?{ f 2 (v) | R v ? X}) for a child X of R u . We need to prove that, for any building edge u of B, f 3 (u) ? ?{ f 3 (v) | R v ? X} for a child X of R u ,

, ?{ f 1 (v) | R v ? Y}), we can affirm that f 3 (u) ? ?{ f 1 (v) | R v ? X} (resp. f 3 (u) ? ?{ f 1 (v) | R v ? Y}) as well, Let X and Y be the children of R u . If f 1 (u) ? ?{ f 1 (v) | R v ? X} (resp. f 1 (u) ?

, By Lemma 19, we can affirm that (V, E(B)) is a MST of both, vol.13

is a MST of (G, f 3 ) as well, which proves the first condition for QF Z(G, f 3 ) to be a flattened hierarchical watershed of (G, w), By Lemma, vol.23 ,

, C(b, a); and 3. if min(a, b) = min(c, d) and max(a, b) < max(c, d) then C(a, b) ? C(c, d)

, By Property 20, we need to prove that there exists a binary partition hierarchy B of (G, w) such that the following statements hold true: 1. (V, E(B )) is a MST of (G, f 3 ); and 2

, Let C be a function from R 2 into R such that, for any two real values x and y, we have C(x, y) = C(y, x)

, As min(a, b) = min(c, d) and max(a, b) = max(c, d), then either we have (i) a = c and b = d which implies that C(a, b) = C(c, d); or (ii) c = b and d = a which implies that C(c, d) = C(b, a), which

, The following three lemmas prove that the conditions 1, 2 and 3 for QF Z(G, f 3 ) to be a flattened hierarchical watershed of (G, w) hold true

n ? 1}, we have: 1. C(0, 0) = 0; and 2. C(a, b) = C(b, a); and 3. if min(a, b) = min(c, d) and max(a, b) < max(c, d) then C(a, b) ? C(c, d); and VII 4, Lemma 29. Let C be a positive function such that ,

, Let f 1 and f 2 be the saliency maps of two hierarchies on V and let G be a subgraph of G such that G is a MST of both (G, f 1 ) and (G, f 2 ). Then G is also a MST of

, C(b, a); and 3. if min(a, b) = min(c, d) and max(a, b) < max(c, d) then C(a, b) ? C(c, d)

, Let f 1 and f 2 be the saliency maps of two hierarchies on V and let B be a binary partition hierarchy of (G, w) such

we have: 1. C(0, 0) = 0; and 2. C(a, b) = C(b, a); and 3. if min(a, b) = min(c, d) and max(a, b) < max(c, d) then C(a, b) ? C(c, d); and 4. if min(a, b) < min(c, d) then C(a, b) < C(c, d). map C( f 1 , f 2 ). Then, for any building edge u of B, there exists a child R of R u such that f 3 (u) ? ?{ f 3 (v) such that R v is included in R}. 1. If f 1 (u) ? ?{ f 1 (v) | R v ? X} and f 2 (u) ? ?{ f 2 (v) | R v ? X}, then, Lemma 31. Let C be a positive function such that ,

, < min

, ?{ f 1 (v) | R v ? Y} and f 2 (u) ? ?{ f 2 (v) | R v ? Y}

,

, B)) is a MST of both (G, ?(H 1 )) and (G, ?(H 2 )). Therefore, by Lemma 29, (V, E(B)) is a MST of (G, f 3 ) as well, which proves that the first condition for QF Z(G, f 3 ) to be a flattened hierarchical watershed of (G, w) holds true. The second and third conditions are the result of Lemmas 30 and 31, of Property 8). By Lemma 19, we can affirm that, vol.18