Jesús Angulo
email: jesus.angulo@mines-paristech.fr

and mathematical morphology

Keywords: Minkowski addition, adjunction, HamiltonJacobi PDE, fuzzy morphology, viscous morphology

( max , min )-convolution and Mathematical

Morphology

Introduction

Let E be the Euclidean R n or discrete space Z n (support space) and let T be a set of grey-levels (space of values). For theoretical reasons it is typically assumed that T = R = R ∪ {-∞, +∞}, but one often has T = [0, M ]. A grey-level image is represented by a function f : E → T , also noted as f ∈ F(E, R), such that f maps each pixel x ∈ E into a grey-level value in T . Given a grey-level image, the two basic morphological mappings F(E, T ) → F(E, T ) are the dilation and the erosion given respectively by

(f ⊕ b)(x) = sup y∈E {f (y) + b(x -y)} , (f b)(x) = inf y∈E {f (y) -b(y -x)} , (1)
where b ∈ F(E, T ) is the structuring function which determines the eect of the operator. The other morphological operators, such as the opening and the closing, are obtained by composition of dilation/erosion [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Heijmans | Morphological image operators[END_REF]. The Euclidean framework has been recently generalized to images supported on Riemannian manifolds [START_REF] Angulo | Riemannian Mathematical Morphology[END_REF]. Operators (1) can be interpreted in nonlinear mathematics as the convolution in (max, +)-algebra (and in its dual algebra) [START_REF] Gondran | Graphs, Dioids and Semirings: New Models and Algorithms[END_REF]. This inherent connection of functional operators [START_REF] Alvarez | Hopf-lax formulas for semicontinuous data[END_REF] with the supremal and inmal convolution of nonlinear mathematics and convex analysis has been extremely fruitful to the state-of-the-art on mathematical morphology (morphological PDE, slope transform, etc.). Nevertheless, the functional operators [START_REF] Alvarez | Hopf-lax formulas for semicontinuous data[END_REF] do not extend all the fundamental properties of the dilation and erosion for sets, as formulated in Matheron's theory. Perhaps the most disturbing for us are, on the one hand, the lack of commutation with level set processing for nonat structuring functions; on the other hand, the limitation of Matheron's axiomatic of granulometry to constant (i.e., at) functions on a convex domain [START_REF] Kraus | Gray-scale granulometries compatible with spatial scalings[END_REF]. In addition, there are some unconventional morphological frameworks, such as the fuzzy morphology [START_REF] Deng | Grey-Scale Morphology Based on Fuzzy Logic[END_REF][START_REF] Maragos | Lattice image processing: a unication of morphological and fuzzy algebraic systems[END_REF][START_REF] Bloch | Duality vs. adjuntion for fuzzy mathematical morphology and general form of fuzzy erosions and dilations[END_REF] or the viscous morphology [START_REF] Vachier | The Viscous Watershed Transform[END_REF][START_REF] Vachier | News from Viscous Land[END_REF][START_REF] Maragos | A PDE Formulation for Viscous Morphological Operators with Extensions to Intensity-Adaptive Operators[END_REF] which do not t in the classical (max, +)algebra. Actually, the (max, +) is not the unique possible alternative to see morphological operators as convolutions. The idea in this paper is to consider the operation of convolution of two functions in the (max, min)-algebra. This is in fact our main motivation: to formally introduce the notion of (max, min)mathematical morphology. As we show in the paper, this framework is not totally new in morphology since some fuzzy morphological operators are exactly the same convolutions that we introduce. But some of the key properties are ignored by in the fuzzy context, and the most important, they are not limited to fuzzy sets. By the way, even if much less considered than the supremal and inmal convolutions, convolutions in (max, min)-algebra have been the object of various studies in dierent branches of nonlinear applied mathematics, from quasi-convex analysis [START_REF] Volle | The Use of Monotone Norms in Epigraphical Analysis[END_REF][START_REF] Seeger | On a convolution operation obtained by adding level sets: classical and new results[END_REF][START_REF] Volle | Duality for the Level Sum of Quasiconvex Functions and Applications[END_REF][START_REF] Gondran | Analyse MINMAX[END_REF][START_REF] Luc | Levels sets Inmal Convolution and Level Addition[END_REF][START_REF] Penot | Approximation of Functions and Sets[END_REF] to viscosity solutions of Hamilton-Jacobi equations [START_REF] Barron | Hopf-Lax formula for ut = H(u, Du) = 0[END_REF][START_REF] Barron | Hopf-Lax formula for ut = H(u, Du) = 0[END_REF][START_REF] Alvarez | Hopf-lax formulas for semicontinuous data[END_REF][START_REF] Van | Hopf-Lax-Oleinik-Type Estimates for Viscosity Solutions to Hamilton-Jacobi Equations with Concave-Convex Data[END_REF]. Interested reader is also referred to the book [START_REF] Gondran | Graphs, Dioids and Semirings: New Models and Algorithms[END_REF] for a systematic comparative study of matrix algebra and calculus in the three algebras (+, ×), (max, +) and (max, min).

The present work is exclusively a theoretical study and thus the practical interest of the operators is not illustrated here. Complete proofs and additional results can be found in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF].

We use the following representation of semicontinuous functions. Given an upper semicontinuous (USC) function f ∈ F(E, R), it can be dened by means of its upper level sets

X + h (f ) as follows f (x) = sup h ∈ R : x ∈ X + h (f ) , or by its strict lower level sets Y - h (f ): f (x) = inf h ∈ R : x ∈ Y - h (f ) , where X + h (f ) = {x ∈ E : f (x) ≥ h} , and Y + h (f ) = {x ∈ E : f (x) > h} ; X - h (f ) = {x ∈ E : f (x) ≤ h} , and Y - h (f ) = {x ∈ E : f (x) < h} .
A continuous function f can be decomposed/reconstructed using either its (strict) upper level sets or its (strict) lower level sets. One has

X + h (f ) c = Y - h (f ).
(f b)(x) = sup y∈R n {f (y) ∧ b(x -y)} , (2)
(f b)(x) = inf y∈R n {f (y) ∨ b c (y -x)} . (3)
We also dene the adjoint infmax f * b and the adjoint supmin f * b convolutions as

(f * b)(x) = inf y∈R n {f (y) ∧ * b(y -x)} , (4)
(f * b)(x) = sup y∈R n {f (y) ∨ * b c (x -y)} , (5)
where ∧ * is the adjoint operator to the minimum ∧ and is given by

f (y) ∧ * b(y -x) = f (y) if b(y -x) > f (y) if b(y -x) ≤ f (y) (6)
and ∨ * the adjoint to ∨:

f (y) ∨ * b c (x -y) = f (y) if b c (x -y) < f (y) ⊥ if b c (x -y) ≥ f (y) (7)
and where, if we dene max g = sup x∈R n g(x) and min g = inf x∈R n g(x), the top and bottom elements for pair of functions f and b correspond to

= (max f ) ∨ (max b) and ⊥ = (min f ) ∧ (min b c ).
Denitions remain valid if we replace R n by a subset E or any subset of discrete space Z n . Similarly, the extended real line R can be replaced by a bounded, eventually discrete, set of intensities [0, M ]. Figure 1 illustrates the four (max, min)-convolutions for a given example of one dimensional functions dened in a bounded interval, i.e., f, b ∈ F(R, [0, M ]).

Duality by complement vs. duality by adjunction. From a morphological viewpoint, their most salient properties are summarized in this proposition (proof in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF]). Proposition 1. The supmin convolution and infmax convolution are dual with respect to the complement. Similarly, the adjoint infmax convolution * and the adjoint supmin * convolution are dual with respect to the complement, i.e., for

f, b ∈ F(R n , R) one has f b = f c b c and f b = f c b c (8) f * b = f c * b c and f * b = f c * b c (9)
The pair ( * , ) forms an adjunction. Similarly, the pair ( , * ) is also an adjunction, i.e., for f, g, b ∈ F(R n , R) one has Commutation with level set processing. We can introduce now the fundamental property of (max, min)-convolutions (proof in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF]). Proposition 2. Let f and b in F(R n , R). Then the four (max, min)-convolutions of f by b obey the following commutation rules of level sets with respect to Minkowski sum and substraction: for all h ∈ R

f b ≤ g ⇐⇒ f ≤ g * b (10) f * b ≤ g ⇐⇒ f ≤ g b (11) (a) (b) (c) (d) (e)
X + h (f b) = X + h (f ) ⊕ X + h (b) (12)
Y - h (f b) = Y - h (f ) ⊕ Y - h ( bc ) (13)
X + h (f * b) = X + h (f ) X + h (b) (14)
Y - h (f * b) = Y - h (f ) Y - h ( bc ) (15)
This expression on strict lower level sets

Y - h for (f b) is valid for lower level sets X - h if (f b) is exact, in the sense that, for each x ∈ dom -(f b), there exists y ∈ R n such that (f b)(x) = f (y) ∨ b c (y -x) (i.e.
, the minimum is attained for any x in the domain) [START_REF] Seeger | On a convolution operation obtained by adding level sets: classical and new results[END_REF][START_REF] Luc | Levels sets Inmal Convolution and Level Addition[END_REF]. In particular, if f and b c are both LSC quasiconvex functions, (f b) and (f * b) are exact, which involves

X - h (f b) = X - h (f )⊕ X - h ( bc ) and X - h (f * b) = X - h (f ) X - h ( bc ).
We need for the sequel an alternative formulation of the infmax and adjoint supmin convolution in terms respectively of Minkowski subtraction and addition ⊕ of level sets. It is simply based on rewriting the infmax convolution using upper level sets:

(f b)(x) = inf h ∈ R : x ∈ Y - h (f b) = sup h ∈ R : x ∈ X + h (f ) Y - h (b c ) . (16)
Analogously, one obtains the following equivalence for the adjoint supmin convolution:

(f * b)(x) = inf h ∈ R : x ∈ Y - h (f ) Y - h ( bc ) = sup h ∈ R : x ∈ X + h (f ) ⊕ Y - h (b c ) . (17)
Therefore, we can write

X + h (f b) = X + h (f ) Y - h (b c ), (18)
X + h (f * b) = X + h (f ) ⊕ Y - h (b c ). (19)
Further properties. Other useful properties of (max, min)-convolutions are proven in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF].

Canonic structuring function. The conic structuring function plays a role similar to the multiscale quadratic structuring function in (max, +)-algebra.

Denition 2. The multiscale conic structuring function is dened as the canonic structuring function in (max, min)-convolution:

c λ (x) = - x λ . (20)
In order to justify this canonicity, let us consider the upper level sets of c λ (x). First, we remind that a ball of radius centered at point x is given by the set B r (x) = {y ∈ R n : x -y ≤ r}.

Proposition 3. The canonic structuring function in (max, min)-convolution satises the semi-group

(c λ c µ ) (x) = c λ+µ (x). (21)
In the case of the L ∞ metric, a dimension separability is obtained for c ∞ λ (x) = -x ∞ /λ; i.e., let us denote the coordinates of point as

x = (x 1 , x 2 , • • • , x n ) and by c λ; i (x) = -|x i |/λ the one dimensional conic structuring function, we have c ∞ λ (x) = (c λ; 1 c λ; 2 • • • c λ; n ) . (22)
It is easy to see this property. We rst note that X + -h (c λ ) = B λh . Second, we remind the Minkowski addition of balls: B r1 ⊕ B r2 = B r1+r2 . Therefore, one has

X + -h (c λ c µ ) = X + -h (c λ ) ⊕ X + -h (c µ ) = B λh ⊕ B µh = B (λ+µ)h .
Dimension separability in L ∞ metric is also a consequence of the Minkowski addition of segments. As a consequence of the L ∞ dimension separability, the classical theory of Minkowski decomposition of structuring elements [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF].

3 Openings, closings using (max, min)-convolutions and granulometries

The adjointness of the pairs ( * , ) and ( , * ) involves that from an algebraic viewpoint both the supmin convolution and the adjoint supmin convolution * are a dilation; both the infmax convolution and the adjoint infmax convolution * are an erosion. Therefore, their composition naturally yields openings and closings. Let us be more precise.

Denition 3. Given any USC function f ∈ F(R n , R), the (max, min)-opening and (max, min)-closing of f by the continuous structuring function b ∈ F(R n , R) are respectively given by

(f ♦ b) = ((f * b) b) , (23)
and

(f b) = ((f * b) b) , (24)
such that their corresponding level sets representations, based on expressions ( 12), [START_REF] Maragos | Lattice image processing: a unication of morphological and fuzzy algebraic systems[END_REF], and ( 13), [START_REF] Maragos | A PDE Formulation for Viscous Morphological Operators with Extensions to Intensity-Adaptive Operators[END_REF], are given by

X + h (f ♦ b) = X + h (f * b) ⊕ X + h (b) = X + h (f ) X + h (b) ⊕ X + h (b) = X + h (f ) • X + h (b) , (25)
Y - h (f b) = Y - h (f * b) ⊕ Y - h bc = Y - h (f ) Y - h bc ⊕ Y - h bc = Y - h (f ) • Y - h bc . (26)
We note that (max, min)-opening is dened from adjunction ( * , ) whereas (max, min)-closing from ( , * ). We can also switch roles and to formulate the so-called second family of dual (max, min)-opening and closing as

(f ♦ * b) = ((f b) * b) , (27)
(f * b) = ((f b) * b) , (28)
which has the following equivalent interpretation in terms of level sets:

Y - h (f ♦ * b) = Y - h (f ) • Y - h bc , (29)
X + h (f * b) = X + h (f ) • X + h (b) . (30)
Besides the duality by complement, classical properties of opening and closing hold in the (max, min) framework as a consequence of the adjunction [START_REF] Heijmans | Morphological image operators[END_REF]. See details in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF].

The extension of the granulometric theory [START_REF] Matheron | Random Sets and Integral Geometry[END_REF] to the framework of (max, +)based morphology was deeply studied in [START_REF] Kraus | Gray-scale granulometries compatible with spatial scalings[END_REF]. In particular, it was proven that one can build grey-level Euclidean granulometries with a multiscale structuring function if and only if structuring function has a convex compact domain and is constant there (i.e., at function).

In the case of (max, min)-openings, we can naturally extend Matheron axiomatic of Euclidean granulometries without the atness limitation (proof in in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF]).

Proposition 4. Given a structuring function b 1 ∈ F(R n , R) such that all its upper level sets X + h (b 1 ) are convex sets, the family of multi-scale (max, min)openings {f ♦ b λ } λ≥1 , where the structuring function at scale λ is given by

b λ (x) = b 1 λ -1 x ,
forms an Euclidean granulometry on any image f ∈ F(R n , R), i.e.,

(f ♦ b λ ) = λ λ -1 f ♦ b 1 , (31)
which involves compatibility with scaling in the spatial domain, in the sense of Matheron's axiomatic dened as follows

(λ f ) (x) = f λ -1 x , ∀λ ≥ 1.
In addition, we have the following semi-group properties,

∀λ 1 , λ 2 ≥ 1 b λ1+λ2 (x) = (b λ1 b λ2 )(x), (32)
((f ♦ b λ1 )♦ b λ2 ) (x) = ((f ♦ b λ2 )♦ b λ1 ) (x) = f ♦ b sup(λ1,λ2) (x) (33)
A good candidate of multi-scale isotropic structuring function leading to (max, min) granulometries is based on the canonic structuring function [START_REF] Vachier | The Viscous Watershed Transform[END_REF], as b

λ (x) = c λ (x) + α, which is equivalent to b λ (x) = λ -1 c 1 (x) + α, λ ≥ 1, α > 0.
4 Hopf-Lax-Oleinik formulas for Hamilton-Jacobi equation u t ± H(u, Du) = 0

We study now the Hopf-Lax-Oleinik type formulas for Hamilton-Jacobi PDE of form u t ± H(u, Du) = 0 and its links to convolutions in (max, min)-algebra. The theory of this equation was developed by Barron, Jensen and Liu [START_REF] Barron | Hopf-Lax formula for ut = H(u, Du) = 0[END_REF][START_REF] Barron | Hopf-Lax formula for ut = H(u, Du) = 0[END_REF]. Other interesting results can be found in paper by Alvarez, Barron and Ishii [START_REF] Alvarez | Hopf-lax formulas for semicontinuous data[END_REF] and the excellent survey paper by Van and Son [23]. The most relevant elements for us can be summarized in the following result.

Proposition 5. Let us consider the two following Cauchy problems (rst-order

Hamilton-Jacobi PDEs):

u t + H 1 (u, Du) = 0, in (x, t) ∈ R n × (0, ∞), u(x, 0) = f (x), ∀x ∈ R n , (34)
and

u t + H 2 (u, Du) = 0, in (x, t) ∈ R n × (0, ∞), u(x, 0) = g(x), ∀x ∈ R n , ( 35
)
where the initial conditions are functions The LSC viscosity solution of (34) is given by

u(x, y) = inf y∈R n f (y) ∨ H 1 x -y t , (36)
and the USC viscosity solution of (35) is

u(x, y) = sup y∈R n f (y) ∧ H 2 x -y t , (37)
where the conjugate operators H and H are dened as

H (q) = inf {γ ∈ R : H(γ, p) ≥ p, q , ∀p ∈ R n } , (38)
H (q) = sup {γ ∈ R : H(γ, p) ≤ p, q , ∀p ∈ R n } . ( 39
)
The simplest case of admissible (A1)-(A4) convex Hamiltonian corresponds to H(γ, p) = γ p such that, using Cauchy-Schwartz inequality, one gets

H (q) = inf {γ ∈ R : γ p ≥ p, q } = q .
The associated concave Hamiltonian is given by H(γ, p) = -γ p , whose conjugate is also H (q) = q . Using this case as a starting point, a prototype of PDE in the framework of operators in (max, min)-algebra can be dened Denition 4. Given any continuous and bounded function

f : E → [a, b] ⊂ R,
the canonic (Hamilton-Jacobi) PDE in (max, min)-morphology is dened as

∂u ∂t = ±u ∇u , x ∈ E, t > 0 u(x, 0) = f (x), x ∈ E (40)
and its (unique weak) solutions at scale t are given by

u(x, t) = sup y∈E f (y) ∧ x -y t (for + sign), (41) u
(x, t) = inf y∈E f (y) ∨ x -y t (for -sign). ( 42
)
Therefore the viscosity solutions of Cauchy problem (40) are a supmin convolution and an infmax convolution using the conic structuring function c λ (x) given by [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF], where the scale parameter is here the time; i.e., λ = t. More precisely, we note that these solutions

u(x, t) = (f (-c t ))(x) (for + sign), u(x, t) = (f c t )(x) (for -sign),
are not adjoint in the sense of Section 2, consequently their composition does not lead to opening or closing.

The model (40) can be generalized to ∂u ∂t = ±αu ∇u , x ∈ E, t > 0 with initial condition u(x, 0) = f (x) and α > 0, such that we easily see that the corresponding solutions are

u(x, t) = (f (-c αt ))(x) (for + sign), u(x, t) = (f c αt )(x) (for -sign),
or in other words, multiplying u by α involves a scaling in time by α. This principle can be a clue to explore the notion of spatially adaptive (max, min)operators based on using a scale depending on space x, i.e., a model of the form u t = ±α(x)u ∇u .

Ubiquity of (max, min)-convolutions in mathematical morphology

It is obvious the connection between (max, min)-convolutions and the distance function or the at morphology. We discuss now links to fuzzy morphology and to viscous morphology. Relationships of (max, min)-convolutions with Boolean random function characterization and geodesic dilation/erosion are discussed in [START_REF] Angulo | Convolution in (max, min)-algebra and its role in mathematical morphology[END_REF].

Links with fuzzy morphology. The state-of-the-art on morphological operators based on fuzzy logic is very extensive, see for instance [START_REF] Bloch | Duality vs. adjuntion for fuzzy mathematical morphology and general form of fuzzy erosions and dilations[END_REF]. Results on fuzzy morphology discussed here are mainly based on Deng and Heijmans [START_REF] Deng | Grey-Scale Morphology Based on Fuzzy Logic[END_REF],

see also [START_REF] Maragos | Lattice image processing: a unication of morphological and fuzzy algebraic systems[END_REF].

In fuzzy logic, the two basic (Boolean) logic operators, the conjunction C(s, t) = s∧t and the implication I(s, t) = s ⇒ t (= ¬s∨t), are extended from the Boolean domain {0, 1} × {0, 1} to the rectangle [0 [START_REF] Alvarez | Hopf-lax formulas for semicontinuous data[END_REF] which is increasing in both arguments and satises C(0, 0) = C(1, 0) = C(0, 1) = 0 and C(1, 1) = 1. A fuzzy implication is decreasing in the rst argument, increasing in the second one and satises I(0, 0) = I(0, 1) = I(1, 1) = 1 and I(1, 0) = 0.

, 1] × [0, 1]. A fuzzy conjunction is a mapping from [0, 1] × [0, 1] into [0,
Given a fuzzy set µ, the dilation and erosion by a fuzzy structuring element ν are then dened as [START_REF] Deng | Grey-Scale Morphology Based on Fuzzy Logic[END_REF]:

δ ν,C (µ)(x) = sup
C KD (a, t) = 0, t ≤ 1 -a t, t > 1 -a ; I KD (a, s) = max(1 -a, s) (46)
It is consequently straightforward to see that the four operators that we have dened in Section 2 are just fuzzy dilations and erosions when they are applied to fuzzy sets (i.e., functions valued in [0, 1]):

δ ν,C GB (µ)(x) = (µ ν) (x) adjoint ←→ ε ν,C GB (µ)(x) = (µ * ν) (x), dual dual ε ν,C KD (µ)(x) = (µ ν) (x) adjoint ←→ δ ν,C KD (µ)(x) = (µ * ν) (x).
Links with viscous morphology. Theory and practice of morphological (at) viscous operators was introduced by Vachier and Meyer [START_REF] Vachier | The Viscous Watershed Transform[END_REF][START_REF] Vachier | News from Viscous Land[END_REF]. The PDE formulation of these operators was done by Maragos and Vachier [START_REF] Maragos | A PDE Formulation for Viscous Morphological Operators with Extensions to Intensity-Adaptive Operators[END_REF].

The idea of viscous operators is to apply a dierent scale (i.e., size) of structuring element at each upper level set. This principle can be seen now as an operator which locally adapts its activity with respect to the intensity. Let us formalize their denition according to [START_REF] Maragos | A PDE Formulation for Viscous Morphological Operators with Extensions to Intensity-Adaptive Operators[END_REF]. For the sake of simplicity, let us consider a nonnegative bounded function f : E → [0, M ]. Viscous operators have been formulated as isotropic transforms, that is based on the use of balls B λ as structuring elements.

Using intensity-adaptive operators and the two viscosity functions, two pairs of viscous dilation and erosion are dened for a given function f :

δ visc ∧ (f ) = δ λ∧(h) (f ) = sup h ∈ [0, M ] : x ∈ X + h (f ) ⊕ B M -h , (47)
ε visc ∧ (f ) = ε λ∧(h) (f ) = sup h ∈ [0, M ] : x ∈ X + h (f ) B M -h , (48)
and

δ visc ∨ (f ) = δ λ∨(h) (f ) = sup h ∈ [0, M ] : x ∈ X + h (f ) ⊕ B h , (49)
ε visc ∨ (f ) = ε λ∨(h) (f ) = sup h ∈ [0, M ] : x ∈ X + h (f ) B h , (50)
such that ε visc ∧ , δ visc Let us introduce the following structuring function:

v(x) = M -x if x ≤ M 0 if x > M such that its complement structuring function is v c (x) = x if x ≤ M and M if x > M . We have X + h (v) = B M -h and Y - h (v c ) = B h .
Hence, viscous dilations and erosions (47)-(50) can be rewritten using the (max, min)-convolution (respectively expressions ( 12), ( 14), ( 18), ( 19)):

δ visc ∧ (f )(c) = (f v) (x) adjoint ←→ ε visc ∧ (f )(x) = (f * v) (x), dual dual ε visc ∨ (f )(x) = (f v) (x) adjoint ←→ δ visc ∨ (f )(x) = (f * v) (x).
In addition to the operator framework, a PDE formulation of viscous dilation and erosion was introduced in [START_REF] Maragos | A PDE Formulation for Viscous Morphological Operators with Extensions to Intensity-Adaptive Operators[END_REF]. The proposed couple of PDEs are particular cases of the Hamilton-Jacobi models discussed above. More precisely, it corresponds to the case of the Hamiltonians given in expressions H 1 (γ, p) = (α + γ) p and H 2 (γ, p) = -(α + γ) p , such that H 1 (q) = H 2 (q) = q -α;

or a pair H 1 (γ, p) = (α -γ) p and H 2 (γ, p) = -(α -γ) p , with H 1 (q) = H 2 (q) = α -q . Therefore solution u(x, t) for + sign of the PDE model is equivalent to viscous dilation δ visc ∧ (f ), but forsign it is not exactly equivalent to the viscous erosion ε visc ∧ (f ). In our terminology, the latter is a case of adjoint infmax convolution while the solution forsign is an infmax convolution with the complemented structuring function.

Conclusion and Perspectives

Operators and lters underlying a formulation as (max, min)-convolutions are common in the state-of-the-art of mathematical morphology. However, their study per se has been neglected. From this epistemological viewpoint, we can conclude that the role of (max, min)-convolutions has been somewhat overshadowed by a multiplicity of viewpoints (fuzzy, viscous, hitting of functions in Choquet capacity, etc.) In order to address this theoretical lack, we have developed in our paper a rigorous formulation and characterization of the four convolution-like operators in (max, min)-algebra.

All the results on (max, min)-convolutions considered here are valid for functions supported in a general Banach space, consequently more general that the Euclidean space R n . In this generalization context, we plan to consider in particular the case of (max, min)-morphology for real-valued images on Riemannian manifolds.

Denition 1 .

 1 Given a structuring function b ∈ F(R n , R), for any function f ∈ F(R n , R) we dene the supmin convolution f b and the infmax convolution f b of f by b as

Fig. 1 .

 1 Fig. 1. Illustration of four (max, min)-convolutions for a given example of one dimensional functions dened in a bounded interval, i.e., f, b ∈ F (R, [0, M ]): (a) original function f (x) and translated structuring function b at point z0; (b) in red, f (y) ∧ b(z0 -y) for all y ∈ R, green triangle represents (f b)(x) the value of the supmin convolution at z0; (c) in red, f (y) ∧ * b(y -z0), green triangle, adjoint infmax at z0: (f * b)(z0); (d) in red, f (y) ∨ b c (y -z0), green triangle, infmax at z0: (f b)(z0); (e) in red, f (y) ∨ * b c (z0 -y), green triangle, adjoint supmin at z0: (f * b)(z0).

y

  {C (ν(x -y), µ(y))} , (43) ε ν,C (µ)(x) = inf y {I (ν(y -x), µ(y))} .

  shown in[START_REF] Deng | Grey-Scale Morphology Based on Fuzzy Logic[END_REF], (I, C) is an adjunction if and only if (ε ν,C , δ ν,C ) is an adjunction.Two particular cases of conjunction and adjoint implication widely used in fuzzy logic are the Gödel-Brower:C GB (a, t) = min(a, t); I GB (a, t) = s, s < a 1, s ≥ a(45)and the Kleen-Dienes:

∧

  and ε visc ∨ , δ visc ∨ form two adjunctions. The pairs ε visc ∨ , δ visc ∧ and ε visc ∧ , δ visc ∨ are dual by complement.

  f, g : R n × R, such that f is a LSC proper function, bounded from below; and g an USC proper function, bounded from above. The Hamiltonians H 1 , H 2 : R × R n → R n are assumed to satisfy the following conditions:(A1) H 1 (γ, p) and H 2 (γ, p) are continuous; (A2) H 1 (γ, p) and H 2 (γ, p) are nondecreasing in γ ∈ R, ∀p ∈ R n ; (A3) H 1 (γ, p) is convex and H 2 (γ, p) is concave in p ∈ R n , ∀γ ∈ R;(A4) H 1 (γ, p) and H 2 (γ, p) are positively homogeneous of degree 1 in p ∈ R n , i.e., H 1 (γ, λp) = λH 1 (γ, p), ∀λ ≥ 0.

(max, min)-convolutions: denition and propertiesIn this Section we dene the alternative convolutions associated to a pair (function f , structuring function b) in the (max, min) mathematical framework. We also study their properties.


Jesús Angulo
email: jesus.angulo@mines-paristech.fr

Morphological PDE and dilation/erosion semigroups on length spaces

Keywords: HamiltonJacobi PDE, HamiltonJacobi semigroup, length space, morphological PDE, morphological semigroup

This paper gives a survey of recent research on Hamilton-Jacobi partial dierential equations (PDE) on length spaces. This theory provides the background to formulate morphological PDEs for processing data and images supported on a length space, without the need of a Riemmanian structure. We rst introduce the most general pair of dilation/erosion semigroups on a length space, whose basic ingredients are the metric distance and a convex shape function. The second objective is to show under which conditions the solution of a morphological PDE in the length space framework is equal to the dilation/erosion semigroups.

Introduction

Let us assume a Lipschitz continuous function f : R n → R. Consider now the following initial-value HamiltonJacobi rst-order partial dierential equation (PDE) u t (x, t) ± H (x, Du(x, t)) = 0, in R n × (0, +∞),

u(x, 0) = f (x), in R n , (1)
Such family of equations usually does not admit classic (i.e., everywhere dierentiable) solutions but can be studied in the framework of the theory of viscosity solutions [START_REF] Crandall | User's guide to viscosity solutions of second order partial dierential equations[END_REF]. It is well known [START_REF] Bardi | On Hopf's formulas for solutions of Hamilton-Jacobi equations[END_REF][START_REF] Evans | Partial dierential equations[END_REF] that if the Hamiltonian has the properties: (i) H(x, p) = H(p) is convex, (ii) superlinear growth in the sense of lim |p|→+∞ H(p)/|p| = +∞, and (iii) H(0) = 0, then the solution of Cauchy problem (1) is given for + andrespectively by the so-called HopfLaxOleinik formulas:

u(x, t) = inf y∈R n f (y) + tL x -y t , u(x, t) = sup y∈R n f (y) -tL x -y t ,
where the Lagrangian L(q) is the one-dimensional LegendreFenchel transform of function H(p), i.e., L(q) = H * (q) = sup

p∈R+ {p q -H(p)} , q ∈ R + . (2)
We note that, by standard results of the LegendreFenchel transform, L is increasing, convex, superlinear and satises L(0) = 0. PDE (1) plays a central role in continuous mathematical morphology [START_REF] Alvarez | Axioms and fundamental equations of image processing[END_REF][START_REF] Arehart | Mathematical morphology: The Hamilton-Jacobi connection[END_REF][START_REF] Brockett | Evolution equations for continuous-scale morphology[END_REF][START_REF] Maragos | Dierential morphology and image processing[END_REF][START_REF] Breuÿ | Highly accurate PDE-based morphology for general structuring elements[END_REF]. In particular, by taking H(p) = 1/2 p 2 , such that L(q) = 1/2 q 2 , a kind of canonic morphological PDE is formulated

∂u ∂t = ± 1 2 ∇u 2 , x ∈ R n , t > 0 u(x, 0) = f (x), x ∈ R n (3)
such that the corresponding viscosity solutions are given by u(x, t) = sup

y∈R n f (y) - x -y 2 2t (for + sign), (4)
u(x, t) = inf y∈R n f (y) +

x -y 2 2t (forsign), [START_REF] Arehart | Mathematical morphology: The Hamilton-Jacobi connection[END_REF] which just correspond to a dilation (f ⊕ b) and an erosion (f b) of function f (x) dened as

(f ⊕ b)(x) = sup y∈R n {f (y) + b(y -x)} , (6)
(f b)(x) = inf y∈R n {f (y) -b(y + x)} , (7)
using as structuring function b(x) the so-called multiscale quadratic (or parabolic) structuring function:

p t (x) = - x 2 2t . (8)
By the way, due to its properties of semigroup, dimension separability and invariance to transform domain [START_REF] Maragos | Slope Transforms: Theory and Application to Nonlinear Signal Processing[END_REF][START_REF] Jackway | Scale-Space Properties of the Multiscale Morphological Dilation-Erosion[END_REF][START_REF] Van Den Boomgaard | The morphological equivalent of Gaussian scalespace[END_REF], the structuring function p t (x) can be considered as the canonic one in morphology, playing a similar role to the Gaussian kernel in linear ltering. Other particularized forms of the HamiltonJacobi model ( 1) cover the at morphology by disks [START_REF] Maragos | Dierential morphology and image processing[END_REF]; i.e., u t = ± ∇u , as well as operators with more general P -power concave structuring functions, i.e., u t = ± ∇u P . For the application of the latter model to adaptive morphology, see [START_REF] Diop | Multiscale Image Analysis Based on Robust and Adaptive Morphological Scale-Spaces[END_REF]. Morphological operators are classically dened for images supported on Euclidean spaces. We have recently introduced mathematical morphology for real valued images whose support space is a Riemannian manifold [START_REF] Angulo | Riemannian Mathematical Morphology[END_REF]. In fact, we have observed that the smoothness of the space (and its Riemannian structure) is not a fundamental requirement, since the counterpart of Euclidean quadratic operators (4) (5) are also sup/inf-convolutions where the Euclidean distance is replaced by the geodesic distance in the Riemannian manifold. Hence, dilation and erosion can be formulated for functions in a more general framework than the Euclidean or even the Riemannian case. We focus here on functions whose domain is a length (or geodesic) space and in particular we are interesting of relating the corresponding dilation/erosion with a HamiltonJacobi PDE formulation.

Morphological PDE on graphs. The approximation of morphological operators using a PDE formulation has been already considered for the non-Euclidean case of weighted graphs [START_REF] Ta | Nonlocal PDEs-Based Morphology on Weighted Graphs for Image and Data Processing[END_REF][START_REF] Elmoataz | Non-Local Morphological PDEs and Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning[END_REF]. The starting point is the denition of a gradient on the graph. Hence, the basic ingredient is an approximation of the rst derivative in a vertex (or node) u in the direction to a vertex v as

√ w uv (f (v) -f (u))
, where w uv is the weight in the edge linking u to v. Then, the gradient of a function f at a vertex u is dened as

∇f (u) = v∈N (u) √ w uv (f (v) -f (u)), N (u)
being the set of vertices linked to u. Using this gradient, a counterpart of the classical morphological PDE is formulated. The weight function in [START_REF] Ta | Nonlocal PDEs-Based Morphology on Weighted Graphs for Image and Data Processing[END_REF] is generally a distance-based kernel used for adaptive/nonlocal lters. In general, this kind of weight does not involve a natural length structure on the graph and this can be a theoretical limitation in order to link such PDE with classical HamiltonJacobi PDE theory. In addition, existence of viscosity solutions, and their semigroups, for those morphological PDEs on graphs were not considered in [START_REF] Ta | Nonlocal PDEs-Based Morphology on Weighted Graphs for Image and Data Processing[END_REF][START_REF] Elmoataz | Non-Local Morphological PDEs and Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning[END_REF].

Numerical schemes for HamiltonJacobi equations. There exists a large state-of-the-art on numerical schemes for HamiltonJacobi equations. The majority of numerical schemes which were proposed to solve HamiltonJacobi equations in Euclidean space are based on nite dierence methods (upwind and centered discretizations, ENO or WENO schemes, etc.) The formulation of numerical discretization of HamiltonJacobi equations on general length spaces is out of the scope of the paper. We can nevertheless cite recent eorts on approximation schemes of HamiltonJacobi PDE on networks [START_REF] Camillia | An approximation scheme for a Hamilton Jacobi equation dened on a network[END_REF][START_REF] Herty | Numerical discretization of Hamilton-Jacobi equations on networks[END_REF].

HamiltonJacobi semigroups on metric, length and geodesic spaces.

During the recent years, a series of works have considered the generalization of the HopfLaxOleinik formula to a class of HamiltonJacobi PDEs on a length space framework. The need of these technical results was motivated by the study of geometric inequalities related to concentration measure. More precisely, connections between logarithmic Sobolev type inequalities and optimal transportentropy inequalities. See the book by Villani [START_REF] Villani | Optimal transport. Old and new[END_REF] for detailed overview on application of HopfLaxOleinik semigroup to optimal transport or papers by Ambrosio and co-workers [START_REF] Ambrosio | Calculus and heat ow on metric measure spaces and applications to spaces with Ricci curvature bounded below[END_REF][START_REF] Ambrosio | Equivalent denitions of BV space and total variation in metric measure spaces[END_REF] for the use of these semigroups in metric space calculus (heat ow, total variation, Ricci curvature bounds, etc.) on metric measure spaces. Nevertheless, up to the best of our knowledge, this theory has not been applied to practical problems in applied mathematics which use HamiltonJacobi PDEs, such as optimal control or mathematical morphology. This series of works were inspired by the seminal contribution by Bobkov et al. [START_REF] Bobkov | Hypercontractivity of HamiltonJacobi equations[END_REF] establishing the equivalence between logarithmic Sobolev inequality and hypercontractivity properties of classical HamiltonJacobi (semigroup) solutions. In our terminology, the semigroup used in [START_REF] Bobkov | Hypercontractivity of HamiltonJacobi equations[END_REF] corresponds to the Euclidean erosion using a quadratic structuring function. The paper by Lott and Villani [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF] is the pioneer work formulating HamiltonJacobi PDE acting on continuous functions on a compact measured length space and for a quadratic Hamiltonian. The approach in the same framework was extended to general con-vex Hamiltonians by Balogh et al. [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF]. A dierent kind of generalization, studied more recently by Gozlan et al. [START_REF] Gozlan | Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities[END_REF] and Ambrosio et al. [START_REF] Ambrosio | Calculus and heat ow on metric measure spaces and applications to spaces with Ricci curvature bounded below[END_REF], involves the general case of a length space without the need of a measure structure. The particular case of the HamiltonJacobi semigroup on Riemannian manifolds is considered in [START_REF] Villani | Optimal transport. Old and new[END_REF]. We can mention also generalizations of HamiltonJacobi semigroups to specic dierential geometry structures such as Heisenberg group [START_REF] Dragoni | Metric HopfLax formula with semicontinuous data[END_REF]. Finally, reader is refereed to [START_REF] Fathi | Weak KAM Theorem in Lagrangian Dynamics[END_REF] for a depth insight to recent progresses on sub-solutions of HamiltonJacobi equations on Riemannian structures based on KAM theory.

Aim of the paper. In this context, the goal of the present paper is to give a survey on this recent theory of HamiltonJacobi PDEs and associated semigroups on length spaces. Therefore, we do not provide new results, except from the adjunction viewpoint, since most of the proofs can be found in the above mentioned measure theory literature. Nevertheless, in our opinion, the paper has a relevant pedagogical interest in the mathematical morphology context since this theory is useful for the generalization of morphological PDEs for images and data supported on non-Euclidean spaces, such as surfaces, graphs, point clouds, and other length spaces which can be obtained by dierent image embeddings [START_REF] Angulo | Riemannian Mathematical Morphology[END_REF].

Preliminaries

Metric, length and geodesic space [START_REF] Burago | A course in metric geometry[END_REF]. A metric space is a set of points X endowed with a distance function d : X × X → [0, ∞). In the paper is assumed that (X, d) is a complete separable metric space, locally compact (every closed ball or subset of X is compact).

A length space is a metric space (X, d) such that for any pair of points x, y ∈ X, we have d(x, y) = inf{Length(σ)}, where the inmum is taken over all rectiable curves σ : [0, 1] → X connecting x with y, i.e., σ(0) = x and σ(1) = y.

A curve σ is called a geodesic if σ has constant speed and if Length(σ| d) is a geodesic space if for every pair of points x, y ∈ X there exists a geodesic σ : [0, 1] → X joining x to y.

[t,t ] ) = d(σ(t), σ(t )), ∀t, t ∈ [0, 1], t ≤ t . A curve σ is a geodesic if for every two points x, y ∈ X, with σ(0) = x and σ(1) = y, one has d(σ(t), σ(t )) = |t -t |d(x, y), ∀t, t ∈ [0, 1]. (X,
Note that every geodesic space is a length space. For the converse, we have the HopfRinow Theorem: Let X be a length space, complete and locally compact, then X is a geodesic space. Doubling measure space [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF]. A Borel measure µ is doubling, if the measure of any open ball is positive and nite, and if there exists a constant

c d ≥ 1 such that µ(B(x, 2r)) ≤ c d µ(B(x, r))
for all x ∈ X and r > 0. Here B(x, r) denotes an open ball of radius r centered in x. A metric measure space (X, d, µ) satises a doubling condition if µ is a Borel doubling measure.

Metric gradient and subgradient [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF][START_REF] Gozlan | Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities[END_REF][START_REF] Ambrosio | Calculus and heat ow on metric measure spaces and applications to spaces with Ricci curvature bounded below[END_REF]. We said that f :

X → R is d- Lipschitz if there exists C ≥ 0 satisfying |f (x)-f (y)| ≤ Cd(x, y), ∀x, y ∈ X.
The least constant C with this property will be denoted by Lip(f ). Lip(X) denotes the set of real-valued Lipschitz functions on X.

Given f : X → R, we dene the metric gradient of f at a point x ∈ X by

|∇f |(x) = lim sup y→x |f (y) -f (x)| d(x, y) . ( 9
) If f is Lipschitz continuous then |∇f | ∈ L ∞ (X).
We further introduce the metric subgradients of f at x dened as

|∇ -f |(x) = lim sup y→x [f (y) -f (x)] - d(x, y) = lim sup y→x [f (x) -f (y)] + d(x, y) , (10)
and

|∇ + f |(x) = lim sup y→x [f (y) -f (x)] + d(x, y) = lim sup y→x [f (x) -f (y)] - d(x, y) , (11)
where a + = max(a, 0) and a -= max(-a, 0).

|∇ -f |(x) is called descending slope and |∇ + f |(x) ascending slope. Notice that |∇ -f |(x) = |∇ + (-f )|(x) and |∇f |(x) = max {|∇ -f |(x), |∇ + f |(x)}.
We can therefore work exclusively with |∇ -f |. Finally, we observe that if d is nite, and (X, d, µ) is doubling, for any

f ∈ Lip(X) then |∇ -f |(x) = |∇ + f |(x) µ-
(x). If f is Lipschitz continuous then |∇ ± f |(x) ≤ Lip(f ), ∀x ∈ X.
Finally, when X is a Riemannian manifold and f is dierentiable at x, metric subgradients |∇ ± f |(x) are equal to the norm of the vector ∇f (x) ∈ T x X (the tangent space at x) [START_REF] Villani | Optimal transport. Old and new[END_REF].

Dilation and erosion on metric spaces

Let us consider a metric space (X, d) and a given bounded function f : X → R. We assume that f is Lipschitz continuous. Let us consider a one-dimensional (shape) function L : R + → R + , being increasing, superlinear, convex of class C 1 such that L(0) = 0. For all scales t > 0, we dene the dilation D L; t f and the erosion E L; t f operators of f on (X, d) according to L as follows

D L; t f (x) = sup y∈X f (y) -tL d(x, y) t , ∀x ∈ X, (12)
E L; t f (x) = inf y∈X f (y) + tL d(x, y) t , ∀x ∈ X. (13)
We adopt the convention D L; 0 f = E L; 0 f = f . In the context of classical mathematical morphology operators ( 6)- [START_REF] Bardi | On Hopf's formulas for solutions of Hamilton-Jacobi equations[END_REF], correspond respectively to the multi-scale

dilation (f ⊕ b t ) (x) = D L; t f (x) and erosion (f b t ) (x) = E L; t f (x) of function f by structuring function b t (x -y) = -tL d(x, y) t .
By the way, we note that by symmetry, one has b t (x -y) = b t (y -x). A typical example of a shape function is L(q) = q P /P , P > 1, such that b t (x -y) = -d(x, y) P P t P -1 .

The canonic shape function corresponds to the case

P = 2: b t (x -y) = -d(x,y) 2 2t .
Properties. The following properties hold for any metric space (X, d).

1. (Adjunction) For any two real-valued functions f and g on (X, d), the pair (E L; t , D L; t ) forms an adjunction, i.e.,

D L; t f (x) ≤ g(x) ⇔ f (x) ≤ E L; t g(x), ∀x ∈ X.

(Duality by involution)

For any function f and ∀x ∈ X, one has

D L; t f (x) = -E L; t (-f )(x); and E L; t f (x) = -D L; t (-f )(x), ∀t > 0. 3. (Increaseness) If f (x) ≤ g(x), ∀x ∈ X, then D L; t f (x) ≤ D L; t g(x)
; and E L; t f (x) ≤ E L; t g(x), ∀x ∈ X, ∀t > 0.

(Extensivity and anti-extensivity)

D L; t f (x) ≥ f (x); and E L; t f (x) ≤ f (x), ∀x ∈ X, ∀t > 0.

(Ordering property) If

0 < s < t then ∀x ∈ X inf X f ≤ E L; t f (x) ≤ E L; s f (x) ≤ f (x) ≤ D L; s f (x) ≤ D L; t f (x) ≤ sup X f.
6. (Convergence) For any function f and ∀x ∈ X, D L; t f (x) and E L; t f (x) converge monotonically to f (x) as t → 0. In particular lim t→0 D L; t f = f and lim t→0 E L; t f = f . 7. (Lipschitz) The maps (x, t) → D L; t f (x) and (x, t) → E L; t f (x) are in Lip(X × R + ). 8. (Semigroup) For any function f and ∀x ∈ X, and for all pair of scales s, t > 0, If X is metric space:

D L; t D L; s f ≤ D L; t+s f ; and E L; t E L; s f ≥ E L; t+s f.
If X is a length space:

D L; t D L; s f = D L; t+s f ; and E L; t E L; s f = E L; t+s f.

Proof. For property 1, on adjunction, we have that the inequality

DL; tf (x) ≤ g(x)
means that

sup y∈X f (y) -tL d(x, y) t ≤ g(x), ∀x ∈ X,
It involves that f (y) -tL (d(x, y)/t) ≤ g(x) for every x, y ∈ X. This is equivalent to rewrite f (y) ≤ g(x) + tL (d(x, y)/t). Therefore, after substitution of z = x, we nally have

f (y) ≤ inf z∈X g(y) + tL d(z, y) t = EL; tg(y).
For the duality of 2, we have that DL; t(-f )(x) is equal to

sup y∈X -f (y) -tL d(x, y) t = -inf y∈X f (y) + tL d(x, y) t = -EL; t(f )(x).
The properties 3 and 4 of increaseness and extensivity/anti-extensivity are obvious from the properties of supremum/inmum.

The proof of ordering property 5 is based on the following semigroup property [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF](Theorem 2.5.(ii)):

For 0 ≤ s < t EL; tf (x) = min y∈X EL; sf (y) + (t -s)L d(x, y) t -s .
Now for a xed z ∈ X, we have

EL; tf (z) = min ζ∈X EL; sf (ζ) + (t -s)L d(ζ, z) t -s ≤ (t -s)L(0) + EL; sf (ζ).
By choosing z = ζ and using L(0) = 0, we have EL; tf (z) ≤ EL; sf (z).

In order to prove the semigroup property 8, following [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF], we consider for the sake of simplicity the case of the canonic shape function L(q) = q 2 /2. Now, triangle inequality implies that for all x, y ∈ X and s, t > 0,

d(x, y) 2 2(t + s) ≤ inf z∈X d(x, z) 2 2t + d(z, y) 2 2s . ( 14
)
The equality in [START_REF] Crandall | User's guide to viscosity solutions of second order partial dierential equations[END_REF] in length spaces comes from choosing a minimal geodesic between x and y, and a point z on this geodesic with d(x, z) = t s+t d(x, y). Finally, from ( 14), we obtain

EL; t+sf (x) = inf y∈X f (y) + d(x, y) 2 2(t + s) = inf y∈X inf z∈X f (y) + d(x, z) 2 2t + d(z, y) 2 2s = EL; tEL; sf (x).
For the a general function L, see [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF].

The proof of properties 6 and 7 on convergence and Lipschitz are not included by the limited length of the paper, see [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF][START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF]. Bibliographic remark. Following [START_REF] Bobkov | Hypercontractivity of HamiltonJacobi equations[END_REF] and [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF], our metric erosion (13) corresponds to the semigroup Q t f , which is the basic ingredient in the theory of geometric inequalities related to concentration measure. The dual and adjoint semigroup (our dilation [START_REF] Brockett | Evolution equations for continuous-scale morphology[END_REF]) is only considered in [START_REF] Gozlan | Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities[END_REF] and is denoted by P t f . Q t f is named as Hamilton-Jacobi semigroup on length spaces in [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF] whereas other works [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF][START_REF] Gozlan | Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities[END_REF][START_REF] Ambrosio | Calculus and heat ow on metric measure spaces and applications to spaces with Ricci curvature bounded below[END_REF] use the most classical terminology from max-plus mathematics on Hilbert spaces: Q t f is the Hopf-Lax-Oleinik semigroup on length (or geodesic) spaces.

We introduce the morphological PDE on a metric space (X, d) as the the following initial-value HamiltonJacobi rst-order equation:

   ∂ ∂t u(x, t) ± H (|∇ -u(x, t)|) = 0, in X × (0, +∞), u(x, 0) = f (x), in X, (15)
where the initial condition f : X → R is a continuous bounded function and H : R + → R + is the Legendre transform of function L(q):

H(p) = max q∈R+ {pq -L(q)} , p ∈ R + .
Our objective now is to show under which conditions the solution of a HamiltonJacobi PDE in the metric space framework is equal to the dilation and erosion semigroups. We rst consider the results from [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF] and [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF].

Theorem 1 [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF][START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF]. The solutions of PDE problem [START_REF] Diop | Multiscale Image Analysis Based on Robust and Adaptive Morphological Scale-Spaces[END_REF] are the dilation [START_REF] Brockett | Evolution equations for continuous-scale morphology[END_REF] and erosion [START_REF] Camillia | An approximation scheme for a Hamilton Jacobi equation dened on a network[END_REF] semigroups:

u(x, t) = D L; t f (x) (for -sign), (16) u(x
, t) = E L; t f (x) (for + sign), [START_REF] Burago | A course in metric geometry[END_REF] in the following cases.

1. If (X, d) is a length space: solutions hold for all x ∈ X and for almost everywhere t > 0. (X,d,µ) satises a doubling condition and supports a local Poincaré inequality: solutions hold for µ-almost everywhere x ∈ X and for all t > 0.

If

Proof. For the sake of pedagogy, let us recall the proof of the solution as an erosion u(x, t) = EL; tf (x) in the case 1. The corresponding one for case 2 can be found in [START_REF] Lott | HamiltonJacobi semigroup on length spaces and applications[END_REF] and [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF], where the role of doubling measure and Poincaré inequality are explained. We rst show that the inequality

∂ ∂t u(x, t) + H(|∇ -u|(x, t)) ≤ 0 (18)
holds for every x ∈ X and a.e. t ∈ R+ for u(x, t) = EL; tf (x). Fix x ∈ X and let t ∈ R+ be a point of dierentiability of u(x, •). If |∇ -u|(x, t) = 0, (18) reduces to ut(x, t) ≤ 0 since H(0) = 0. This clearly holds since u(x, •) is non increasing. We can thus assume that |∇ -u|(x, t) > 0, and there exists a sequence xn → x for which u(xn, t) < u(x, t) and |∇ -u|(x, t) = limn→∞ u(x,t)-u(xn,t) d (xn,x) . For the moment, consider any positive sequence (hn) with hn → 0. By the semigroup property [START_REF] Balogh | Functional Inequalities and HamiltonJacobi Equations in Geodesic Spaces[END_REF]

Since H(p) = max q∈R + {pq -L(q)}, ∀p ∈ R+, for each n it is possible to choose hn > 0 such that As xn → x and hn → 0, letting n → ∞ gives us [START_REF] Elmoataz | Non-Local Morphological PDEs and Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning[END_REF].

H u(x,
The converse inequality to [START_REF] Elmoataz | Non-Local Morphological PDEs and Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning[END_REF] can be written as

lim inf s→0 + EL; t+sf (x) -EL; tf (x) s ≥ -H |∇ -EL; tf |(x) . (22)
Let us x x ∈ X and t ∈ R+. Since (x, t) → EL; tf (x) is a Lipschitz function, the limit inferior in [START_REF] Herty | Numerical discretization of Hamilton-Jacobi equations on networks[END_REF]

For each n we choose a point yn ∈ X for which the minimum is attained. The superlinearity of L implies that yn → x. As EL; tf (x) is decreasing in t, we have E L; t+hn f (x) ≤ EL; tf (x), and hence

EL; tf (yn) ≤ hnL d(x, y) hn + EL; tf (yn) ≤ EL; tf (x). (25)
Since H(p) = max q∈R + {pq -L(q)}, we have H(p) + L(q) ≥ pq, ∀p, q ∈ R+. Together with [START_REF] Maragos | Slope Transforms: Theory and Application to Nonlinear Signal Processing[END_REF] Combining inequalities ( 18) and ( 22), we obtain the equality.

Theorem 1 tell us that the solutions of the morphological PDE are the dilation and erosion for all x ∈ X, X being a length space, and for all t outside a set N t of measure 0. In fact, it has been proven more recently [START_REF] Gozlan | Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities[END_REF] that the result holds without the need of measure theory.

Theorem 2 [START_REF] Gozlan | Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities[END_REF]. In a geodesic space (X, d), the solutions ( 16)-( 17) hold for all x ∈ X and for all t > 0.

Finally, in analogy to the Euclidean case, the canonic morphological PDE in a length space (X, d) is given by

∂ ∂t u(x, t) = ± 1 2 |∇ -u(x, t)| 2 , x ∈ X, t > 0 u(x, 0) = f (x), x ∈ X, (26)
such that the corresponding semigroup solutions are given by

u(x, t) = sup y∈X f (y) - d(x, y) 2 2t (for + sign), (27) u
(x, t) = inf y∈X f (y) + d(x, y) 2 2t (for -sign). (28)
Bibliographic remark. We note that the case of real-valued extended functions f : X → R, R =, with R ∪ {+∞, -∞}, requires a more technical treatment, see Section 3 in Ambrosio et al. [START_REF] Ambrosio | Calculus and heat ow on metric measure spaces and applications to spaces with Ricci curvature bounded below[END_REF].

Conclusions and perspectives

We have introduced the most general pair of dilation/erosion operators on a metric space, whose basic ingredients are the metric distance and a convex shape function. We have stated that the families of scale-space dilations {D L; t } t>0 and erosions {E L; t } t>0 are semigroups acting on bounded functions only for length spaces. We have introduced the morphological PDE on length spaces and reviewed the theoretical results which provide us a complete transposition from the Euclidean to the geodesic counterpart, linking the morphological PDE to its viscous solutions as dilation/erosion semigroups. The theory of this paper can be used in many practical situations under the assumption of working on a geodesic space, but without the need of any smoothness of the space or curvature constraints. Discretization and numerical schemas for the morphological PDE on useful cases such as graphs and meshes will be considered in future work. The starting point can be the recent approximation schemes of HamiltonJacobi PDE on networks [START_REF] Camillia | An approximation scheme for a Hamilton Jacobi equation dened on a network[END_REF][START_REF] Herty | Numerical discretization of Hamilton-Jacobi equations on networks[END_REF].

From a theoretical viewpoint, we plan in our perspectives to explore three dierent lines. First, Eikonal equation is another HamiltonJacobi PDE which is the basic ingredient for morphological segmentation (computation of a weighted distance function and watershed segmentation [START_REF] Meyer | Multiscale Morphological Segmentations Based on Watershed, Flooding, and Eikonal PDE[END_REF]), the corresponding PDE on length spaces is therefore important for us. Second, we will focuss on the particular case of metric spaces of non-positive curvature and CAT (0) spaces [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF]. That includes combinatorial spaces such as trees and simplicial complexes. Third, we will study the counterpart of the theory for bounded functions on ultrametric spaces, which are also relevant to mathematical morphology (dendograms and hierarchies).

  almost everywhere in X [3](Proposition 2.6). Clearly, we notice that |∇ -f |(x) ≤ |∇f |(x), thus the metric subgradient is a ner notion than the gradient norm and |∇ -f |(x) vanishes if f has a local minimum at x. In a sense, |∇ -f |(x) measures the downward pointing component of f near x: local variation of f taking into account only values less than f

  is nite and we can choose a positive sequence (hn) such that hn → 0 and

	lim inf s→0 +	EL; t+sf (x) -EL; tf (x) s	= lim n→∞	E L; t+hn f (x) -EL; tf (x) hn	.	(23)
	Next, applying again the semigroup property (19) we can write
		E L; t+hn f (x) = min y∈X	hnL	d(x, y) hn	+ EL;

tf (y) .

  Notice that, if u(x, t) = EL; tf (x), and t is a point of dierentiability of t → u(x, t) for a xed x, then it follows that ∂ ∂t u(x, t)+ H(|∇ -u|(x, t)) ≥ 0. Since u is Lipschitz continuous, the above inequality holds for all x ∈ X and a.e. t ∈ R+.

	Together with (24) this implies
	E L; t+hn f (x) -EL; tf (x) hn	=	1 hn	hnL	d(x, yn) hn	+ EL; tf (yn) -EL; tf (x)
						≥ -H	[EL; tf (x) -EL; tf (yn)]+ d(x, yn)	.
	Letting now n → ∞ and using (23) we obtain
	lim inf s→0 +	EL; t+sf (x) -EL; tf (x) s	≥ lim sup n→∞	-H	[EL; tf (x) -EL; tf (yn)]+ d(x, yn)
								≥ -H |∇ -EL; tf |(x) .
		this implies that
		H	EL; tf (x) -EL; tf (yn) d(x, yn)	+ L	d(x, yn) hn	≥	EL; tf (x) -EL; tf (yn) hn	,
	and we have
	L	d(x, yn) hn	+	EL; tf (yn) -EL; tf (x) hn	≥ -H	[EL; tf (x) -EL; tf (yn)]+ d(x, yn)	.