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, Since f is one-side increasing for ?, by the statement 3 of Definition 3, there is a child Y of parent(X) such that f (u) ? ?{f (v) | R v ? Y }. Hence, there is a child Y of parent(X) such that f (u) ? (Y ). Then, we have f (u) ? (X) or f (u) ? (sibling(X)), Thus, we have (X) ? (sibling(X))

, We will prove by induction that this lemma holds true for any dominant region for f and ?. In the base step, we consider that parent(X) is V . In the inductive step, we show that, if the property holds true for parent(X), then it also holds true for X. Please note that, if parent(X) is not a dominant region for f and ?, the property holds for parent(X) as proven in the previous case. (a) Base step: if parent(X) is V , then ?(X) = ?(V ) = (V ) + 1 (first case of Definition 10). We can see that (V ) ? (X) because

, Since ?(X) = ?(parent(X)), we have ?(X) ? (parent(X)). We can affirm that, for any edge v in E ? such that R v ? X, we also have R v ? parent(X). Hence, (parent(X)) ? (X). Therefore, ?(X)

, Given the following propositions: (a) u is a watershed-cut edge (b) ?(H)(u) > 0 we will prove that (a) implies (b), and that not (b) implies not (a)

, one minimum of w. Therefore, the extinction value of both children of R u is non-zero and, consequently, the persistence value ?(u) of u is non-zero. Moreover, by Lemma 20, in this case we have ?(H)(e) = ?(e) for any building edge e for ?. Thus, ?(H)(u) is nonzero. On the other hand, if u is not a watershed-cut edge for ?, then there is a child X of R u which does not contain any minimum of w. Therefore, the extinction value of X is equal to 0: (X) = 0. Since, by definition ?(u) = min{ (X), (sibling(X))} and the minimal extinction value is zero

, Let v be the building edge of a region Z ? X. Then, we can say that the extinction value of both children of Z is less than or equal to the extinction value (X). Hence, ?(v) ? (X) and, then, ?(v) ? ?(u), By Lemma, vol.20

, Let H be a hierarchy on V and let ? be an altitude ordering such that ?(H) is one-side increasing for ?. Then the hierarchy H is a hierarchical, Lemma, vol.46

,

, Let ? be an altitude ordering for w and let H be a hierarchy on V such that ?(H) is one-side increasing for ?

, As ?(H) is order to prove that (V, E ? ) is a MST of (G, ?(H)), we will prove that, for any MST G of (G, ?(H)), the sum of the weight of the edges in G is greater than or equal to ?. Let G be a MST of (G, ?(H)). As G is a MST of (G, ?(H)), by the condition 1 of Lemma 21, Proof Let ? denote the sum of the weight of the edges in E ? in the map ?(H): ? = e?E ? ?(H)(e)

As ?(H) is the saliency map of H, we have that ,

Since the partitions P i and P i?1 are distinct, then there exists a region in P i which is not in P i?1 . Therefore, there is a region X of P i which is composed of a several regions {R 1 , R 2 , . . . } of P i?1 . Then, there are two adjacent vertices x and y such that x and y are in distinct regions in {R 1, n ? 1} is a subset of the range of ?(H) ,

Thus, there exists an edge u = {x, y} in E ? such that ?(H)(u) = i. Hence, the sum of the weight of the edges of G is at least 1 + · · · + n ? 1, which is equal to ?. Therefore, the graph (V, E ? ) is a MST of (G, ?(H)). -side increasing maps established in Lemma 47. In order to prove Property 14, we establish some auxiliary lemmas on MSTs and saliency maps. In the following, we state a well-known property of spanning trees in Lemma 48. Let x and y be two vertices in V and let ? = (x 0, the lowest j such that x and y belong to the same region of P j is i ,

, Let u = {x, y} be an edge in E \ E(G ) and let ? be the path from x to y (resp. y to x) in G . The graph G is a MST of (G, f ) if and only if f (u) ? f (v) for any edge v in ?. The following lemma characterizes MSTs of saliency maps, Lemma 48. Let G be a spanning tree of a weighted graph (G, f )

, Let u = {x, y} be an edge in E \ E(G ) and let ? be the path from x to y (resp. y to x) in G . Let v be an edge of greatest weight in ?. The graph G is a MST of (G, f ) if and, Lemma 49. Let f be the saliency map of a hierarchy on V and let G be a spanning tree of (G, f )

, Proof We will first prove the forward implication of this lemma. Let G be a MST of (G, ?(H)). Then, by Lemma 48

, in the ?-level set of (G, ?(H)), the vertices x and y are connected, ?(H)(u). Then, given ? = ?(H)(v)

,

, by Definition 13, there is a hierarchical watershed H w of (G, w) such that H is a flattening of H w . By Theorem 4, there is an altitude ordering ? for w such that ?(H w ) is one-side increasing for ?. Let ? be the altitude ordering for w such that ?(H w ) is one-side increasing for ?, By Lemma, vol.22

, ?(H w )). Then, any partition of H is a partition of QF Z(G , ?(H w )). By the definition of saliency maps, we can affirm that any partition of QFZ(G, ?(H)) is a partition of QF Z(G , ?(H w )). In the following

, By contradiction, let us assume that G is not a MST of (G, ?(H)). Then, by Lemma 49, there is an edge u = {x, y} such that u is in E \ E(G ) and such that ?(H)(u) is different from the greatest weight among the edges in the path ? from x to y in (G , ?(H)). Let v be an edge of greatest weight in ?. As H is equal to QFZ(G, ?(H)), we may affirm that ?(H)(u) is lower than ?(H)(v) because, otherwise, the vertices x and y would be connected in the ?-level set of

, Lemma 51, as H is a flattening of H w , we may conclude that ?(H w )(u) < ?(H w )(v). Hence, the weight ?

, As H w is one-side increasing for ?, by the second condition of Definition 3, for any watershed-cut edge u = {x, y} for ?, we have ?(H w )(u) = 0. Then, for any partition P of H w , x and y belong to the same region of P. Therefore, as any partition of H is a partition of H w , we can say that, for any partition P of H, x and y belong to the same region of P. Hence, the lowest ? such, We will now prove the second condition for H to be a flattened hierarchical

, By the third statement of Definition 3, we have that, for any edge u in E ? , there exists a child R of R u such that ?(H w )(u) ? ?{?(H w )(v) | R v ? R}

,

, Then H is a flattened hierarchical watershed of (G, w)

, In order to prove Lemma 53, we first state two auxiliary lemmas. From Property 10 of [10], we can deduce the following lemma linking binary partition hierarchies and MSTs

, Let B be a binary partition hierarchy of (G, w), Lemma, vol.54

, By Property 12 of [10] linking hierarchical watersheds and hierarchies induced by an altitude ordering and a sequence of minima, and by Lemma 21, we infer the following lemma

, Let G be a MST of (G, w) and let H be a hierarchical watershed of (G , w). Then H is also a hierarchical

, Lemma 53) Let H be a hierarchy on V such that there is an altitude ordering ? for w such that

,

, To this end, we will prove that there is a hierarchical watershed H w of (G, w) such that any partition of H is also a partition of H w . Let G denote the graph (V, E ? )

, By the definition of f , we have f (u) = 0 and, for any edge v such that R v ? X, we have f (v) = 0. Hence, f (u) ? ?{f (v) such that R v is included in X}. Otherwise, let us assume that u is a watershed-cut edge for ?. Then there is at least one minimum of w included in each child of R u . By the hypothesis 3, there is a child X of R u such that ?(H)(u) ? ?{?(H)(v) such that R v is included in X}. Let X be the child of R u such that ?(H)(u) ? ?{?(H)(v) such that R v is included in X}, no minimum of w included in X. Hence, none of the building edges of the descendants of X is a watershed-cut edge for ?

, QFZ(G , f ) is a hierarchical watershed of (G , w) (resp. (G, w)). Now, we only need to prove that any partition of H is a partition of QF Z(G , f ), Hence, f is one-side increasing for ? and, as stated previously

Let G ?,?(H) be the ?-level set of (G , ?(H)). Let ? be the greatest value in {f (u) | u ? E(G ?,?(H) )}. We will prove that the ?-level set of (G , f ) is equal to the ?-level set of (G , ?(H)) ,

, Since the minimum value of ? is zero, we can say that ?(H)(u) > 0 and, by the hypothesis 2, u is a watershed-cut edge for ?. Let v be an edge in the ?-level set of (G , ?(H)). Since ?(H)(u) > ?(H)(v), if v is a watershed-cut edge for ?, then v ? 2 u and f (u) > f (v). Otherwise, if v is not a watershedcut edge for ?, by the definition of f , we have f (v) = 0 and f (u) > f (v). Thus, for any edge v in the ?-level set of, Then, ?(H)(u) > ? and, for any edge v in the ?-level set of (G , ?(H)), we have ?(H)(u) > ?(H)(v)

As the partitions of H are given by the set of connected components of the level sets of (G , ?(H)), we can affirm that any partition of H is also a partition of QFZ(G , f ). Therefore, there is a hierarchical watershed H w = QFZ(G , f ) of (G , w) (resp. (G, w)) such that any partition of H is also a partition of H w . Then ,