Index

• k – steep
  • flowing path, 133
• , 63
• , 63
• (E), 49
• ε_ne, 63
• ε_ne, 76
• ϕ_n, 64
• δ_en, 76
• θ (p)
  • see steepness measure, 133
• ∞ – steep
  • digraph, 134
  • flowing path, 133, 156, 192
• γ_e, 64

A, B, C

• adjunction, 53
  • closing, 54
  • dilation, 53
  • erosion, 53
  • opening, 54
• anti-extensive, 52
• antisymmetry, 49
• arc, 60, 99, 190
  • inversion, 193
• arrow, 61, 188
  • bidirectional, 61
• authorized pruning, 120
• bidirectional
  • flowing edge, 72
  • flowing path, 72
• black
  • holes, 103, 172
  • holes detection, 110, 196
• bounds
  • lower bound, 50
• button hole, 145, 183
• catchment zone, 68, 70, 72
  • of a dead end, 102
  • of a node-weighted graph, 68
  • of an edge-weighted graph, 70
• closing, 53
• cocycle, 62
• commutes, 52
• complementation, 52
• complete lattice, 50
• concatenation
  • of paths, 61
• connected component, 179
• core expanding flooding algorithm, 156, 157, 159, 161, 165, 168
• cycle, 62

D, E, F

• dead
  • end, 102
  • leaves model, 114
• decreasing operator, 52
• digital elevation model (DEM), 148
• digraph, 60
  • ∞ – steep, 170, 192
• digraphs
  • black holes, 145
  • flat zones, 145
  • flow digraph, 130
  • gravitational digraphs, 131
  • regional minima digraph, 158
• dilation, 52
  • between node and edge weights, 63, 76, 77
• directed
  • erosion, 190
  • graph, 99
  • path, 62, 99
• downstream, 109
• dual operator, 54
• edge, 60
  • flat zone, 69
• edge-weighted graph, 60
  • flowing, 69
• edge weights opening, 64, 77
• equivalence classes
  • even zone, 71
  • node flat zones, 67, 69
• equivalence relation, 67, 69
• erosion, 52
  • between edge and node weights, 63, 76, 77
• even zone, 71
• extensive, 52
• family of functions, 56, 62
• flat zone
  • of a digraph, 103
  • plateaus, 171
• flow digraph, 130, 156
• flowing
  • adjunction, 77
  • graph, 76, 82, 84, 86
  • path, 67, 70
• flowing edge
  • in a node-weighted graph, 67
  • in an edge-weighted graph, 69
• flowing graph, 94, 95, 99
  • coupling edge and node weights, 79
  • e-flowing graph
    • ↻ G(ν, nil), 82, 93
  • n-flowing graph
    • ↓ G(nil, η), 95

G, H, I

• Galois adjunction, 53
• geodesic distance, 179
• gradient operator, 64
• graph, 60
  • e-flowing, 74, 99
  • flowing, 76
  • n-flowing, 75, 99
  • subgraph, 61
• gravitational digraph, 106, 121, 123, 131
• greatest lower bound, 50
• grids, 188
• hierarchical queue, 159, 160
  • core expanding algorithm, 169
  • driven algorithms, 151, 181
• homotopic thickening, 179
• hydographic model, 88
  • edge unstable filling, 91, 92
  • node unstable filling, 91, 92
  • totally unstable filling, 89
• idempotent, 52
• increasing operator, 52
• infimum
  • in a lattice, 50
  • of functions, 56
• invariants
  • of an operator, 55
  • of γ_e, 85
  • of φ_n, 83
• involution, 52

K, L, M

• k-steep
  • digraph, 134
• label, 66, 107, 192
  • downstream propagation, 108, 152, 172
  • priority order relation, 107
  • upstream propagation, 108
• least upper bound, 50
• lexdist, 156
• lexicographic
  • preorder relation, 131
  • weight, 132, 156, 157
• minimum spanning tree, 104

N, O, P

• node, 60
  • closing, 64, 77
  • flat zone, 67
  • weights closing
• opening, 53
• operator, 52
  • anti-extensive, 52
  • extensive, 52
  • idempotent, 52
  • increasing, 52
• out-arrow, 188, 190
• overlapping
  • core, 116, 197
  • of catchment zones, 116
  • seeds, 116, 197
• partial
  • digraph, 61
  • graph, 61
  • order relation, 49
• partially ordered set, 50
• partitions
  • dead leaves partitions, 116, 150
  • watershed partitions, 170
• pass point, 199
• path, 61
  • directed path, 62
  • undirected path, 61
• perpetuum mobile, 105
• plateau
  • see flat zone, 171
• poset
  • see partially ordered set, 50
• pruning, 120
  • comet, 146
  • directed erosion
    • , 138, 139
  • ↓^k , 141
  • factorization, 147
  • infinite, 142
  • iterated, 141
  • operator, 192
  • step, 140
  • scissor operator
    • ↓², 138, 139

R, S, T

• Random Access Memory (RAM), 179
• reflexivity, 49, 67
• regional minimum, 72
  • edge-weighted graph, 69
  • node-weighted graph, 67
• scissor, 190
• skeleton by zones of influence, 179
• smooth zone, 101
• smoothly linked, 101
• steepness, 133
  • ∞ – steep, 133, 142, 151, 170
  • measure, 133
• subgraph, 61
• supremum, 50
  • of functions, 56
• symmetry, 67
• tanks and pipes network, 88
• topographic
  • distance, 181
  • length, 181
• total order relation, 50
• transitivity, 49
  • equivalence relation, 67

U, V, W

• unstable
  • edge, 89
  • node, 89
• upper bound, 50
• upstream, 108
• vertex, 60
• waterfall hierarchy, 201
• watershed partition, 198