Nicolas Boutry
email: nicolas.boutry@lrde.epita.fr

Laurent Najman
email: laurent.najman@esiee.fr

Thierry Géraud
email: thierry.geraud@lrde.epita.fr

Some equivalence relation between persistent homology and morphological dynamics

Keywords: Mathematical Morphology, Morse Theory, Computational Topology, Persistent Homology, Dynamics, Persistence

In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n ≥ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.

Fig. 1: Low sensibility of dynamics to noise (extracted from [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF]).

Introduction

In Mathematical Morphology [START_REF] Najman | Mathematical Morphology: from Theory to Applications[END_REF]47,[START_REF] Serra | Mathematical Morphology and its Applications to Image Processing[END_REF], dynamics [START_REF] Grimaud | La géodésie numérique en Morphologie Mathématique. Application à la détection automatique des microcalcifications en mammographie numérique[END_REF][START_REF] Grimaud | New measure of contrast: the dynamics[END_REF][START_REF] Vachier | Extraction de caractéristiques, segmentation d'image et Morphologie Mathématique[END_REF], defined in terms of continuous paths and optimization problems, represents a very powerful tool to measure the significance of extrema in a gray-level image (see Figure 1). Thanks to dynamics, we can efficiently select markers of objects in an image. These markers (that do not depend on the size or on the shape of objects) help to select relevant components in an image; hence, this process is a way to filter objects depending on their contrast, whatever the scale of the objects, and is often combined with the watershed [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF][START_REF] Vincent | Watersheds in digital spaces: an efficient algorithm based on immersion simulations[END_REF] for image segmentation. This contrasts with convolution filters often used in digital signal processing or morphological filters [START_REF] Najman | Mathematical Morphology: from Theory to Applications[END_REF]47,[START_REF] Serra | Mathematical Morphology and its Applications to Image Processing[END_REF] where geometrical properties do matter. Fig. 2: The dynamics of a minimum of a given function can be computed thanks to a flooding algorithm (extracted from [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF]).

Note that there exists an interesting relation between flooding algorithms and the computation of dynamics (see Figure 2). Indeed, when we flood the topographical view of a function, at a given level, two basins merge, and the dynamics of the highest minima of the two basins is the difference between the current level of water and the altitude of this highest minima.

Similarly, in Persistent Homology [START_REF] Edelsbrunner | Persistent Homology -A survey[END_REF][START_REF] Edelsbrunner | Topological persistence and simplification[END_REF] well-known in Computational Topology [START_REF] Edelsbrunner | Computational Topology: an Introduction[END_REF], we can find the same paradigm: topological features whose persistence is high are "true" when the ones whose persistence is low are considered as sampling artifacts, whatever their scale. An example of application of persistence is the filtering of Morse-Smale complexes [START_REF] Edelsbrunner | Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds[END_REF][START_REF] Edelsbrunner | Morse-Smale complexes for piecewise linear 3-manifolds[END_REF][START_REF] Günther | Efficient computation of 3D Morse-Smale complexes and Persistent Homology using Discrete Morse Theory[END_REF] used in Morse Theory [START_REF] Willard Milnor | Morse Theory[END_REF][START_REF] Forman | A user's guide to Discrete Morse Theory[END_REF] where pairs of extrema of low persistence are canceled for simplification purpose. This way, we obtain simplified topological representations of Morse functions. A discrete counterpart of Morse theory, known as Discrete Morse Theory can be found in [START_REF] Forman | A Discrete Morse Theory for cell complexes[END_REF][START_REF] Jöllenbeck | Minimal resolutions via Algebraic Discrete Morse Theory[END_REF][START_REF] Forman | A user's guide to Discrete Morse Theory[END_REF][START_REF] Forman | Morse Theory for cell complexes[END_REF].

As detailed in [START_REF] Tamal | Stability of critical points with interval persistence[END_REF], pairing by persistence of critical values can be extended in a more general setting to pairing by interval persistence of critical points. The result is that it is possible to perform function matching based on their critical points, and then to pair all critical points of a given function (see Figure 2 in [START_REF] Tamal | Stability of critical points with interval persistence[END_REF]) where persistent homology does not succeed. However, due to the modification of the definition introduced in [START_REF] Tamal | Stability of critical points with interval persistence[END_REF], this matching is not applicable when we consider usual threshold sets.

In this paper, we prove that the relation between Mathematical Morphology and Persistent Homology is strong in the sense that pairing (of minima) by dynamics and pairing 1-saddles by persistence is equivalent (and then dynamics and persistence of the corresponding pair are equal) in n-D (n ≥ 1), when we work with Morse functions. For n = 1, the proof is much simpler (with some extra condition on the limits of the domain), but contains the essence of the proof for n ≥ 1, which is more technical. In order to ease the reading, we provide the complete proofs for both cases, first for the 1D case and then for the n-D case. This paper is the exten-sion of [START_REF] Boutry | An equivalence relation between Morphological Dynamics and Persistent Homology in 1D[END_REF] (which contains the 1D case) and [START_REF] Boutry | An equivalence relation between morphological dynamics and persistent homology in n-D[END_REF] (which generalizes [START_REF] Boutry | An equivalence relation between Morphological Dynamics and Persistent Homology in 1D[END_REF] to the n-D case, n ≥ 1).

The plan of the paper is the following: Section 2 recalls the mathematical background needed in this paper, Section 3 provides sketches of the equivalence of pairing by dynamics and by persistence in 1D and in n-D, Section 4 contains the complete proof of the 1D equivalence, while Section 5 contains the complete proof of the n-D equivalence. In Section 6, we discuss several research directions opened by the results of this paper. Section 7 concludes the paper.

Mathematical pre-requisites

We call path from x to x both in R n a continuous mapping from [0, 1] to R n . Let Π 1 , Π 2 be two paths satisfying Π 1 (1) = Π 2 (0), then we denote by Π 1 <> Π 2 the join between these two paths. For any two points x 1 , x 2 ∈ R n , we denote by [x 1 , x 2 ] the path:

λ ∈ [0, 1] → (1 -λ).x 1 + λ.x 2 .
Also, we work with R n supplied with the Euclidean norm:

. 2 : x → x 2 = n i=1 x 2 i .
In the sequel, we use lower threshold sets coming from cross-section topology [START_REF] Meyer | Skeletons and perceptual graphs[END_REF][START_REF] Bertrand | Topological approach to image segmentation[END_REF][START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF] of a function f defined for some real value λ ∈ R by:

[f < λ] = x ∈ R n f (x) < λ , and [f ≤ λ] = x ∈ R n f (x) ≤ λ .

Morse functions

We call Morse functions the real functions in C ∞ (R n ) whose Hessian is not degenerated at critical values, that is, where their gradient vanishes. A strong property of Morse functions is that their critical values are isolated.

In particular, we call D-Morse functions the Morse functions which tend to ±∞ when the 2-norm of their argument tends to +∞. Note that this last property will only be used to treat the 1D case in this paper.

Lemma 1 (Morse Lemma [START_REF] Audin | Morse Theory and Floer Homology[END_REF]).

Let f : C ∞ (R n ) → R be a Morse function. When x * ∈ R n is a critical point of f ,
then there exists some neighborhood V of x * and some diffeomorphism ϕ :

V → R n such that f is equal to a ℝ ℝ x min
x max Fig. 3: Example of pairing by dynamics: the abscissa x min of the red point is paired by dynamics relatively to f with the abscissa x max of the green point on its left because the "effort" needed to reach a point of lower height than f (x min ) (like the two black points) following the graph of f is minimal on the left.

second order polynomial function of

x = (x 1 , . . . , x n ) on V : ∀ x ∈ V , f •ϕ -1 (x) = f (x * )-x 2 1 -x 2 2 -• • • -x 2 k +x 2 k+1 +• • • +x 2 n .
We call k-saddle of a Morse function a point x ∈ R n such that the Hessian matrix has exactly k strictly negative eigenvalues (and then (n -k) strictly positive eigenvalues); in this case, k is sometimes called the index of f at x. We say that a Morse function has unique critical values when for any two different critical values x 1 , x 2 ∈ R n of f , we have f (x 1 ) = f (x 2 ). (See Appendix A for a discussion about this hypothesis.)

Pairing by dynamics (1D)

Let f : R → R be a D-Morse function with unique critical values. For x min ∈ R a local minimum of f , if there exists at least one abscissa x min ∈ R of f such that f (x min ) < f (x min ), then we define the dynamics [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF] of x min by: dyn(x min ) := min

γ∈C max s∈[0,1] f (γ(s)) -f (x min ),
where C is the set of paths γ : [0, 1] → R verifying γ(0) := x min and verifying that there exists some s ∈ ]0, 1] such that f (γ(s)) < f (x min ).

Let us now define γ * as a path of C verifying:

max s∈[0,1] f (γ * (s))-f (x min ) = min γ∈C max s∈[0,1] f (γ(s))-f (x min ),
then we say that this path is optimal. The real value x max paired by dynamics to x min (relatively to f ) is the local maximum of f characterized by:

x max := γ * (s * ), ℝ ℝ x min 2 x max
x min Fig. 4: Example of pairing by persistence: the abscissa x max of the local maximum in red is paired by persistence relatively to f with the abscissa x min of the local minimum in green, since its image by f is greater than the image by f of the abscissa x min 2 of the local minimum drawn in pink.

with f (γ * (s * )) = max s∈[0,1] f (γ * (s)). We obtain then:

f (x max ) -f (x min ) = dyn(x min ).
Note that the local maximum x max of f does not depend on the path γ * (see Figure 3), and its value is unique (by hypothesis on f ), which shows that in some way x max and x min are "naturally" paired by dynamics.

Pairing by persistence (1D)

From now on, we denote by R := {+∞, -∞} ∪ R the complete real line, and by cl R (A) the closure in R of the set A ⊆ R.

Algorithm 1: Pairing by persistence of x max .

x min := ∅; [x - max , x + max ] := cl R (CC([f ≤ f (x max )], x max )); if x - max > -∞ x + max < +∞ then x min -:= rep([x - max , x max ], f ); x min + := rep([x max , x + max ], f ); if x - max > -∞ && x + max < +∞ then x min := arg max x∈{xmin -,xmin + } f (x); if x - max > -∞ && x + max = +∞ then x min := x min -; if x - max = -∞ && x + max < +∞ then x min := x min + ; return x min ;
Let f : R → R be a D-Morse function with unique critical values, and let x max be a local maximum of f . Let us recall the 1D procedure [START_REF] Edelsbrunner | Persistent Homology -A survey[END_REF] which pairs (relatively to f ) local maxima to local minima (see Algorithm 1). Roughly speaking, the representatives x min and x min + are the abscissas where connected components of respectively

[f ≤ (f (x min -)] and [f ≤ (f (x min + )]
"emerge" (see Figure 4), when x max is the abscissa where two connected components of [f < f (x max )] "merge" into a single component of [f ≤ f (x max )]. Pairing by persistence associates then x max to the value x min belonging to {x min -, x min + } which maximizes f (x min ). The persistence of x max relatively to f is then equal to Per(x max ) := f (x max ) -f (x min ).

Pairing by dynamics (n-D)

From now on, f : R n → R is a Morse function with unique critical values. Let x min be a local minimum of f . Then we call set of descending paths starting from x min (shortly (D xmin )) the set of paths going from x min to some element

x < ∈ R n satisfying f (x < ) < f (x min ). The effort of a path Π : [0, 1] → R n (relatively to f ) is equal to: Effort(Π) := max ∈[0,1], ∈[0,1] (f (Π( )) -f (Π( ))).
A local minimum x min of f is said to be matchable if there exists some x < ∈ R n such that f (x < ) < f (x min ). We call dynamics of a matchable local minimum x min of f the value: (see the left side), which allows us to compute C sad , to deduce the components C I i whose closure contains x sad , and to decide which representative is paired with x sad by persistence by choosing the one whose height is the greatest. We can also observe (see the right side) the merge phase where the two components merge and where the component whose representative is paired with x sad dies. and we say that x min is paired by dynamics (see Figure 5) with some 1-saddle x sad ∈ R n of f when:

dyn(x min ) = min Π∈(Dx min ) max ∈[0,1] (f (Π( )) -f (x min )) ,
dyn(x min ) = f (x sad ) -f (x min ).
An optimal path Π opt is an element of (D xmin ) whose effort is equal to min Π∈(Dx min ) (Effort(Π)). Note that for any local minimum x min of f , there always exists some optimal path Π opt such that: Effort(Π opt ) = dyn(x min ).

Thanks to the uniqueness of critical values of f , there exists only one critical point of f which can be paired with x min by dynamics.

Dynamics are always positive, and the dynamics of an absolute minimum of f is set at +∞ (by convention).

Pairing by persistence (n-D)

Let us denote by clo the closure operator, which adds to a subset of R n all its accumulation points, and by CC(X) the connected components of a subset X of R n . We also define the representative of a subset X of R n relatively to a Morse function f the point which minimizes f on X:

rep(X) = arg min x∈X f (x).
Definition 1. Let f be some Morse function with unique critical values, and let x sad be the abscissa of some 1saddle point of f . Now we define the following expressions. First,

C sad = CC([f ≤ f (x sad )], x sad )
denotes the component of the set [f ≤ f (x sad )] which contains x sad . Second, we denote by:

{C I i } i∈I = CC([f < f (x sad )]) the connected components of the open set [f < f (x sad )].
Third, we define

{C sad i } i∈I sad = C I i | x sad ∈ clo(C I i )
the subset of components C I i whose closure contains x sad . Fourth, for each i ∈ I sad , we denote

rep i = arg min x∈C sad i f (x)
the representative of C sad i . Fifth, we define the abscissa

x min = rep i paired with i paired = arg max i∈I sad f (rep i ),
thus x min is the representative of the component C sad i of minimal depth. In this context, we say that x sad is paired by persistence to x min . Then, the persistence of x sad is equal to:

Per(x sad ) = f (x sad ) -f (x min ).
3 Sketches of the proofs (1D vs. n-D)

Pairing by dynamics implies pairing by persistence

Let us start from the 1D case (see Figure 7). We assume (see Table 1) that we have some Morse function f defined on the real line and that the critical values are unique, that is, for two different extrema x 1 , x 2 of f , we have f (x 1 ) = f (x 2 ). Furthermore, we assume that the abscissas {x min , x max } with x max > x min are paired by dynamics, that is, starting from x min and following the graph of f , the lower effort to reach a lower value is on the right side. Using these properties, we want to show that x max and x min are paired by persistence.

1D proof: Let us proceed in three steps. First, we want to show that x min is the representative of the basin [x - max , x max ] of level f (x max ) containing it. This is easily proven by contradiction: if x min is not the representative of this basin, there exists some x * in it where f (x * ) < f (x min ), and then the dynamics of x min is lower than f (x max ) -f (x min ), which is impossible by hypothesis. Now that we know that x min represents the basin [x - max , x max ], we can show that f (x min ) is greater than the image by f of the representative of [x max , x + max ] corresponding also to the lower threshold set [f ≤ f (x max )].

x max x max + -x min x max Fig. 7: Pairing by dynamics implies pairing by persistence in 1D: when x min (in black) is paired with x max (in purple) by dynamics, we observe easily that x min is the representative of the basin where it lies. Furthermore, the optimal path descending lower than f (x min ) goes on the right side and goes through x max (since we look for a minimal effort and f (x 2 max ) is greater than f (x max )). This implies that the right basin contains a representative lower than f (x min ). Since CC([f ≤ f (x max )], x max ) is made of the two described basins, we obtain easily that x max is paired with x min by persistence.

By assuming the contrary, we would imply that any descending path starting from x min would go outside the component

[x - max , x + max ] = CC([f ≤ f (x max ), x max ]
), which means that we would obtain a dynamics of x min greater than f (x max ) -f (x min ), which is impossible.

Since we have obtained that x min is the representative of the highest basin starting for the extrema x max , we can conclude easily that x max is paired with x min by persistence.

n-D proof: The proof in n-D, n ≥ 2, is very similar, except that we have more complex notations. Indeed, we study 1-saddles instead of maxima; the path between the two points is not "unique" anymore; and we do not have anymore a natural order between two abscissas.

We cannot define x - max and x + max , but instead we can define the closed connected component

CC([f ≤ f (x max )], x max ) containing x max . Also, we cannot de- fine ]x - max , x max [ or ]x max , x + max [ but instead we can define the connected components C I i which are com- ponents of [f < f (x sad )], and the components C sad i of [f < f (x sad )
] with the additional property that their closure contains x sad . Last point, we do not need anymore the condition that the studied function tends to infinity when the norm of the abscissa tends to infinity, but the consequence is that the proof is a little more complex.

After having introduced these notations, we can follow the same three steps as before. We first prove that Table 1: Sketches of the 1D/n-D proofs that pairing by dynamics implies pairing by persistence.

Hypotheses:

f is a D-Morse function f is a Morse function f has unique critical values
x min is a local minimum of f

x min and x max/sad are paired by dynamics

x max > x min x min = x sad Notations [x - max , x + max ] = cl R (CC([f ≤ f (x max )], x max )) C sad = CC([f ≤ f (x sad )], x sad ) {C I i } i∈I = CC([f < f (x sad )]) {C sad i } i∈I sad = C I i | x sad ∈ clo(C I i ) Step 1: ∃ i ∈ I s.t. x min ∈ C I i x min represents [x - max , x max ] with x min representing C I i (otherwise dyn(x min ) < f (x max/sad ) -f (x min
) which leads to a contradiction)

C I i belongs to {C sad i } i∈I sad then x min represents some C sad imin Step 2: f (rep([x max , x + max ], f )) < f (x min ) ∀ i = i min , f (rep(C sad i , f ) < f (x min ) (otherwise dyn(x min ) > f (x max/sad ) -f (x min ) which leads to a contradiction)
Step 3:

x min and x max/sad are paired by persistence x min , paired to x sad by dynamics, is the representative of some C I i (otherwise we would obtain that the dynamics of x min is lower than f (x sad ) -f (x min ) since we can reach a point on the graph of f which is lower than f (x min )). Then, the proof that this C I i is in fact one of the C sad i follows from the fact that otherwise, any descending path of x min must go out of C I i to reach a lower value than f (x min ), and then the dynamics of x min would be greater than f (x sad ) -f (x min ). Now that we know that x min belongs to some C sad i , we can use the property that there exists exactly two basins in the component CC([f ≤ f (x sad )], x sad ) (since we work with a Morse function). By assuming that x min is not the highest representative among the open components C sad i , we obtain one more time that any path starting from x min must go outside

CC([f ≤ f (x sad )], x sad )
to descend lower than f (x min ), which would lead to a greater dynamics than f (x sad ) -f (x min ). Thus, x min is the highest representative among the ones of the components {C sad i } i .

We conclude one more time that x sad is paired to x min by persistence when x min is paired to x sad by dynamics.

Pairing by persistence implies pairing by dynamics

We assume as usual that f is a Morse function (see Table 2), that its critical values are unique. Let us prove that when some maximum of f in the 1D case (or some 1-saddle of f in the n-D case) is paired by persistence to some minimum of this same function f , then this minimum is paired with this maximum (resp. this 1saddle) by dynamics.

1D proof: Let us start with the 1D case (see Figure 8). By considering that some maximum x max is paired with some minimum x min by persistence (with Table 2: Sketches of the 1D/n-D proofs that pairing by persistence implies pairing by dynamics.

Hypotheses:

f is a D-Morse function f is a Morse function f has unique critical values
x max/sad is a local maximum/1-saddle of f

x max/sad and x min are paired by persistence

x max > x min x min = x sad Notations: [x - max , x + max ] = CC([f ≤ f (x max )], x max ) C sad = CC([f ≤ f (x sad )], x sad ) x ∀ min = rep([x - max , x + max ], f ) {C I i } i∈I = CC([f < f (x sad )]) {C sad i } i∈I sad = C I i | x sad ∈ clo(C I i ) i min ∈ I sad s.t. x min represents C sad imin
Step 1:

γ := [x min , x ∀ min ] Card(I sad ) > 1 with f (x ∀ min ) < f (x min ) ⇒ ∃ i < ∈ I sad , ∃ x < ∈ C sad i< , s.t. f (x < ) < f (x min )
x min is matchable

Step 2:

γ is a descending path ∃ γ 1 from x min to x sad in C sad imin ∀ i ∈ I sad \ {i min }, ∃ γ 2 from x sad to x < ⇒ γ := γ 1 <> γ 2 is a descending path the dynamics of γ is equal to f (x max ) -f (x min ) ⇒ dyn(x min ) ≤ f (x max/sad ) -f (x min )
Step 3:

If x min is paired by dynamics with x * dyn(x min ) < f (x sad ) -f (x min ) (H) Then x * > x min ⇒ ∃ a descending γ from x min in C sad imin x < := inf{x > x min ; f (x) < f (x min )} ⇒ x min does not represent C sad imin x < > x max ⇒ (H) is false γ optimal path ⇒ {x min , x max , x < } ∈ γ dyn(x min ) ≥ f (x max/sad ) -f (x min )
Step 4:

dyn(x min ) = f (x max/sad ) -f (x min )
x max/sad and x min are paired by dynamics x min < x max ), we obtain at the same time several properties (by definition of the pairing by persistence):

we can draw the threshold set [f ≤ f (x max )] at level f (x max ), we know that it draws a connected component

CC([f ≤ f (x max )], x max ) containing x max that we can define as [x - max , x + max ] with x - max < x max < x + max , -we know then that x min is the representative of [x - max ,
x max ] and we can define some x ∀ min as being the representative of [x max , x + max ], with f (x min ) > f (x ∀ min ). . From these facts, we deduce easily that x min is matchable since f (x ∀ min ) < f (x min ). We also deduce that there exists a descending path from x min to x max to x ∀ min which lies inside [x - max , x + max ] and then its associated effort is equal to f (x max ) -f (x min ), which means that the dynamics of x min is lower than or equal to this same value. Additionally, we can show that every optimal path connects x min to x max and thus the dynamics of x min is greater than or equal to f (x max ) -f (x min ). It is then easy to conclude that the dynamics of x min is equal to f (x max ) -f (x min ), and then by uniqueness of the critical values, x min is paired with x max by dynamics. Now let us prove that x min is paired by dynamics to x max in four steps. First, we know that there exists some path γ :

[0, 1] → [x min , x ∀ min ] : λ → (1 -λ)x min + λx ∀ min ] joining x min to x ∀ min with f (x ∀ min ) < f (x min ), then x min is matchable.
Then, the second step is straightforward: since γ reaches some x ∀ min with an altitude lower than the one of x min , it is a descending path. Furthermore, the effort associated to γ is equal to f (x max ) -f (x min ), since we have to reach (x max , f (x max )) when we start from (x min , f (x min )) to be able to go down to

(x ∀ min , f (x ∀ min )).
Then the optimal effort associated to x min , that is the dynamics of x min , is lower than or equal to f (x max )f (x min ). Now, for the third step, we assume that x min is paired with some x * < x min , which is clearly impossible: otherwise dynamics of x min would be greater than f (x max )-f (x min ) (we would need to go outside the connected component [x - max , x + max ] to reach some altitude lower than f (x min )). Then x min is paired with some maximum x * greater than x min . Now, we define x < as the "first" abscissa of altitude lower than f (x min ) on the right side of x min ; obviously this abscissa is greater than x max since x min is the representative of the basin [x - max , x max ]. Since any optimal descending path starting from x min goes through the abscissas x min , x max and then x < , its associated effort is greater than or equal to f (x max ) -f (x min ).

The fourth step combines the previous properties and leads to the conclusion that the dynamics of x min is equal to f (x max ) -f (x min ), which means that the maxima associated to x min by dynamics is x max (by uniqueness of the critical values).

n-D proof: The main steps of the n-D proof are very similar to the 1D case. However, the notations are very different, due to the fact that the number of path from one point to another in R n is infinite (and there is no "left" nor "right"). Starting from the 1-saddle x sad paired by persistence to x min , we have to use the following notations:

we define the closed component

C sad = CC([f ≤ f (x sad )], x sad ), -we define also the open components {C imin } i of [f < f (x sad )],
whose subset {C sad i } i corresponds to these components whose closure contains x sad , we call i min the index of the component C sad imin that x min represents.

The first step consists of recalling that the number of components of C sad i is equal to two, then greater than one, and thus there exists some index i < and some abscissa x < ∈ C sad i< such that f (x < ) < f (x sad ) (since pairing by persistence associates x sad to the local minimum of the highest altitude). Thus, x min is matchable.

As a second step, we construct a path γ 1 from x min to x sad in C sad imin and another path γ 2 from x sad to x < in the component C sad i containing it, from which we deduce a descending path γ := γ 1 <> γ 2 associated to x min . Thus, the effort associated to γ is lower than or equal to f (x sad ) -f (x min ) (since this path has not yet been shown to be optimal). The third step uses a proof by contradiction. We assume that the dynamics of x min is lower than f (x sad )f (x min ); we call this hypothesis HYP. Then, HYP implies that there exists a descending path inside the component C sad imin , which implies that x min does not represent C sad imin , which is impossible (it contradicts the hypotheses). Then, the dynamics of x min is greater than or equal to f (x sad ) -f (x min ). As for the 1D case, the fourth steps concludes: since the dynamics of x min is equal to f (x sad ) -f (x min ) thanks to the combination of the previous steps, the only possible local maximum paired by dynamics to x min is x sad .

Pairings by dynamics and by persistence are equivalent in 1D

In this section, we prove that under some constraints, pairings by dynamics and by persistence are equivalent in the 1D case.

Proposition 1. Let f : R → R be a D-Morse function with unique critical values. Now, let us assume that a local minimum x min ∈ R of f is paired with a local maximum x max of f by dynamics. We assume without loss of generality that x min < x max (the reasoning is the same for the opposite assumption). Also, we denote by (x - max , x + max ) ∈ R 2 the two values verifying:

[x - max , x + max ] = cl R (CC([f ≤ f (x max )], x max )).
Then the following properties are true: Let us prove (P 1); we proceed by reductio ad absurdum. When x min is not the lowest local minimum of f on the interval [x - max , x max ], then there exists another local minimum x * ∈ [x - max , x max ] of f (see Figure 10) which satisfies f (x * ) < f (x min ) (x * and x min being distinct local extrema of f , their images by f are not equal). Then, because the path joining x min and x * belongs to C (defined in Subsection 2.2), we have:

(P1) x min = rep([x - max , x max ], f ), (P2) When x + max is finite, x min 2 := rep([x max , x + max ], f ) satisfies f (x min 2 ) < f (x min ), (
dyn(x min ) ≤ max{f (x) -f (x min ) ; x ∈ iv(x * , x min )}. Let us call x * * := arg max x∈[iv(xmin,x * )] f (x), we can de- duce that f (x * * ) < f (x max ) since x * * ∈ iv(x * , x min ) ⊆ ]x - max , x max [. In this way, dyn(x min ) ≤ f (x * * ) -f (x min ),
which is lower than f (x max ) -f (x min ); this is a contradiction since x min and x max are paired by dynamics.

(P 1) is then proved. Now let us prove (P 2). Let us assume that x + max is finite and let x min 2 be the representative of [x max , x + max ] relatively to f . Let us assume that f (x min 2 ) > f (x min ). Note that we cannot have equality of f (x min 2 ) and f (x min ), since x min and x min 2 are both local extrema of f . Then we obtain Figure 11. Since with x ∈ [x max , x + max ], we have f (x) ≥ f (x min 2 ) > f (x min ), and because x min is paired with x max by dynamics with x min < x max , then there exists a value x on the right of x max where f (x) is lower than f (x min ). In other words, there exists:

x < := inf{x ∈ [x max , +∞[ ; f (x) < f (x min )}
such that for some arbitrarily small value ε > 0, f (x < + ε) < f (x min ). Since x < > x + max , any path γ joining x min to x < goes through a local maximum x 2 max defined by

x 2 max := arg max x∈[x + max ,x < ] f (x)
which satisfies f (x 2 max ) > f (x + max ). Then the dynamics of x min is greater than or equal to f (x 2 max ) -f (x min ) which is greater than f (x max ) -f (x min ). We obtain a contradiction. Then we have f (x min 2 ) < f (x min ). The proof of (P 2) is done.

Thanks to (P 1) and (P 2), we obtain directly (P 3) by applying Algorithm 1. Proposition 2. Let f : R → R be a D-Morse function with unique critical values. Now, let us assume that a local minimum x min ∈ R of f is paired with a local maximum x max of f by persistence. We assume without loss of generality that x min < x max (the reasoning is the same for the opposite assumption). Then, x max and x min are paired by dynamics.

Proof: We denote by (x - max , x + max ) ∈ R 2 the two values verifying:

[x - max , x + max ] = cl R (CC([f ≤ f (x max )], x max )).
Since x min is paired by persistence to x max with x min < x max (see Figure 12), then:

x min = rep([x - max , x max ], f ) ∈ R,
and, by Algorithm 1, we know that x - max > -∞ (then x - max is finite). When x + max < +∞ (Case 1), the representative x ∀ min of [x max , x + max ] relatively to f is exists in ]x max , x + max [ and is unique, and its image by f is lower than f (x min ). When x + max = +∞ (Case 2), lim x→+∞ f (x) = -∞, and then there exists one more time an abscissa x ∀ min ∈ R whose image by f is lower than f (x min ). So, in both cases, there exists a (finite) value

x ∀ min ∈]x max , x + max [ verifying f (x ∀ min ) < f (x min )
. This way, we know that x min is paired with some abscissa in R by dynamics.

In Case 1, we know that the path defined as:

γ : λ ∈ [0, 1] → γ(λ) := (1 -λ)x min + λx ∀ min
belongs to the set of paths C defining the dynamics of x min (see Subsection 2.2). Then,

dyn(x min ) ≤ max{f (x) -f (x min ) ; x ∈ γ([0, 1])}, which is lower than or equal to f (x max ) -f (x min ) since f is maximal at x max on [x - max , x + max ].
Then we have the following property:

dyn(x min ) ≤ f (x max ) -f (x min ).
In Case 2, since f (x) is lower than f (x max ) for x ∈ ]x max , +∞[, then one more time we get dyn(x min ) ≤ f (x max ) -f (x min ). Let us call this property (P ).

Even if we know that there exists some local maximum of f which is paired with x min by dynamics, we do not know whether the abscissa of this local maximum is lower than or greater than x min . Then, let us assume that there exists a local maximum x * < x min (lower case) which is associated to x min by dynamics. We denote this property (H) and we depict it in Figure 13. Since f (x) is greater than or equal to f (x min ) for x ∈ [x - max , x min ], (H) implies that x * < x - max . Then, we can observe that the local maximum x 1 of f of maximal abscissa in [x * , x - max ] satisfies f (x 1 ) > f (x max ), which implies that dyn(x min ) ≥ f (x 1 ) -f (x min ) > f (x max ) -f (x min ) (since we go through x 1 to reach x * ), which contradicts (P ). (H) is then false. In other words, we are in the upper case: the local maximum paired by dynamics to x min belongs to ]x min , +∞[, let us call this property (P ). Now let us define:

x < := inf{x > x min ; f (x) < f (x min )}, x min x max x max - ℝ ℝ x *
x¹ Fig. 13: The proof that it is impossible to obtain a local maximum x * < x min paired with x min by dynamics when x min is paired with x max > x min by persistence.

(see again Figure 12) and let us remark that x < > x max (because x min is the representative of f on [x - max , x max ]). Since we know by (P ) that a local maximum x > x min of f is paired by dynamics with x min , then the image of every optimal path belonging to C contains {x < }, and then contains [x min , x < ]. Indeed, an optimal path in C whose image would not contain {x < } would then contain an abscissa x < x - max and then we would obtain dyn(x min ) > f (x max ) -f (x min ), which would contradict (P ). Now, the maximal value of f on [x min , x < ] is equal to f (x max ), then dyn(x min ) = f (x max )-f (x min ). The only local maximum of f whose value is f (x max ) is x max , then x max is paired with x min by dynamics relatively to f . Theorem 1. Let f : R → R be a D-Morse function with a finite number of local extrema and unique critical values. A local minimum x min ∈ R of f is paired by dynamics to a local maximum x max ∈ R of f iff x max is paired by persistence to x min . In other words, pairings by dynamics and by persistence lead to the same result. Furthermore, we obtain Per(x max ) = dyn(x min ).

Proof: This theorem results from Propositions 1 and 2.

Note that pairing by persistence has been proved to be symmetric in [START_REF] Cohen-Steiner | Extending persistence using Poincaré and Lefschetz duality[END_REF] for Morse functions defined on manifolds: the pairing is the same for a Morse function and its negative. Fig. 14: Every optimal descending path goes through a 1-saddle. Observe the path in blue coming from the left side and decreasing when following the topographical view of the Morse function f . The effort of this path to reach the minimum of f is minimal thanks to the fact that it goes through the saddle point at the middle of the image.

The n-D equivalence

Let us make two important remarks that will help us in the sequel. Lemma 2. Let f : R n → R be a Morse function and let x min be a local minimum of f . Then for any optimal path Π opt in (D xmin ), there exists some * ∈]0, 1[ such that it is a maximum of f • Π opt and at the same time Π opt ( * ) is the abscissa of a 1-saddle point of f . Proof : This proof is depicted in Figure 14. Let us proceed by counterposition, and let us prove that when a path Π in (D xmin ) does not go through a 1-saddle of f , it cannot be optimal.

Let Π be a path in (D xmin ). Let us define * ∈ [0, 1] as one of the positions where the mapping f • Π is maximal:

* ∈ arg max ∈[0,1] f (Π( )),
and x * = Π( * ). Let us prove that we can find another path Π in (D xmin ) whose effort is lower than the one of Π.

At x * , f can satisfy three possibilities:

-When we have ∇f (x * ) = 0 (see the left side of Figure 15), then locally f is a plane of slope ∇f (x * ) , and then we can easily find some path Π in (D xmin ) with a lower effort than Effort(Π). More precisely, let us fix some arbitrary small value ε > 0 and draw the closed topological ball B(x * , ε), we can define three points: Thanks to these points, we can define a new path Π :

min = min{ | Π( ) ∈ B(x * , ε)}, max = max{ | Π( ) ∈ B(x * , ε)}, x B = x * -ε. ∇f (x * ) ∇f (x * ) .
Π| [0, min ] <> [Π( min ), x B ] <> [x B , Π( max )] <> Π| [ max,1] .
By doing this procedure at every point in [0, 1] where f • Π reaches its maximal value, we obtain a new path whose effort is lower than the one of Π. -When we have ∇f (x * ) = 0, then we are at a critical point of f . It cannot be a 0-saddle, that is, a local minimum, due to the existence of the descending path going through x * . It cannot be a 1-saddle neither (by hypothesis). It is then a k-saddle point with k ∈ [2, n] (see the right side of Figure 15). Using Lemma 1, f is locally equal to a second order polynomial function (up to a change of coordinates ϕ s.t. ϕ(x * ) = 0):

f •ϕ -1 (x) = f (x * )-x 2 1 -x 2 2 -• • •-x 2 k +x 2 k+1 +• • •+x 2 n .
Now, let us define for some arbitrary small value ε > 0:

min = min{ | Π( ) ∈ B(0, ε)}, max = max{ | Π( ) ∈ B(0, ε)},
and

B =    x i∈[1,k] x 2 i ≤ ε 2 and ∀j ∈ [k + 1, n], x j = 0    \{0}.
This last set is connected since it is equal to a kmanifold (with k ≥ 2) minus a point. Let us assume without loss of generality that Π( min ) and Π( max ) belong to B (otherwise we can consider their orthogonal projections on the hyperplane of lower dimension containing B but the reasoning is the same). Thus, there exists some path Π B joining Π( min ) to Π( max ) in B, from which we can deduce the path

Π = Π| [0, min] <> Π B <> Π| [ max ,1]
whose effort is lower than the one of Π since its image is inside [f < f (x * )]. Since we have seen that, in any possible case, Π is not optimal, it concludes the proof. Proposition 3. Let f be a Morse function from R n to R with n ≥ 1. When x * is a critical point of index 1, then there exists ε > 0 such that:

Card (CC(B(x * , ε) ∩ [f < f (x * )])) = 2,
where Card is the cardinality operator.

Proof : The case n = 1 is obvious, let us then treat the case n ≥ 2 (see Figure 16). Thanks to Lemma 1 and thanks to the fact that x sad is the abscissa of a 1saddle, we can say that (up to a change of coordinates and in a small neighborhood around x sad ) for any x:

f (x) = f (x sad ) + x T .
-1 0 0 I n-1 .x, where I n-1 is the identity matrix of dimension (n -1) × (n -1). In other words, around x sad , we obtain that:

[f < f (x sad )] = x -x 2 1 + n i=2 x 2 i < 0 = C + ∪ C -,
with: We assume that the 1-saddle point of f whose abscissa is x sad is paired by persistence to a local minimum x min of f . Then, x min is paired by dynamics to x sad .

C + =    x x 1 > n i=2 x 2 i    , and
C -=    x x 1 < - n i=2
Proof : Let us assume that x sad is paired by persistence to x min , then we have the hypotheses described in Definition 1. Let us denote by C min the connected component in {C i } i∈I sad satisfying that x min = rep(C imin ). Since x sad is the abscissa of a 1-saddle, by Proposition 3, we know that Card(I sad ) = 2, then there exists: x < = rep(C < ) with C < the component C i with i ∈ I \ {i min }, then x min is matchable. Let us assume that the dynamics of x min satisfies: dyn(x min ) < f (x sad ) -f (x min ). (HYP) This means that there exists a path Π < in (D xmin ) such that: max

∈[0,1] f (Π < ( )) -f (x min ) < f (x sad ) -f (x min ),
that is, for any ∈ [0, 1], f (Π < ( )) < f (x sad ), and then by continuity in space of Π < , the image of [0, 1] by Π < is in C min . Because Π < belongs to (D xmin ), there exists then some x < ∈ C min satisfying f (x < ) < f (x min ). We obtain a contradiction, (HYP) is then false. Then, we have dyn(x min ) ≥ f (x sad ) -f (x min ).

Because for any i ∈ I sad , x sad is an accumulation point of C i in R n , there exist a path Π m from x min to x sad such that:

∀ ∈ [0, 1],Π m ( ) ∈ C sad , ∀ ∈ [0, 1[,Π m ( ) ∈ C min .
In the same way, there exists a path Π M from x < to x sad such that:

∀ ∈ [0, 1],Π M ( ) ∈ C sad , ∀ ∈ [0, 1[,Π M ( ) ∈ C < .
We can then build a path Π which is the concatenation of Π m and → Π M (1-), which goes from x min to x < and goes through x sad . Since this path stays inside C sad , we know that Effort(Π) ≤ f (x sad ) -f (x min ), and then dyn(x min ) ≤ f (x sad ) -f (x min ).

By grouping the two inequalities, we obtain that dyn(x min ) = f (x sad ) -f (x min ), and then by uniqueness of the critical values of f , x min is then paired by dynamics to x sad . We assume that the local minimum x min of f is paired by dynamics to a 1-saddle of f of abscissa x sad . Then, x sad is paired by persistence to x min .

Proof : Let us assume that x min is paired to x sad by dynamics. Let us recall the usual framework relative to persistence:

C sad = CC([f ≤ f (x sad )], x sad ), {C I i } i∈I = CC([f < f (x sad )]), {C sad i } i∈I sad = C I i |x sad ∈ clo(C I i ) , ∀i ∈ I sad , rep i = arg min x∈C sad i f (x).
By Definition 1, x sad is paired to the representative rep i of C sad i which maximizes f (rep i ).

1. Let us show that there exists some index i min such that x min is the representative of a component C sad imin of {C sad i } i∈I sad . (a) First, x min is paired by dynamics with x sad and dyn(x min ) is greater than zero, then f (x sad ) > f (x min ), then x min belongs to [f < f (x sad )], then there exists some i min ∈ I such that x min ∈ C imin (see Equation (2) above). (b) Now, if we assume that x min is not the representative of C imin , there exists then some x < in C imin satisfying that f (x < ) < f (x min ), and then there exists some Π in (D xmin ) whose image is contained in C imin . In other words, dyn(x min ) ≤ Effort(Π) < f (x sad ) -f (x min ), which contradicts the hypothesis that x min is paired with x sad by dynamics. (c) Let us show that i min belongs to I sad , that is,

x sad ∈ clo(C imin ). Let us assume that:

x sad ∈ clo(C imin ). (HYP2)
Every path in (D xmin ) goes outside of C imin to reach some point whose image by f is lower than f (x min ) since x min has been proven to be the representative of C imin . Then this path intersects the boundary ∂ of C imin . Since by (HYP2), x sad does not belong to the boundary ∂ of C imin , any optimal path Π * in (D xmin ) goes through one 1-saddle x sad2 = arg max ∈[0,1] f (Π * ( )) (by Lemma 2) different from x sad and satisfying then f (x sad2 ) > f (x sad ). Thus, dyn(x min ) > f (x sad )f (x min ), which contradicts the hypothesis that x min is paired with x sad by dynamics. Then, we have:

x sad ∈ clo(C imin ).
2. Now let us show that f (x min ) > f (rep(C sad i )) for any i ∈ I sad \ {i min }. For this aim, we prove that there exists some i ∈ I sad such that f (rep(C sad i )) < f (x min ) and we conclude with Proposition 3. Let us assume that the representative r of each component C sad i except C min satisfies f (r) > f (x min ), then any path Π of (D xmin ) has to go outside C sad to reach some point whose image by f is lower than f (x min ). We obtain the same situation as before (see (1.c)), and then we obtain that the effort of Π is greater than f (x sad ) -f (x min ), which leads to a contradiction with the hypothesis that x min is paired with x sad by dynamics. We have then that there exists i ∈ I sad such that f (rep(C sad i )) < f (x min ). Thanks to Proposition 3, we know then that x min is the representative of the components of [f < f (x sad )] whose image by f is the greatest. 3. It follows that x sad is paired with x min by persistence.

Perspectives: a research program linking Topological Data Analysis and MM

This paper is a step towards exploring the possible interactions between Topological Data Analysis (TDA) and MM. In this section, we detail some ideas for a research program linking these two fields. As a very first example, let us look at Fig. 17, which provides an illustration of an image analysis pipeline originally performed in the context of topological data analysis using the library called Topology Toolkit [START_REF] Tierny | The topology toolkit[END_REF][START_REF] Talha Bin Masood | An overview of the topology toolkit[END_REF] (shortly TTK). In the original publication [START_REF] Lukasczyk | Localized topological simplification of scalar data[END_REF], the steps are the following 1. The original data (microscopy image of cells and their nuclei) are simplified with a small threshold of persistence (Fig. 17 Thanks to the result of this paper and some previous work, we can translate this process in mathematical morphology. The filtering by persistence belong to a class of morphological filters called connected filters [START_REF] Salembier | Connected operators[END_REF], with a criterion named dynamics. The Morse-Smale complexe is replaced by the watershed [START_REF] Čomić | Morse-smale decompositions for modeling terrain knowledge[END_REF][START_REF] Čomić | Computing a discrete Morse gradient from a watershed decomposition[END_REF]. The persistence curve is called a granulometric curve [START_REF] Matheron | Random sets theory and its applications to stereology[END_REF]. Hence, from a morphological perspective, the same example can be done using Higra [START_REF] Perret | Higra: Hierarchical graph analysis[END_REF], a (morphological) library that computes the various steps, and this leads to the following description.

1. A connected filter with a small dynamics threshold is first applied on the original data (Fig. It is worthwhile to explore the differences between the two approaches. In mathematical morphology, there is no persistence diagram. On the other hand, there exist saliency maps [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF][START_REF] Najman | On the equivalence between hierarchical segmentations and ultrametric watersheds[END_REF][START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF]. Intuitively, a saliency map can be obtained by filtering the original image/data for all values of the criterion (here, dynamics), and stacking (summing) the watersheds of all the filtered images. A contour that is persistent is present many times in the stack, and has a high value in the resulting saliency map. Fig. 18 shows the saliency map of the original data of Fig. 17 for the dynamics criterion.

In TDA, only a few criteria other than dynamics have been studied [START_REF] Carr | Simplifying flexible isosurfaces using local geometric measures[END_REF] but MM has many more, see [1] for a few of them. There exist also several ways to simplify using non-increasing criteria [START_REF] Salembier | Antiextensive connected operators for image and sequence processing[END_REF][START_REF] Erik R Urbach | Connected shape-size pattern spectra for rotation and scale-invariant classification of gray-scale images[END_REF][START_REF] Xu | Connected filtering on tree-based shape-spaces[END_REF].

The links between Morse-Smale Complex and watershed [START_REF] Čomić | Morse-smale decompositions for modeling terrain knowledge[END_REF][START_REF] Čomić | Computing a discrete Morse gradient from a watershed decomposition[END_REF] need to be explored, specifically in the Fig. 17: An example of segmentation of a microscopy image of cells and their nuclei [START_REF] Lukasczyk | Localized topological simplification of scalar data[END_REF] with the topological data analysis framework. The very same example can be seen as an application of the morphological data analysis framework (see text). Fig. 18: Saliency map corresponding to Fig. 17. In this image, the contours that are the more persistent are darker than the others (see text for details.) context of Discrete Morse Theory [START_REF] Forman | Morse Theory for cell complexes[END_REF]. We envision doing such a study based on watershed cuts [START_REF] Cousty | Watershed cuts: Minimum spanning forests and the drop of water principle[END_REF], see also [START_REF] Cousty | Collapses and watersheds in pseudomanifolds of arbitrary dimension[END_REF] that highlights some links between the watershed and topology.

Many other comparisons should be done. To mention one of those, the contour tree [START_REF] Freeman | On searching a contour map for a given terrain elevation profile[END_REF] from TDA is closely related to the tree of shapes [START_REF] Caselles | Geometric Description of Images as Topographic Maps[END_REF] from MM. Comparing those trees and the algorithms for computing them from TDA [START_REF] Carr | Computing contour trees in all dimensions[END_REF][START_REF] Gueunet | Task-based augmented contour trees with fibonacci heaps[END_REF] and from MM [START_REF] Géraud | A quasi-linear algorithm to compute the tree of shapes of n-D images[END_REF][START_REF] Crozet | A first parallel algorithm to compute the morphological tree of shapes of nd images[END_REF][START_REF] Carlinet | The tree of shapes turned into a max-tree: a simple and efficient linear algorithm[END_REF] would be rewarding. In particular, the morphological algorithms for computing the tree of shapes, which are quasi-linear whatever the dimension of the space, are based on the ones for computing the tree of upper or lower level sets, called the component trees [START_REF] Carlinet | A comparative review of component tree computation algorithms[END_REF], and seem more efficient than the ones from TDA.

Conclusion

In this paper, we have proved that persistence and dynamics lead to the same pairings in n-D, n ≥ 1, which implies that they are equal whatever the dimension. Concerning the future works, we propose to investigate the relationship between persistence and dynamics in the discrete case [START_REF] Forman | Morse Theory for cell complexes[END_REF] (that is, on complexes). We will also check under which conditions pairings by persistence and by dynamics are equivalent for functions that are not Morse. Furthermore, we will examine if the fast algorithms used in MM like watershed cuts, Betti numbers computations or attribute-based filtering are applicable to PH. Conversely, we will study if some PH concepts can be seen as the generalization of some MM concepts (for example, dynamics seems to be a particular case of persistence).

More generally, we believe that exploring the links and differences between TDA and MM would benefit to the two communities.

A Ambiguities occurring when values are not unique Fig. 19: Ambiguities can occur when critical values are not unique for pairing by dynamics and for pairing by persistence.

As depicted in Figure 19, the abscissa of the blue point can be paired by persistence to the abscissas of the orange and/or the red points. The same thing appears when we want to pair the abscissa of the pink point to the abscissas of the green and/or blue points. This shows how much it is important to have unique critical values on Morse functions. This point is discussed in detail in [START_REF] Bertrand | On the dynamics[END_REF], where it is shown that a strict total order relation on the set of minima allows for a good definition of the dynamics.

Fig. 5 :

 5 Fig. 5: Pairing by dynamics on a Morse function: the red and blue paths are both in (D xmin ) but only the blue one reaches a point x < whose height is lower than f (x min ) with a minimal effort.

Fig. 6 :

 6 Fig.6: Pairing by persistence on a Morse function: we compute the plane whose height is reaching f (x sad ) (see the left side), which allows us to compute C sad , to deduce the components C I i whose closure contains x sad , and to decide which representative is paired with x sad by persistence by choosing the one whose height is the greatest. We can also observe (see the right side) the merge phase where the two components merge and where the component whose representative is paired with x sad dies.

2 ℝFig. 9 :

 29 Fig. 9: A D-Morse function where the local extrema x min and x max are paired by dynamics.

  P3) x max and x min are paired by persistence. Proof: Figure 9 depicts an example of D-Morse function where x min and x max are paired by dynamics.

Fig. 10 :

 10 Fig. 10: Proof of (P 1).

Fig. 11 :

 11 Fig.11: Proof of (P 2) in the case where x + max is finite.

Fig. 12 :

 12 Fig. 12: A D-Morse function f : R → R where the local extrema x min and x max are paired by persistence relatively to f .

Fig. 15 :

 15 Fig.15: How to compute descending paths of lower efforts. The initial path going through x * (the little gray ball) is in red, the new path of lower effort is in green (the non-zero gradient case is on the left side, the zerogradient case is on the right side).

Fig. 16 :

 16 Fig. 16: A 1-saddle point leads to two open connected components. At a 1-saddle point whose abscissa is x sad (at the center of the image), the component [f ≤ f (x sad )] is locally the merge of the closure of two connected components (in orange) of [f < f (x sad )] when f is a Morse function.

, 2 ,

 2 where C + and C -are two open connected components of R n . Indeed, for any pair (M, M ) of C + , we have x from which we define N = (x M 1 , 0, . . . , 0) T ∈ C + and N = (x M 1 , 0, . . . , 0) T ∈ C + from which we deduce the path[M, N ] <> [N, N ] <> [N , M ] joining M to M in C + .The reasoning with C -is the same. Since C + and C -are two connected (separated) disjoint sets, the proof is done.

5. 1 Theorem 2 .

 12 Pairing by persistence implies pairing by dynamics in n-D Let f be a Morse function from R n to R.

5. 2 Theorem 3 .

 23 Pairing by dynamics implies pairing by persistence in n-D Let f be a Morse function from R n to R.

.a) 2 .

 2 The Morse-Smale complex leads to an oversegmentation ((Fig.17.b) 3. The persistence curve (Fig.17

  .c) is the number of persistent pairs as a function of their persistence. The vertical dashed line on the left corresponds to the level of simplification of Fig. 17.a and b. The vertical dashed line on the right corresponds to the level of simplification of Fig. 17.e and f. 4. The diagram of persistence (Fig. 17

.d) 5 .

 5 The image is simplified (Fig.17

  .e) with a threshold corresponding to the vertical dashed line on the right of Fig.17

.c. 6 .

 6 The Morse-Smale complex separatrices of Fig.17

  .e provides 1 maximum per nuclei, while the nuclei are the maxima of the same image (Fig.17

  .f).

.a) 2 .

 2 The watershed of Fig.17

  .a is oversegmented (see Fig.17.b) 3. The granulometric curve (Fig.17

  .c) provides the number of maximum as a function of the dynamics 4. A connected filter of Fig.17

  .a with a dynamics threshold corresponding to the vertical dashed line on the right of Fig.17.c leads to Fig.17

.e. 5 .

 5 The watershed of Fig.17

  .e gives one region per cell, while the nuclei are the maxima of the same image (Fig.17.f).

  set of f which contains x max . By definition of pairing by persistence, we know that the representative of the component [x - max , x max ] is x min drawn in purple (since x min < x max ) and we call x ∀ min (drawn in red) the representative of the component [x max , x + max ]

	x min	x max	x min \/
	-x max		x max +
	Fig. 8: Pairing by persistence implies pairing by dy-
	namics in 1D: starting from the local maximum x max
	(in black), we define the component [x -max , x + max ] of
	the lower threshold

Acknowledgements

The authors would like to thank Julien Tierny for many interesting discussions and for providing us Fig. 17

.