and! ε consider for each node p the arcs having p as origin and the weights of their extremities. The operator # k considers for each node the owing paths of origin p counting k nodes, and cuts the rst arc of all paths which are not at least k steep. The union of the owing paths having their origin in a set X of nodes and counting each k nodes is called comet(x, k). The subgraph of! G spanned! by the nodes of comet(x, k) is called comet(x, k). The preceding analysis shows that applying # k to!! G or to comet(x, k) has the same e ect on the arcs having their origin in X, cutting the rst arcs of owing paths which are not at least k steep. Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

35 To which catchment zone belongs the node with a red rim? A: edge weighted graph G ; B: the edge weighted owing digraph #! G ; C: node weighted owing digraph with labeled regional minima ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

36 Fernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37

37 ernand Meyer (Centre de Morphologie Mathématique) Pruning digraphs for reducing catchment zones 19 March / 37