, E(B)) is a MST of
, if u is not a watershed-cut edge for B, then ?(H)(u) = 0in E(B), there exists a child R of R u such that ?(H)(u) ? ?{?(H)(v) such that R v is
, 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, ) ? ?{ f 1 (v) | R v ? X} (resp. f 1 (u) ? ?{ f 1 (v) | R v ? Y})
, By Lemma 29, we can affirm that (V, E(B)) is a MST of both, vol.16
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.33 ,
,
, Let H 1 and H 2 be two hierarchical watersheds of (G, w) and let B be a binary partition hierarchy of (G, w) such that both ?(H 1 ) and ?(H 2 ) are one-side increasing for B. Let C be a positive function from R 2 into R such that
, 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)
, We want to prove that the hierarchy QF Z(G, f 3 ) is a flattened hierarchical watershed of (G, w). By Property 30, 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)
, 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 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
, Since G is a spanning tree, by Lemma 34, the graph (V, E(G ) ? {u}) contains a cycle ? which includes the edge u. Let ? be the cycle of (V, E(G ) ? {u}) which includes the edge u. As G is a MST of (G, f 1 ) and of (G, f 2 ), by Lemma 35, for any edge v in the cycle ?, we have f 1 (v) ? f 1 (u) and f 2 (v) ? f 2 (u). Therefore, for any edge v in the cycle ?, we have min( f 1 (v), f 2 (v)) ? min( f 1 (u), f 2 (u)) and max( f 1 (v), f 2 (v)) ? max( f 1 (u), f 2 (u)). Then, we should consider the three following cases: 1. If min( f 1 (v), f 2 (v)) < min
= min( f 1 (u), f 2 (u)) and max( f 1 (v), f 2 (v)) < max ,
, = min
, (v)) = max( f 1 (u), f 2 (u)), then, by Lemma 38, we have C( f 1 (v), f 2 (v)) = C
= f 3 (v) ? C( f 1 (u), f 2 (u)) = f 3 (u). Hence, for any edge v in the cycle ?, we have f 3 (v) ? f 3 (u). Thus, by Lemma 35, p.1 ,
, Lemma 40. Let C be a positive function such that
, 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)
, Lemma 41. Let C be a positive function such that
, 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
, ?{ f 1 (v) | R v ? X} and f 2 (u) ? ?{ f 2 (v) | R v ? X}, then
, Therefore, min( f 1 (e), f 2 (e)) ? min
, Otherwise, we have min
, As f 1 (u) ? f 1 (v) and f 2 (u) ? f 2 (e), we have max
(e)). Otherwise, we have max, = max( f 1 (e), f 2 (e)) then, by Lemma, vol.38, p.1 ,
In this case, and as min( f 1 (u), f 2 (u)) = f 1 (u), we have min( f 1 (u), f 2 (u)) > f 1 (v), and thus min( f 1 (u), f 2 (u)) > min ,
, ?{ f 1 (v) | R v ? Y} and f 2 (u) ? ?{ f 2 (v) | R v ? X}
, ?{ f 1 (v) | R v ? Y} and f 2 (u) ? ?{ f 2 (v) | R v ? Y}
,
, By Lemma 29, we can affirm that (V, E(B)) is a MST of both (G, ?(H 1 )) and (G, ?(H 2 )). Therefore, by Lemma 39, (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 40 and 41
, 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)
, Since m ? n, we can prove that those fours statements hold true for any a, b, c and d in {0, . . . , m ? 1}. The proof of the first and second statements are trivial, order to prove the third and fourth statements, we state Lemmas 42 and 43
, Lemma 42. Let C(x, y) = x m y m x m +y m and let a, b and d be natural numbers