Loïc Mazo
email: loic.mazo@unistra.fr

Nicolas Passat

Michel Couprie

Christian Ronse

Digital imaging: a unified topological framework

Keywords: digital imaging, digital topology, cubical complexes, homotopy, fundamental group 1 Introduction

In this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on Z n and equipped with a standard pair of adjacencies) as an image defined in the space of cubical complexes (F n ). In particular, it is shown that all the standard pairs of adjacencies (namely the (4, 8) and (8, 4)-adjacencies in Z 2 , the (6, 18), (18, 6), (6, 26), and (26, 6)-adjacencies in Z 3 , and more generally the (2n, 3 n -1) and (3 n -1, 2n)-adjacencies in Z n ) can then be correctly modelled in F n . Moreover, it is established that the digital fundamental group of a digital image in Z n is isomorphic to the fundamental group of its corresponding image in F n , thus proving the topological correctness of the proposed approach. From these results, it becomes possible to establish links between topology-oriented methods developed either in classical digital spaces (Z n ) or cubical complexes (F n ), and to potentially unify some of them.

sound digital topological models. Indeed, in order to segment, process, or analyse digital images in various application fields, it is often fundamental to be able to preserve, get back or integrate topological information.

Basically, an n-dimensional (digital) binary image can reasonably be considered as a subset of Z n or, equivalently, as a function from Z n to the set of values {0, 1}. However, the actual structures visualised in such images generally correspond to objects of the real world, and are thus continuous (at least at the macroscopic scale where they are considered). Consequently, they are objects of R n , and not of Z n .

In order to deal with this continuous/discrete issue, research efforts have essentially focused (since the first works, proposed nearly fifty years ago [START_REF] Rosenfeld | Sequential operations in digital picture processing[END_REF][START_REF] Duda | Graphical data processing research study and experimental investigation[END_REF]) on specific and pragmatic questions related to topology, namely the definition of a notion of adjacency relation, and the induced notions of connectivity and arcs. These notions lead, in particular, to high-level concepts of topology, such as homotopy, with natural applications to "homotopy type-preserving" image processing.

The first solution proposed to model the topology of a digital image in Z n was to consider that two points (also called xels) are adjacent if there are neighbours in the n-D cubic grid naturally induced by Z n (possibly enriched by some well chosen sets of "diagonals"). In this framework, partial solutions have been found to model the above topological properties, for instance by defining dual adjacencies for the object (composed of the xels of value 1) and the background (composed of the xels of value 0) [START_REF] Duda | Graphical data processing research study and experimental investigation[END_REF], thus enabling to define, from these adjacency relations, the notions of connectivity [START_REF] Rosenfeld | Connectivity in digital pictures[END_REF] and of digital fundamental group [START_REF] Kong | A digital fundamental group[END_REF], which permits to compare objects from a topological point of view. This historical approach is known as digital topology [START_REF] Kong | Digital topology: introduction and survey[END_REF].

Other kinds of discrete spaces, enabling to model more explicitly the continuous topological properties of digital images, have also been proposed. Among these alternative approaches of topology modelling, one can cite connected ordered topological spaces [START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF], abstract cell complexes [START_REF] Kovalevsky | Finite topology as applied to image analysis[END_REF] (which are globally equivalent) or orders [START_REF] Bertrand | New notions for discrete topology[END_REF]. Broadly speaking, they propose to put some "topological glue" between the xels of digital images in order to better define the topological links with their continuous analogues.

By comparison to these (more sophisticated) approaches, digital topology -proposed directly on Z n -may appear as the less satisfactory solution to accurately deal with topological properties of binary images. Nevertheless, digital topology remains the most commonly used framework for developing image processing tools dealing with topological issues. Indeed, since digital topology is directly defined in Z n , methods relying on it will also provide final results in Z n (namely, the actual support of the processed images), which generally constitutes a desired property in most applications. Moreover, a large literature has already been devoted to homotopy-type preservation in digital topology, especially thanks to the concept of simple point [START_REF] Bertrand | A new characterization of threedimensional simple points[END_REF][START_REF] Bertrand | A Boolean characterization of three-dimensional simple points[END_REF][START_REF] Fourey | A concise characterization of 3D simple points[END_REF][START_REF] Couprie | New characterizations of simple points in 2D, 3D and 4D discrete spaces[END_REF]. In this context, only very few methods have been based on alternative models (e.g., for image segmentation based on cubical complexes [START_REF] Cointepas | Cellular complexes: A tool for 3D homotopic segmentation in brain images[END_REF] or orders [START_REF] Daragon | Segmentation du néo-cortex cérébral depuis des données IRM dans le cadre de la topologie des ordres[END_REF]; parallel skeletonisation based on cubical complexes [START_REF] Bertrand | Two-dimensional thinning algorithms based on critical kernels[END_REF]), while digital topology has led to the design of quite numerous methods, devoted (non exhaustively) to deformable models (see, e.g., [START_REF] Han | A topology preserving level set method for geometric deformable models[END_REF][START_REF] Bai | Topology-preserving geometric deformable model on adaptive quadtree grid[END_REF]), segmentation (see, e.g., [START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF][START_REF] Dokládal | Liver blood vessels extraction by a 3-D topological approach[END_REF]), image correction (see, e.g., [START_REF] Aktouf | A three-dimensional holes closing algorithm[END_REF][START_REF] Passat | Using multimodal MR data for segmentation and topology recovery of the cerebral superficial venous tree[END_REF]), or image warping (see, e.g., [START_REF] Faisan | Topologypreserving warping of binary images according to one-to-one mappings[END_REF]).

Because of this intensive use of digital topology, it may be important to guarantee that there exists an actual compatibility between digital topology and the other proposed topological approaches (and more generally with the "continuous" topology). This requires, in particular, to be able to embed a binary image initially defined in Z n into a richer space (while respecting the chosen adjacencies in Z n ) while preserving certain topological characteristics of objects (see, e.g., [START_REF] Kong | A theory of binary digital images[END_REF][START_REF] Ayala | Digital lighting functions[END_REF]).

The "richer space" that is used here is F n , namely the space of cubical complexes, which is together a connected ordered topological space, a cellular space and an order (i.e., a poset). Though it is commonly admitted that there exists a strong link between digital topology and cubical complexes [START_REF] Kong | Topology-preserving deletion of 1's from 2-, 3-and 4dimensional binary images[END_REF], since complexes are closed objects, the images handled in F n correspond generally to images defined in Z n with a (3 n -1, 2n)-adjacency pair. In [START_REF] Bertrand | A model for digital topology[END_REF], a method is proposed to include and improve digital topology in the framework of posets, but the case of the [START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF][START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF]-and (18, 6)-adjacency pairs is not considered. In [START_REF] Ayala | Digital lighting functions[END_REF][START_REF] Ayala | Homotopy in digital space[END_REF][START_REF] Ayala | Digital homotopy with obstacles[END_REF], the authors give a way to embed digital pictures in a space of complexes according to the kind of connectivity that has been chosen in Z n . However, they do not use an intrinsic topology on complexes which are just a step between Z n and R n . This leads them to define specific notions of connectedness and digital homotopy in F n . Thereby, in this article, we propose a complete framework to correctly embed a binary digital image in the topological space F n , according to the choice of adjacencies which has been made in Z n .

The sequel of the article is organised as follows. Section 2 recalls background notions and useful notations. Section 3 describes the mapping enabling to associate a binary digital image defined on Z n to an equivalent image defined in the space of cubical complexes (F n ). A detailed study of such images is then proposed. Section 4, which is devoted to homotopy in binary images, presents the main contribution of this work. It provides the proof that the connected components and the digital fundamental group of the digital images in Z n are preserved in F n when using the mapping described in Section 3. Section 5 concludes this article by summarising the contributions and presenting further works. Auxiliary properties (used in the proofs of Sections 3 and 4) are proposed in Appendix A.

Background notions

This section provides the minimal set of background notions required to make this article globally self-contained, and then more comprehensible for the reader. Some of them are standard notions of lattice theory (Section 2.1) and can also be found in [START_REF] Birkhoff | Lattice Theory[END_REF][START_REF] Blyth | Lattices and Ordered Algebraic Structures[END_REF]. The reader interested in topological notions (Section 2.2) can find complementary information in any lecture book on general topology [START_REF] Munkres | Topology[END_REF][START_REF] Viro | Elementary Topology: Problem Textbook[END_REF] or on algebraic topology [START_REF] Maunder | Algebraic Topology[END_REF][START_REF] May | A Concise Course in Algebraic Topology[END_REF][START_REF] Hatcher | Algebraic Topology[END_REF]. The main cubical complexes notions (Section 2.3) are described, e.g., in [START_REF] Bertrand | Two-dimensional thinning algorithms based on critical kernels[END_REF][START_REF] Passat | An introduction to simple sets[END_REF], while a good introduction to digital topology may be found in [START_REF] Kong | Digital topology: introduction and survey[END_REF].

Posets

Let X be a set. A binary relation on X is a partial order if it is reflexive, antisymmetric, and transitive. A partially ordered set (or poset) is an ordered pair (X, ) where the relation is a partial order on X. We write x < y when x y and x y. The relation defined on X by x y if y

x is a partial order on X called the dual order. The covering relation ≺, associated to , is defined by: x ≺ y (say "y covers x") if x < y and there is no z such that x < z < y. We set:

x ↑ = {y ∈ X | x y} ; x ↑+ = {y ∈ x ↑ | y ↑ = {y}} ; x ↓ = {y ∈ X | y x} ; x ↓⋆ = x ↓ \ {x} = {y ∈ X | y < x} ; x ≺ = {y ∈ X | x ≺ y} . (1) An element x ∈ X is minimal if x ↓ = {x} and maximal if x ↑ = {x}. An element x ∈ X is the minimum of X if x ↑ = X and is the maximum of X if x ↓ = X.
If the minimum (resp., the maximum) exists, then it is unique.

Topological spaces

Let X be a set, the elements of which will be called points. A topology on X is a collection U of subsets of X, called open sets, such that: Let (X, ) be a poset. The set U defined by U = {U ⊆ X | ∀x ∈ U, x ↑ ⊆ U} is a topology on X which is called the Alexandroff topology. In this topology, the set x ↑ is the smallest open set containing x (or the smallest neighbourhood of x, called the star of x) and the set x ↓ is the smallest closed set containing x (the closure of x). Two points x, y ∈ X are adjacent if x y or y x. A sequence

(z i ) r i=0 (r ≥ 0) of elements of X is an arc in X (from z 0 to z r ) if for all i ∈ [[1, r]], z i-1 and z i are distinct and adjacent 1 . A subset Y of X is connected if for all x, y ∈ Y, there exists an arc in Y from x to y 2 . A connected component of a subset Y of X is a maximal (for inclusion) connected subset of Y. The closure Y ↓ of a subset Y ⊆ X is the smallest closed set including Y. The interior Y • of a subset Y ⊆ X is the largest open set included in Y (it

Cubical complexes

Let Z be the set of integers. Let

F 1 0 = {{a} | a ∈ Z} and F 1 1 = {{a, a + 1} | a ∈ Z}. Let n ≥ 1. Let f ⊂ Z n . If f is the Cartesian product of m elements of F 1
1 and nm elements of F 1 0 , we say that f is a face or an m-face (of Z n ), m is the dimension of f , and we write dim( f ) = m (some faces of Z 2 are depicted in Figure 1). We denote by F n the set composed of all faces of Z n ; this set is the (n-D) space of cubical complexes. We denote by F n k (0 ≤ k ≤ n) the set composed of all k-faces of Z n . The couple 1 For the sake of readability, a discrete interval will be noted [[a, b]], while a continuous one will be noted [a, b]. 2 In a poset, this definition is equivalent to the classical one for which a set is connected if it cannot be split in two open (or closed) non empty sets.

(F n , ⊆) is a poset. Let F ⊆ F n be a set of faces 3 . Let f ∈ F be a face. The face f is a facet of F if f is maximal in F. In particular, if x = (x i ) n i=1 ∈ Z n , the set ẋ = n i=1 {x i , x i + 1} is a facet of F n .
Some technical lemmas, essentially devoted to cubical complexes, and needed in the sequel are provided in Appendix A.

Digital topology

Let x = (x i ) n i=1 and y = (y i ) n i=1 be two distinct points in Z n (also called n-xels, or simply xels). The points x and y are 2n-adjacent if n i=1 |x i -

y i | = 1. They are (3 n -1)-adjacent if max n i=1 {|x i -y i |} = 1.
When n = 3, the points x and y are 18-adjacent if they are 26-adjacent and n i=1 |x i - [1, r]], z i-1 and z i are α-adjacent. The integer r is the length of γ. A subset X ⊆ Z n is α-connected, if for all x, y ∈ X, there exists a digital α-path from x to y in X. In order to retrieve some topological features in binary digital images (such as the notion of hole), it is necessary to use pairs of adjacencies, one for the object X and one for the background Z n \ X. The suitable pairs are (2n, 3 n -1) and (3 n -1, 2n) (plus, when n = 3, [START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF][START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF] and [START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF][START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF]).

y i | ≤ 2. Let α ∈ {2n, 3 n -1} (or possibly α = 18 if n = 3). Any point in Z n is α-adjacent to α other points. A sequence γ = (z i ) r i=0 (r ≥ 0) of points in X ⊆ Z n is a (digital) α-path 4 (from z 0 to z r ) if for all i ∈ [
3 Connectivity: from Z n to F n A (digital) image λ on Z n is a function from Z n to {0, 1}. A (complex) image µ on F n is a function from F n to {0, 1}.
The object (resp. the background) associated to the image θ : X → {0, 1} (with

X = Z n or F n ) is the set θ -1 ({1}) (resp. θ -1 ({0})).
If µ is a complex image, then we write x∈X µ(x) (resp. The poset (F n , ⊆) is equipped with its Alexandroff topology.

All along the article, we provide intuitive representations of (abstract) objects of F n by using different boxes for faces of different dimensions. Furthermore, in F 2 , the faces of the object are depicted in black while the faces of the background are in white, or are not shown, and, in F 3 , we use different colours for the faces of the object (blue: 3-faces; green: 2-faces; yellow: 1-faces; red: 0-faces) while the background is not represented.

3.1 One-to-one correspondence between images on Z n and F n When two faces g, h ∈ F n cover a face f ∈ F n and their smallest neighbourhoods do not intersect (i.e., g ↑ ∩ h ↑ = ∅), we say that their are opposite with respect to the face f (see Figure 2). We denote opp( f ) the set of all {g, h} for g opposite to h w.r.t. f . Intuitively, the face f is required to "locally connect" the faces g and h. When f is a facet, we have opp( f ) = ∅.

µ( f ) = {a,b}∈opp( f ) µ(a) ∨ µ(b) if ε(m) = 1 {a,b}∈opp( f ) µ(a) ∧ µ(b) if ε(m) = -1
Examples of regular images are depicted in Figure 3. 5 By abuse of notation, we will often write a connectivity function ε : [ [1, n]] → {-1, 1} by exhaustively providing its values, i.e., by denoting (ε(i)) n i=1 instead of ε, as done for instance in Figure 4(b,c). We will also use a dot symbol (.) when an ε(i) value does not influence the behaviour of the ε function on a given object, as done for instance in Figure 5(a). For the sake of simplicity, we will also write ε = 1 (or ε = -1) when ε is a constant function, as done for instance in Figure 4(a,d).

(2) = -1, ε(1) = 1, (b) on F 3 with ε(3) = -1, ε(2) = ε(1) = 1.
For each connectivity function ε : [ [1, n]] → {-1, 1}, we define the function ζ ε : {0, 1} Z n → {0, 1} F n which maps any digital image λ to the unique ε-regular image ζ ε (λ) such that, for each a ∈ Z n , we have ζ ε (λ)(ȧ) = λ(a). It is obvious that, for each ε, the function ζ ε is a bijection between the set of digital images {0, 1} Z n and the subset of ε-regular images of {0, 1} F n . Moreover, thanks to the choice of the connectivity function ε, we can accurately "carve" an image in F n to model the desired connectivity in Z n (see Figure 4). In particular, we can get the usual pairs of adjacencies 6 (see Figures 4567and Table 1). In Section 4, the correspondences given in Table 1 will be justified by two theorems establishing that, by following these links, we preserve the connected components and fundamental groups.

When the function ε is constant, Definition 1 can be simplified. Note that the case ε = -1 corresponds to the 2nadjacency in Z n while the case ε = 1 corresponds to the (3 n -1)-adjacency in Z n . Proposition 1 Let µ : F n → {0, 1} be an ε-regular image. Let f be a face of F n . (a) ε = ( . , . , -1) (a) ε(3

(a) ε = -1 (b) ε = (1, -1) (c) ε = (-1, 1) (d) ε = 1
(b) ε(3) = 1, ε(2) = -1 (c) ε(3) = ε(2) = 1, ε(1) = -1 (d) ε = 1
) = ε(2) = -1, ε(1) = 1 (b) ε(3) = ε(2) = ε(1) = -1
Fig. 6 The (subtle) difference between the connectivity functions (1, -1, -1) and (-1, -1, -1).

(a) ε(3) = -1, ε(2) = 1 (b) ε(3) = ε(2) = -1 (c) ε(3) = ε(1) = 1, ε(2) = -1 (d) ε(3) = 1, ε(2) = ε(1) = -1
-1) (3 n -1, 2n) Connectivity ε = -1 ε = 1 ε = -1 ε(3) = -1 ε(3) = 1 ε = 1 ε = -1 ε = 1 function ε(2) = 1 ε(2) = -1 ε(1) = -1 ε(1) = 1
Table 1 Correspondence between pairs of adjacencies in Z n and connectivity functions.

(i) If ∀m > dim( f ), ε(m) = -1, then we have µ( f ) = f ≺a µ(a) = a∈ f ↑+ µ(a) (ii) If ∀m > dim( f ), ε(m) = 1, then we have µ( f ) = f ≺a µ(a) = a∈ f ↑+ µ(a) In particular, if ε = -1 (resp. ε = 1) then µ( f ) = a∈ f ↑+ µ(a) (resp. µ( f ) = a∈ f ↑+ µ(a)) for all f ∈ F n .
Proof The proof is restricted to part (i), since part (ii) can be easily obtained by duality. We build the proof by induction

on k = n -dim( f ) ≥ 1. When k = 1, there are two facets g, h such that f ↑+ = f ≺ = {g, h} and opp( f ) = {{g, h}}. Thus, since ε(n) = -1, we have µ( f ) = {a,b}∈opp( f ) µ(a) ∧ µ(b) = µ(g) ∧ µ(h) = a∈ f ↑+ µ(a) = f ≺a µ(a). Suppose now that, for some k ∈ [[1, n -1]], ∀m ≥ n -k, ε(m) = -1 and for each face b of dimension n -k, µ(b) = b≺a µ(a) = a∈ f ↑+ µ(a).
Let f be a face of dimension nk -1. From Definition 1, we have µ( f ) = {a,b}∈opp( f ) µ(a) ∧ µ(b), and from the induction hypothesis, it then comes

µ( f ) = {a,b}∈opp( f )                 c∈a ↑+ µ(c)         ∧         c∈b ↑+ µ(c)                 and thus µ( f ) = {a,b}∈opp( f ) c∈a ↑+ ∪b ↑+ µ(c) From Lemma 7(ii) (see Appendix A), we have f ↑+ = a ↑+ ∪ b ↑+ for all (a, b) ∈ opp( f ).
Hence, we obtain :

µ( f ) = {a,b}∈opp( f ) c∈ f ↑+ µ(c) = c∈ f ↑+ µ(c)
Using again the induction hypothesis, it finally comes

f ≺a µ(a) = f ≺a b∈a ↑+ µ(b) = b∈ f ↑+ µ(b) = µ( f )
Hence, the result holds.

Duality

Let θ : X → {0, 1} with (X = Z n or F n ) be an image. We define the negative image ¬θ : X → {0, 1} of θ by ¬θ(x) = ¬(θ(x)) = 1θ(x), for all x ∈ X.

In the following, ε denotes an arbitrary connectivity function.

Proposition 2 Let µ : F n → {0, 1} be an ε-regular image. Then ¬µ is a (-ε)-regular image. Proof Let f ∈ F n m-1 , with 1 ≤ m ≤ n. Let us suppose that ε(m) = 1. Then, (¬µ)( f ) = ¬(µ( f )) = ¬( {a,b}∈opp( f ) µ(a) ∨ µ(b)).
From De Morgan's law, we infer:

(¬µ)( f ) = (a,b)∈opp( f ) ¬(µ(a)) ∧ ¬(µ(b)) = (a,b)∈opp( f ) (¬µ)(a) ∧ (¬µ)(b). The case ε(m) = -1 is similar. Whence, ¬µ is (-ε)-regular. Let µ : F n → {0, 1} be an ε-regular image. We define the image -µ : F n → {0, 1} by (-µ)( f ) = µ( f ) for all f ∈ F n
n and µ is (-ε)-regular, as exemplified in Figure 8.

Proposition 3 Let µ : F n → {0, 1} be an ε-regular image. Then we have ¬(-µ) = -(¬µ). µ ¬ ------→ ¬µ -           - -µ ¬ ------→ -(¬µ) Proof Let f ∈ F n n . We have (¬(-µ))( f ) = ¬((-µ)( f )) = ¬(µ( f )) = (¬µ)( f ) = (-(¬µ))( f ). Furthermore, -µ and ¬µ are (-ε)-regular. So, ¬(-µ) and -(¬µ) are ε-regular. Thus, ¬(-µ) = -(¬µ).
From the above definitions and propositions, we straightforwardly derive the following result.

Proposition 4 Let λ : Z n → {0, 1} be a digital image. Let ε : [[1, n]] → {-1, 1} be a connectivity function. Then, we have ¬(ζ ε (λ)) = ζ -ε (¬λ). λ ¬ ------→ ¬λ ζ ε           ζ -ε ζ ε (λ) ¬ ------→ ζ -ε (¬λ)
Propositions 3 and 4 are illustrated in Figures 8 and9 Note that (b,d) have been rotated of -π/2 along the (nearly) vertical axis in order to ease the visualisation of (d).

Remark 1 This proposition establishes that, for a given connectivity function ε (and the associated pair of adjacencies (α, β)), all the properties valid for λ -1 ({1}) and µ -1 ({1}) are also valid for λ -1 ({0}) and µ -1 ({0}) for the opposite connectivity function -ε (and the associated pair of adjacencies (β, α)). Broadly speaking, this means that the notions of object and background can be switched without loss of generality, provided that the pair of adjacencies (β, α) is also switched accordingly.

Image values and connectivity functions

The values of the facets of an ε-regular image on F n are directly provided by its associated digital image on Z n . How-ever, the values of the lower-dimensional faces depend on the connectivity function ε. In this section, we state some general properties related to links between these values and the considered connectivity functions.

The following lemma is a straightforward consequence of Definition 1.

Lemma 1 Let

µ : F n → {0, 1} be an ε-regular image. Let f ∈ µ -1 ({1}) be such that dim( f ) < n. Then, either ε(dim( f )+ 1) = -1 and there exists {a, b} ∈ opp( f ) such that µ(a) = µ(b) = 1, or ε(dim( f ) + 1) = 1 and for all {a, b} ∈ opp( f ), µ(a) = 1 or µ(b) = 1.
By applying ndim( f ) times Lemma 1, we deduce the following proposition.

Proposition 5 Let µ : F n → {0, 1} be an ε-regular image. Let f ∈ µ -1 ({1}). Then, for each k ∈ [[dim( f ) + 1, n]] there exists a face g ∈ f ↑ ∩ F n k such that µ(g) = 1.
The next proposition -easy to prove by induction-ensures that in a region of facets of uniform values, the image has a constant value in all dimensions.

Proposition 6 Let µ : F n → {0, 1} be a regular image. Let x ∈ {0, 1} and f ∈ F n . If, for each facet g in f ↑+ , µ(g) = x, then µ( f ) = x.
The following proposition establishes a partial converse of Proposition 1. In the sequel, we write f ↑+ 1 for the set of facets in the star of f which have value 1:

f ↑+ 1 = {g ∈ F n n | f g and µ(g) = 1}. Proposition 7 Let µ : F n → {0, 1} be an ε-regular image. Let f ∈ µ -1 ({1}). If there exists a unique facet in f ↑+ 1 , then ε(k) = 1 for all k ∈ [[dim( f ) + 1, n]].
Proof The proof is made par induction on ndim( f ). For dim( f ) = n, the property is obvious. Let us now suppose that the property is true for any face of dimension greater or equal to a given m ∈ [ [1, n]]. Let f ∈ F n m-1 be such that f ∈ µ -1 ({1}) and there exists a unique facet g in f ↑+ 1 . Let h be a face in f ≺ . If h ∈ f ≺ \ g ↓ (and such a face exists), none of the facets in h ↑ are in µ -1 ({1}). Thus, from Proposition 5, we derive that h µ -1 ({1}) and, by Lemma 1, we know that not all the faces h in f ≺ can verify h µ -1 ({1}). Hence, there exists h ∈ f ≺ ∩ g ↓ such that h ∈ µ -1 ({1}) and, from the induction hypothesis, we deduce that ε Hereafter, we study the value of an intersection of facets. This is an important issue, which will especially be considered in Section 4.

(k) = 1 for all k ≥ m + 1. Let (a, b) ∈ opp( f ). From Lemma 7(ii),
Proposition 8 Let µ : F n → {0, 1} be an ε-regular image. Let f, g be two distinct facets of µ -1 ({1}) with f ∩ g ∅. If ε(k) = 1 for all k ∈ [[dim( f ∩ g) + 2, n]], then µ( f ∩ g) = 1. Conversely, if µ( f ∩ g) = 1 and the only facets in ( f ∩g) ↑+ 1 are f and g, then ε(k) = 1 for all k ∈ [[dim( f ∩ g) + 2, n]]. Proof We set h = f ∩ g and d = dim(h) + 1. Since µ( f ) = µ(g) = 1 and µ(a) = b∈a ↑+ µ(b) for each face a such that dim(a) ∈ [[d, n -1]] (Proposition 1
), all the k-faces included in f or g, with k ≥ d, are in µ -1 ({1}). In particular, thanks to Lemma 6, we derive h ≺ ⊆ µ -1 ({1}) and, therefore, whatever is the value of ε(d), h ∈ µ -1 ({1}). Conversely, any face a ∈ h ≺ is either included in f or in g (Lemma 6), but not in both of them. Since the only facets in ( f ∩g) ↑+ 1 are f and g, we can use Proposition 7 to conclude that ε(k) = 1 for all k ≥ d + 1. 10 for examples). Let f, g be two facets of

Proposition 9 Let µ : F n → {0, 1} be an ε-regular image. Let f, g be two distinct facets of µ -1 ({1}) with f ∩ g ∅. If µ( f ∩ g) = 1, then there exists an arc q = (q i ) 2r i=0 (r ≥ 0) from f to g in ( f ∩ g) ↑ ∩ µ -1 ({1}) such that, for all i ∈ [[0, r]], q 2i ∈ F n n and for all i ∈ [[0, r -1]], q 2i+1 = q 2i ∩ q 2(i+1) ∈ F n m with m ∈ [[k, n -1]] where k ∈ [[0, n -1]] is the lowest integer such that ε(i) = 1 for all i ∈ [[k + 2, n]]. Proof Let µ be an ε-regular image and k ∈ [[0, n -1]] be an integer such that ε(i) = 1 for all i ∈ [[k + 2, n]] and ε(k + 1) = -1 (see Figure
µ -1 ({1}) such that f ∩ g ∅ and µ( f ∩ g) = 1. The proof is carried out by induction on k -dim( f ∩ g). If k ≤ dim( f ∩ g),
the property is obvious since we can set q = ( f, f ∩ g, g).

We suppose now that the property is true for any facets i, j

such that dim(i ∩ j) ≥ k -m for some m ∈ [[0, k -1]]. Let f, g be such that dim( f ∩ g) = k -m -1.
There exists a face a such that f ∩ g ≺ a and µ(a) = 1 (Lemma 1) and a is included in f or in g (Lemma 6) but not in both of them. Since, ε(k + 1) = -1 and k + 1 ≥ dim(a) + 1, we derive from Proposition 7 that there exists a facet h, h f and h g, in a ↑ ∩ µ -1 ({1}). It is easy to see that as f, g, h are three facets in F n such that f ∩ g ⊂ h, then dim( f ∩ h) and dim(g ∩ h) are strictly greater than dim( f ∩ g) so we can use the induction hypothesis: there exist two arcs in ( f ∩ g) ↑ ∩ µ -1 ({1}), q 1 from f to h and q 2 from h to g, whose faces of odd rank are facets of F n and faces of even rank i are the intersection of the facets of ranks i -1 and i + 1 with a dimension belonging to [[k, n -1]]. Therefore, there exists an arc from f to g in ( f ∩ g) ↑ ∩ µ -1 ({1}) whose faces satisfy all the conditions of the proposition.

Computing values directly from facets

The aim of this section is to find the number of facets which must have the value 1 in the star of a face to ensure that this face also has value 1. In F 2 , the answer is straightforward. In F 3 , it requires to carefully study a particular configuration (depicted in Figure 11), however, it can be answered, as stated hereafter. In higher dimensional spaces, the particular configurations to study are too numerous to get a useful result.

a) ε(3) = -1 (b) k = n -1 = 2 (c) ε(3) = 1, ε(2) = -1 (d) k = n -2 = 1
Let f ∈ F n , with n ≥ 3. If dim( f ) = n -3, the poset ( f ↑ , ⊆) has a unique minimum, namely f , and 8 maximal elements, namely the facets forming f ↑+ . From an adjacency point of view, these facets are geometrically organised as the 8 vertices of a cubical structure. When f ↑+ 1 (i.e., the facets of f ↑+ whose values are equal to 1) is organised as in the configuration depicted in Figure 11(a) (up to rotations and symmetries), we say that f ↑+ 1 is a trihedron. We define Card -(E) = 3 and Card

+ (E) = 5, if E is a trihedron, and Card -(E) = Card + (E) = Card(E) otherwise.
For each connectivity function ε, we define recursively the function δ ε by

δ ε : [[0, n]] → [[1, 2 n ]] 0 → 1 i + 1 → 2δ ε (i) -1 if ε(n -i) = 1 2δ ε (i) if ε(n -i) = -1
It is easy to check that, for all m ∈ [[0, n]], we have

δ ε (m) = 1 + m k=1 (1 -ε(n -k + 1))2 m-k-1
Proposition 10 Let µ : 2 Necessary and sufficient conditions to obtain µ( f ) = 1 (see Corollary 1).

F n → {0, 1} be an ε-regular image. Let f be a k-face of F n (with n -3 ≤ k ≤ n -1). (a) (b) ε(3) = 1, ε(2) = ε(1) = -1 (c) ε(3) = -1, ε(2) = ε(1) = 1 (d) ε(3) = ε(1) = -1, ε(2) = 1
dim( f ) = n -2 ε (. . . , 1, 1) (. . . , -1, 1) (. . . , 1, -1) (. . . , -1, -1) C ε Card( f ↑+ 1 ) ≥ 1 Card( f ↑+ 1 ) ≥ 2 Card( f ↑+ 1 ) ≥ 3 Card( f ↑+ 1 ) ≥ 4 dim( f ) = n -3 ε (. . . , 1, 1, 1) (. . . , -1, 1, 1) (. . . , 1, -1, 1) (. . . , -1, -1, 1) C ε Card( f ↑+ 1 ) ≥ 1 Card( f ↑+ 1 ) ≥ 2 Card( f ↑+ 1 ) ≥ 3 Card( f ↑+ 1 ) ≥ 4 f ↑+ 1 not a trihedron ε (. . . , 1, 1, -1) (. . . , -1, 1, -1) (. . . , 1, -1, -1) (. . . , -1, -1, -1) C ε Card( f ↑+ 1 ) ≥ 5 Card( f ↑+ 1 ) ≥ 6 Card( f ↑+ 1 ) ≥ 7 Card( f ↑+ 1 ) = 8 or f ↑+ 1 a trihedron Table
(i) If Card -( f ↑+ 1 ) ≥ δ ε (n -k), then µ( f ) = 1. (ii) If µ( f ) = 1, then Card + ( f ↑+ 1 ) ≥ δ ε (n -k). Proof We set m = n -dim( f ) = n -k (note that 1 ≤ m ≤ 3). If m = 1, then δ ε (n -k) = δ ε (1) is equal to 1 if ε(n) = 1 and to 2 if ε(n) = -1.
Since, in the case m = 1, f ↑+ contains exactly two faces, which are opposite facets with respect to f , the statements (i) and (ii) are obviously true for m = 1. Now let us suppose that m ∈ [ [START_REF] Duda | Graphical data processing research study and experimental investigation[END_REF][START_REF] Rosenfeld | Connectivity in digital pictures[END_REF]] and that the statements are true for m -1.

First case:

ε(dim( f ) + 1) = -1. If Card -( f ↑+ 1 ) ≥ δ ε (m) = 2δ ε (m -1)
, then we deduce from Lemma 8 that there exist two faces g and h, opposite with respect to f , such that Card(g ↑+ 1 ) ≥ δ ε (m -1) and Card(h ↑+ 1 ) ≥ δ ε (m -1). Since the dimension of the faces g and h is greater than or equal to n -2, we get Card(g ↑+ 1 ) = Card -(g ↑+ 1 ) and Card(h ↑+ 1 ) = Card -(h ↑+ 1 ) so we can use the induction hypothesis to derive that µ

(g) = µ(h) = 1. Recalling that ε(dim( f ) + 1) = -1, we have µ( f ) = {a,b}∈opp( f ) µ(a) ∧ µ(b) ≥ µ(g) ∧ µ(h) = 1,
and thus µ( f ) = 1. Then (i) holds. Conversely, if µ( f ) = 1, thanks to Lemma 1 we know that there exists {g, h} ∈ opp( f ) such that µ(g) = µ(h) = 1. Hence, by the induction hypothesis, Card + (g ↑+ 1 ) = Card(g ↑+ 1 ) ≥ δ ε (m -1) and Card + (h ↑+ 1 ) = Card(h ↑+ 1 ) ≥ δ ε (m -1). Since g and h are opposite with respect to f , we derive from Lemma 7 that Card( Lemma 7) in each couple of opposite faces with respect to f there exists a face g such that Card -(g ↑+ ) = Card(g ↑+ 1 ) ≥ δ ε (m -1), that is, thanks to the induction hypothesis, in each couple of opposite faces with respect to f there exists a face whose value is equal to 1. Thus, µ( f

f ↑+ 1 ) = Card(g ↑+ 1 ) + Card(h ↑+ 1 ). Hence, Card + ( f ↑+ 1 ) ≥ Card( f ↑+ 1 ) ≥ 2δ ε (m -1) = δ ε (m). Then (ii) holds. Second case: ε(dim( f ) + 1) = 1. If Card -( f ↑+ 1 ) ≥ δ ε (m) = 2δ ε (m -1) -1, then (from
) = {a,b}∈opp( f ) µ(a) ∨ µ(b) ≥ {a,b}∈opp( f ) 1 = 1.
Then (i) holds. Conversely, if µ( f ) = 1, thanks to Lemma 1 and the induction hypothesis, we know that in each couple of opposite faces with respect to f there exists a face g such that Card + (g ↑+ ) = Card(g ↑+ 1 ) ≥ δ ε (m -1). We deduce from Lemma 9 that Card(

f ↑+ 1 ) ≥ 2δ ε (m -1) -1 = δ ε (m). Then (ii) holds.
From Proposition 10 and Definition 1 (needed when f ↑+ 1 is a trihedron), we derive the following corollary.

Corollary 1 Let µ : F n → {0, 1} be an ε-regular image. Let f be a k-face of F n (with n -3 ≤ k ≤ n -1). Then µ( f ) = 1 iff the set f ↑+ 1 satisfies the condition C ε given in Table 2.

Regular images and regular open/closed sets

We have defined the object (resp. the background) of a regular image µ : F n → {0, 1} as the set µ -1 ({1}) (resp. µ -1 ({0})).

We show in this section that they are topologically regular, i.e., they do not have thin parts nor thin holes (by "thin", we mean of lower dimension than the surrounding space).

Lemma 2 Let µ : F n → {0, 1} be an ε-regular image. Let x ∈ {0, 1} Then, we have

((µ -1 ({x})) ↓ ) • ⊆ µ -1 ({x}) ⊆ ((µ -1 ({x})) • ) ↓ Proof
The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). Let E be a subset of F n . The two following chains of implications are easy to establish:

f ∈ (E ↓ ) • ⇒ f ↑ ⊆ E ↓ ⇒ f ↑+ ⊆ E ↓ ⇒ f ↑+ ⊆ E ∃g ∈ f ↑+ , g ∈ E ⇒ ∃g f, g ∈ E • ⇒ f ∈ (E • ) ↓
Now, we take E = µ -1 ({1}). From Proposition 6, we have

f ↑+ ⊆ E ⇒ f ∈ E. From Proposition 5, we also have f ∈ E ⇒ ∃g ∈ f ↑+ , g ∈ E.
Hence, the assertion holds:

E ↓ • ⊆ E ⊆ E •↓ .
Proposition 11 Let µ : F n → {0, 1} be an ε-regular image. Let x ∈ {0, 1}. Then (µ -1 ({x}))

• is a regular open set and

(µ -1 ({x})) ↓ is a regular closed set.
Proof The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). Let E = µ -1 ({1}). From Lemma 2, we have (E ↓ )

• ⊆ E ⊆ (E • ) ↓ . Hence, we readily derive:

E ↓ •↓ ⊆ E ↓ ⊆ E •↓ ↓ = E •↓ ⊆ E ↓ •↓
where the last inclusion comes from E ⊆ E ↓ . Therefore, we get the equality E ↓ •↓ = E ↓ which means that E ↓ is a regular closed set. The proof of the equality

E •↓ • = E • is similar. Corollary 2 Let µ : F n → {0, 1} be an ε-regular image. If ε = -1, then µ -1 ({1}) (resp. µ -1 ({0})) is a regular open (resp. closed) set. If ε = 1, then µ -1 ({1}) (resp. µ -1 ({0})) is a regular closed (resp. open) set.
Proof The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). If ε = -1, we readily have

g f ⇒ g ↑+ ⊆ f ↑+ ⇒ h∈g ↑+ µ(h) ≥ h∈ f ↑+ µ(h) ⇒ µ(g) ≥ µ( f ). Thus, g f and f ∈ µ -1 ({1}) imply g ∈ µ -1 ({1}), that is µ -1 ({1}) is an open set. It follows that µ -1 ({1}) = (µ -1 ({1})) • is a regular open set. Similarly if ε = 1, g f implies µ(g) ≥ µ( f ). Hence, µ -1 ({1}) is a regular closed set.

Paths and (digital) fundamental groups

In this section, we study how the functions ζ ε defined in Section 3 behave relatively to the classical notions of path in Z n and F n . The main results are Theorems 3 and 4. Theorem 3 states that ζ ε induces a bijection between the set of the connected components of the object (resp. background)

λ -1 ({1}) ⊆ Z n (resp. λ -1 ({0}) ⊆ Z n ) associated to an image λ : Z n → {0, 1}
, and the set of the connected components of the object (resp. background) µ -1 ({1}) ⊆ F n (resp. µ -1 ({0}) ⊆ F n ) associated to the regular image µ : F n → {0, 1}, defined by µ = ζ ε (λ), the function ε being chosen with respect to a given pair of adjacencies in Z n . Theorem 4 states that ζ ε induces an isomorphism between the digital fundamental group of λ -1 ({1}) (resp. λ -1 ({0})) and the fundamental group of µ -1 ({1}) (resp. µ -1 ({0})).

Background notions on paths and arcs

We recall the classical definitions of path equivalence, fundamental group in a topological space (in particular, in a poset), and of digital path equivalence and digital fundamental group in Z n .

The fundamental group of topological spaces

Let X be a topological space. A path p in X is a continuous function p : [0, 1] → X. Two paths p, q in X are equivalent if they have the same extremities (i.e., p(0) = q(0) and p(1) = q(1)) and p can be continuously deformed to fit q, that is if there exists a continuous map h : [0, 1] × [0, 1] → X such that, for all t ∈ [0, 1], h(t, 0) = p(t) and h(t, 1) = q(t), and, for all u ∈ [0, 1], h(0, u) = p(0) = q(0) and h(1, u) = p(1) = q(1) (the map h is called a path-homotopy). This relation on paths is actually an equivalence relation. We write [p] for the equivalence class of p. If p, q are two paths in X such that p(1) = q(0) we can define the product p • q by

(p • q)(t) = p(2t) if t ∈ [0, 1 2 ] q(2t -1) if t ∈ [ 1 2 , 1]
This product is well defined on equivalence classes by [p] • [q] = [p • q]. Let x be a point of X. A loop at x is a path in X which starts and ends at x. The product of two loops at x is a loop at x and the set π(X, x) of equivalence classes of loops at x is a group for this product. It is called the fundamental group of X (with basepoint x).

Finite paths in posets

In a poset X, a function f : [0, 1] → X is a step function if there exist finitely many intervals

(I i ) r i=0 (r ≥ 0) such that f is constant on each interval I i and [0, 1] = r i=0 I i . If for all i ∈ [[1, r]], sup(I i-1 ) = inf(I i ) and f (I i-1 )
f (I i ), we write f = r i=0 x i 1 I i where {x i } = f (I i ). A finite path in X is a path in X which is a step function. The sequence (I i ) r i=0 is called the intervals sequence of p and the sequence (x i ) r i=0 the track of p. A finite path is regular if there is no singleton in its intervals sequence. Two comparable paths (for the componentwise order), finite or not, with same extremities are equivalent. In the sequel, we denote by the componentwise order on paths. The next four results have been proved in [START_REF] Mazo | Paths, homotopy and reduction in digital images[END_REF].

Proposition 12 The track of a finite path is an arc and any arc is the track of a regular finite path.

Proposition 13 A step function p = r i=0 x i 1 I i is a finite path in X iff for all i ∈ [[0, r -1]], x i is adjacent to x i+1 and x i x i+1 ⇔ sup(I i ) ∈ I i .

The product of two arcs (x i ) r i=0 and (y i ) s i=0 is defined by (x i ) r i=0 .(y i ) s i=0 = (x 0 , . . . , x r , y 1 , . . . , y s ) provided that x r = y 0 .

An arc χ = (x i ) r i=0 (r ≥ 2) is an elementary stretching (in X) of an arc χ ′ if for some j ∈ [[1, r -1]], χ ′ = (x i ) r i=0,i j or (x j-1 = x j+1 and χ ′ = (x i ) r i=0,i j-1,i j
). An arc χ is a deformation of an arc χ ′ if there exists a sequence (χ i ) s i=0 of arcs in X such that χ 0 = χ, χ s = χ ′ and for any i ∈ [ [1, s]], either χ i is an elementary stretching of χ i-1 or vice versa. Let x be a point in X. "Being a deformation or equal" is an equivalence relation in the set of arcs in X from x to x. The set of equivalence classes, denoted by ρ(X, x), is a group for the arc product.

Theorem 1 Two finite paths are equivalent iff they have the same track or the track of one of them is a deformation of the track of the other.

Theorem 2 Let x ∈ X. The fundamental group π(X, x) of X with basepoint x is isomorphic to the group ρ(X, x).

The digital fundamental group of Z n

A discrete analogue of the concept of fundamental group has been proposed in digital topology [START_REF] Kong | A digital fundamental group[END_REF]. Let n ∈ {2, 3} and X ⊆ Z n . The definition of the product for digital paths is straightforward but not so is the notion of equivalence between digital paths or loops. Two paths (a i ) r i=0 , (b i ) s i=0 (r, s ≥ 0) with same extremities are directly equivalent (in X) 7 if they differ only in a unit lattice cube of Z n provided that, if n = 3 and the pair of adjacencies is [START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF][START_REF] Bertrand | A model for digital topology[END_REF], the cube must not contain two diametrically opposite points not in X.

Finally, two paths p 0 , p t (t ≥ 0) with same extremities are equivalent (in X) if there is a sequence (p i ) t i=0 of paths such that, for all i ∈ [[1, t]], p i is directly equivalent to p i-1 . 7 In [START_REF] Fourey | A concise characterization of 3D simple points[END_REF], we find another (equivalent) form of the previous definition. Two paths p, q are directly equivalent if there exist four paths p 0 , p 1 , p ′ 1 , p 2 such that p = p 0 .p 1 .p 2 , q = p 0 .p ′ 1 .p 2 , p 1 and p ′ 1 have the same extremities and are both included in a 2 × 2 square if n = 2 or n = 3 and the pair of adjacencies is [START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF][START_REF] Bertrand | A model for digital topology[END_REF], or in a 2 × 2 × 2 cube otherwise.

Mapping paths in Z n onto arcs in F n

Let χ and χ ′ be two arcs in F n . We write8 χ χ ′ if there exist two paths p p ′ in F n whose tracks are χ and χ ′ (all paths in F n considered in the sequel are regular finite paths).

Definition 2

Let ω be an adjacency relation on Z n and γ = (p i ) r i=0 (r ≥ 0) be an ω-path in Z n . We define the arc ζ(γ) in F n by ζ(γ) = (q j ) 2r j=0 with q j = ṗ j 2 if j is even and q j = q j-1 ∩ q j+1 if j is odd, for all j ∈ [[0, 2r]].

It is obvious that the sequence of faces ζ(γ) defined above is actually an arc in F n which is itself the track of a regular finite path in F n (Proposition 12).

The following proposition states that ζ associates to a path in the object (resp. in the background), of a digital image λ, an arc in the object (resp. in the background) of the complex image 9 ζ ε (λ) under the condition that the connectivity function ε has been well chosen. The main consequence of this proposition is that the images of the connected components of the digital object (resp. background) are included in the connected components of the image of the object (resp. background).

Proposition 14 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β) (see Table 1). Let x ∈ {0, 1}.

Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Let γ be an ω-path in λ -1 ({x}). Then, ζ(γ) is an arc in µ -1 ({x}).
Proof The proof is proposed for (α, β) = (3 n -1, 2n) and (n = 3 and (α, β) = [START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF][START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF]). The cases (α, β) = (2n, 3 n -1) and (n = 3 and (α, β) = (6, 18)) are obtained by duality (see Remark 1). Case (α, β) = (3 n -1, 2n). Here,

ε(k) = 1 for all k ∈ [[1, n]]. If γ = (p i ) r i=0 (r ≥ 0) is an α-path in λ -1 ({1}), from Propo- sition 8, we derive directly that µ( ṗ i-1 ∩ ṗi ) = 1 for all i ∈ [[1, r]], that is ζ(γ) is an arc in µ -1 ({1}). If γ is a β-path in λ -1 ({0}), then dim( ṗ i-1 ∩ ṗi ) = n -1 and thus µ( ṗ i-1 ∩ ṗi ) = µ( ṗ i-1 ) ∨ µ( ṗi ) = 0 ∨ 0 = 0 for all i ∈ [[1, r]]. So, ζ(γ) is an arc in µ -1 ({0}).
Case n = 3 and (α, β) = [START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF][START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF]. The connectivity function ε is such that ε(3) = 1 and ε(2) = -1. If γ is an 18-path in λ -1 ({1}), then all the faces of γ have a dimension greater or equal to 1, thereby we can use the direct part of Proposition 8 to conclude that ζ(γ) is an arc in µ -1 ({1}). If γ is a 6-path in λ -1 ({0}), we use the proof of the case (α, β) = (3 n -1, 2n).

The following proposition is straightforward.

Proposition 15

Let ω be an adjacency relation on Z n . The corresponding function ζ is a homomorphism for the paths product and the arc product: for all ω-paths γ, γ

′ ∈ Z n , ζ(γ.γ ′ ) = ζ(γ).ζ(γ ′ ).
The injectivity of ζ is obvious since two distinct n-xels a, b ∈ Z n are associated to distinct facets ȧ, ḃ ∈ F n . Proposition 16 establishes the surjectivity up to deformations: any arc χ from ȧ to ḃ in an object (resp. in the background) of the complex image is the deformation of an arc ζ(γ) for some path γ from a to b of the object (resp. background) of its associated digital image (if the complex image is associated to such a digital image). Lemma 3 is the basic block for the proof of Proposition 16.

Lemma 3 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β). Let x ∈ {0, 1}. Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Let p be a path in µ -1 ({x}) whose track is ( f, g, h) with f ⊃ g ⊂ h. Let a, b ∈ Z n be two xels in λ -1 ({x}) such that ȧ ⊃ f and h ⊂ ḃ. Then, there exists an ω-path γ in λ -1 ({x}) from a to b and a path q, q p, whose track is ζ(γ).

Fig. 12 In grey and red: an arc in F 3 in a (-1, 1, -1)-regular image. In this arc, two consecutive grey facets share a 0-face (in red) but their intersection does not belong to the object.

Proof The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). If a = b, we set γ = (a) and q = ȧ1 [0,1] and we are done. Let us now suppose that a b. We set g ′ = ȧ ∩ ḃ. Let p = f 1 I + g1 J + h1 K (where I, J, K are intervals included in [0, 1]) be a path. Since ȧ ⊃ f ⊃ g ⊂ h ⊂ ḃ, we have g ⊂ g ′ . If ε = -1, since g ∈ µ -1 ({1}), from Proposition 1 we derive that all facets in g ↑ have value 1 and, therefore, all facets in g ′↑ have value 1. Thus, g ′ is in µ -1 ({1}). If ε = 1, still from Proposition 1, we derive that g ′ ∈ µ -1 ({1}) since there exists at least one facet in µ -1 ({1}) ∩ g ′↑ . If n = 3 and ε(3) = 1, we derive from Proposition 8 that if dim(g ′ ) ≥ 1, then g ′ ∈ µ -1 ({1}) and if dim(g ′ ) = 0, then g ′ = g so, g ′ ∈ µ -1 ({1}). In all the previous cases, we set γ 0 = (a, b) and q 0 = ȧ1 I + g ′ 1 J + ḃ1 K . Observe, before examining the last case, that we cannot claim that γ 0 is an ω-path, but just a (3 n -1)-path, since we do not know if a and b are ω-adjacent. Now, let us consider the case n = 3 and ε = (-1, 1, -1) which corresponds to (α, β) = [START_REF] Khalimsky | Computer graphics and connected topologies on finite ordered sets[END_REF][START_REF] Mangin | From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations[END_REF]. In this case, if dim(g ′ ) = 1 and dim(g) = 0, it may happen that g ′ µ -1 ({1}) (see Figure 12). Let us suppose that g ′ µ -1 ({1}) (and thus, dim(g ′ ) = 1 and dim(g) = 0). From Proposition 10 and Table 2, we know that there are at least 6 facets in g ↑+ 1 and at most 2 facets in g ′↑+ 1 (and actually, there are two facets in g ′↑+ 1 , ȧ and ḃ). It follows that the facet ċ, whose intersection with ȧ is g, is in µ -1 ({1}) (see Figure 12 to look at the only possible configuration), and ċ shares a 2-face with ḃ. We call this 2-face h ′ . It is plain that h ′ ∈ µ -1 ({1}). So, we set γ 0 = (a, c, b) and q 0 = ȧ1 (a,c,b), we build in λ -1 ({1}) an ω-path γ from a to b. Since this χ is in y ↑ , one can readily build in µ -1 ({1}) a path q q 0 whose track is ζ(γ). Hence, the result holds.

I +g1 J 1 + ċ1 J 2 + h ′ 1 J 3 + ḃ1 K where J 1 ∪ J 2 ∪ J 3 = J, max(J 1 ) = inf(J 2 ),
Proposition 16 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β). Let x ∈ {0, 1}. Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Let a, b ∈ Z n . Let χ be an arc from the facet ȧ to the facet ḃ in µ -1 ({x}). Then, there exists an ω-path γ from a to b in λ -1 ({x}) such that ζ(γ) is a deformation in µ -1 ({x}) of χ in µ -1 ({x}). Moreover, if p is a path in µ -1 ({x}) whose track is χ, there exists a path q in µ -1 ({x}), q p, whose track is ζ(γ).

Proof The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). Let χ = (q i ) r i=0 (r ≥ 0) be an arc from the facet ȧ to the facet ḃ in µ -1 ({1}) and p = r i=0 q i 1 I i be a path whose track is χ (Proposition 12). We build the proof by induction on r, the size of χ. If r = 0, the statement is obvious. Let us now suppose that the property is true whenever the size of χ is strictly less than an integer r ≥ 2 (r = 1 is impossible). Let j ∈ [[1, r -1]] be the lowest integer such that q j is a local minimum of χ, i.e., such that q j-1 ⊃ q j ⊂ q j+1 (remember that q 0 = ȧ and q r = ḃ are facets) and let k ∈ [[ j + 1, r]] be the lowest integer such that q k is a local maximum of χ, i.e. q k-1 ⊂ q k ⊃ q k+1 . Let t be the centre of the interval I k and ϕ : [0, 1] → [0, 1] be the bijective function which puts t on 1 2 and is linear between 0 and t and between t and 1. We have p

• ϕ -1 = ( k i=0 q i 1 (2ϕ)(I ′ i ) ).( r i=k q i 1 (2ϕ-1)(I ′′ i ) )
where

I ′ i = I i if 0 ≤ i < k, I ′ k = I k ∩ [0, t], I ′′ k = I k ∩ [t, 1] and I ′′ i = I i if k < i ≤ r.
From Proposition 5, we derive that there exists a xel c ∈ Z n , such that c ∈ λ -1 ({1}) and q k ⊆ ċ (if q k is a facet of F n , ċ = q k ). As ċ1 (2ϕ-1)(I ′′ k ) + r i=k+1 q i 1 (2ϕ-1)(I ′′ i ) is a path from ċ to ḃ whose length is strictly lower than r, by the induction hypothesis, there exists an ω-path γ (2) from c to b in λ -1 ({1}) and a path q (2) from ċ to ḃ such that ζ(γ (2) ) is the track of q (2) and q (2) ċ1 (2ϕ-1)(I ′′ k ) + r i=k+1 q i 1 (2ϕ-1)(I ′′ i ) r i=k q i 1 (2ϕ-1)(I ′′ i ) . On the other side, from Lemma 3, we can define in λ -1 ({1}) an ω-path γ (1) from a to c and a path in µ -1 ({1}), q (1) ȧ1 (2ϕ)(I) + q j 1 (2ϕ)(I ′ j ) + ċ1 (2ϕ)(J) ) ). Eventually, we set γ = γ (1) .γ (2) and q = q (1) .q (2) . As ζ is a morphism (Proposition 15), ζ(γ) is the track of q and from Lemma 10, we have q p • ϕ -1 , thereby q • ϕ p. Of course, q • ϕ has the same track as q, so the result holds: since q • ϕ p, the track ζ(γ) of q • ϕ is a deformation of the track χ of p (Theorem 1).

k i=0 q i 1 (2ϕ)(I ′ i ) , where I = j-1 i=0 I ′ i and J = k i= j+1 I ′ i , whose track is ζ(γ ( 1
Theorem 3 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β). Let x ∈ {0, 1}. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Then the function which associates to the xel a ∈ Z n the facet ȧ ∈ F n induces a oneto-one correspondence between the connected components of λ -1 ({x}) and the connected components of µ -1 ({x}) Proof The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). Let ≡ λ (resp. ≡ µ ), be the equivalence relation defined on λ -1 ({1}) (resp. µ -1 ({1})), by a ≡ λ b (resp. a ≡ µ b), iff a and b belong to the same connected component of λ -1 ({1}) (resp. µ -1 ({1})), i.e., there exists a path from a to b in λ -1 ({1}) (resp. µ -1 ({1})). From Proposition 14, we can define the quotient map ζ from the set of connected components of λ -1 ({1}) to the set of connected components of µ -1 ({1}). Let C be a connected component of µ -1 ({1}) and f be a face in C. From Proposition 5, we derive that f is at least included in one facet ȧ ∈ µ

-1 ({1}) so ȧ = ζ(a) ∈ C (for ( f, a) is an arc). Hence, C = ζ(C a )
where C a is the connected component of λ -1 ({1}) including a and, as we have not made any hypothesis on C, ζ is surjective. Let a, b ∈ Z n be to xels such that ȧ ≡ µ ḃ. Proposition 16 indicates that there exists an ω-path from a to b in λ -1 ({1}) so a ≡ λ b. This establishes the injectivity of ζ, so ζ is a bijection.

Fundamental groups

The aim of this section is to compare the digital fundamental group defined by Kong [START_REF] Kong | A digital fundamental group[END_REF] for subsets of Z n , n ∈ {2, 3}, with the fundamental group of subspaces of F n . Thanks to Theorem 2, we can use arcs as well as paths in F n in order to perform this comparison.

Proposition 17 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to

(α, β). Let x ∈ {0, 1}. Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Let γ, γ ′ be two equivalent ω- paths in λ -1 ({x}). Then, ζ(γ ′ ) is a deformation of ζ(γ) in µ -1 ({x}).
Proof The proof is proposed for x = 1. The case x = 0 is obtained by duality (see Remark 1). Since ζ is a homomorphism (Proposition 15) and "being a deformation" is an equivalence relation, we derive from the definition of digital homotopy equivalence (see section 4.1.3) that it suffices to establish the lemma for two digital paths γ, γ ′ having the sames extremities and included in a 2 × 2 square or in a 2 × 2 × 2 cube. That is, the loop ζ(γ.γ ′-1 ) can be deformed (in µ -1 ({x})) in a constant arc. So, let γ be a closed path in λ -1 ({1}) from a to a (a ∈ Z n ) included in a unit square (if n = 2 or n = 3 and β = 26), or in a unit cube. If the length of γ is less than or equal to 2 then the property is obvious. Now, we assume that the length of γ is greater than or equal to 3. We denote by m the face in F n laying at the centre of the unit square or cube C including ζ(γ) and we observe that the arc ζ(γ) is included in m ↑ . Since at least three facets of C belongs to µ -1 ({1}), we derive from Table 2 that, if α = 3 n -1 or (n = 3 and α = 18), then m ∈ µ -1 ({1}) and we can easily build10 a sequence of elementary stretchings from (a) to ζ(γ) 11 . Hence, from now, we suppose that α = 2n. To end the proof, we check all the loops γ which can be drawn in a unit square or a unit cube in Z n (up to rotations and symmetries). We can suppose without loss of generality that ζ(γ) is a simply closed arc in the sense that each face in ζ(γ) is passed through only once. The reader can check (in Figure 13) that in each case the arc ζ(γ) can be deformed (in µ -1 ({1})) in a constant arc by introducing m in ζ(γ).

Lemma 4 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β). Let x ∈ {0, 1}. [START_REF] Cointepas | Cellular complexes: A tool for 3D homotopic segmentation in brain images[END_REF] We use the dual order in F n to represent the unit square/cube: the facets are depicted by points, the (n -1)-faces by edges and so on (the face m is then seen as the square {a, b, c, d} or the cube {a, b, c, d, e, f, g, h}). The arc ζ(γ) is drawn like a solid black curve. Since ζ ε (λ) is a regular image, whatever the connectivity function ε, when two 4/6-adjacent vertices (which are actually facets) of the depicted polyhedron are on ζ(γ), the edge between them belongs to λ -1 ({1}) and when the four edges of a square are on ζ(γ) then the square between them belongs to λ -1 ({1}). (a) All the vertices, and thus m, are in λ -1 ({1}). (b, c, d) These three figures are only concerned by (α, β) = (6, 18), i.e., ε = (-1, 1, -1). In each figure, the length of γ is greater than or equal to 6, so, from Table 2

Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Let a, b ∈ Z n . Let χ, χ ′ be two arcs in µ -1 ({x}) from ȧ to ḃ (a, b ∈ Z n ). If χ ′ is
m ∈ λ -1 ({1}). ( f i ) i 0 i=0 .( f i 0 , f, f i 0 +1 ).( f i ) r i=i 0 +1 with i 0 ∈ [[0, r -1]].
We can assume without loss of generality that f i 0 ⊂ f i 0 +1 . If f ⊂ f i 0 +1 , from Proposition 16, we can build an ω-path γ in λ -1 ({x}) from a to b and two paths p = r i=0 f i 1 I i and q in µ -1 ({x}) whose tracks are χ and ζ(γ) and such that q p. We set y = max(I i 0 ) = inf(I i 0 +1 ) (since f i 0 ⊂ f i 0 +1 , the interval I i 0 is closed on the right (Proposition 13)). Let δ > 0 be a real such that y-2δ ∈ I i 0 and y+2δ ∈ I i 0 +1 (δ exists since p is regular). We set

p ′ = i 0 -1 i=0 f i 1 I i + f i 0 1 I + f 1 I ′ + f i 0 +1 1 I ′′ + r i=i 0 +2 f i 1 I i where I ′ = [y -δ, y + δ], if f ⊂ f i 0 and I ′ = ]y, y + δ] if f i 0 ⊂ f ⊂ f i 0 +1 , I = I i 0 \ I ′ and I ′′ = I i 0 +1 \ I ′ . Proposition 13
ensures that p ′ is actually a path and it is plain that the track of p ′ is χ ′ and p ′ p. We have thereby p ′ p q. Hence, ζ(γ) χ and ζ(γ) χ ′ . If f i 0 +1 ⊂ f , still using Proposition 16, we start to build an ω-path γ ′ in λ -1 ({x}) from a to b and two paths p ′ = i 0 i=0 f i 1 I i + f 1 I + r i=i 0 +1 f i 1 I i , q ′ in µ -1 ({x}) whose tracks are χ ′ and ζ(γ ′ ) and such that q ′ p ′ . We set p = i 0 i=0 f i 1 Proof Let p, q, p ′ , q ′ be four paths in µ -1 ({x}) whose tracks are respectively χ, χ, ζ(γ), ζ(γ ′ ) and such that p p ′ and q q ′ . If p q, since these two regular finite paths have the same tracks, the intervals of their intervals sequences are homeomorphic (Proposition 13). Hence, there exists a homeomorphism ϕ, piecewise linear, such that q = p•ϕ. We set p ′′ = q ′ • ϕ -1 . As q ′ q, we have p ′′ p. Obviously, the track of p ′′ is ζ(γ ′ ). We set p = r i=0 f i 1 I i , p ′ = s i=0 g i 1 J i and p ′′ = t i=0 h i 1 K i (r, s, t ≥ 0). We prove the lemma by induction on r, the size of χ. For r = 0, the result is obvious. Let us now suppose that the property is true whenever the size of χ is strictly lower than r (r ≥ 2 for r cannot be 1). Let j ∈ [ [1, r]] be the lowest integer such that f j is a local minimum of χ, i.e., such that f j-1 ⊃ f j ⊂ f j+1 (remember that f 0 = ȧ and f r are facets) and let j ′ ∈ [[ j + 1, r]] be the lowest integer such that f j ′ is a local maximum of χ. As f j ′ is a local maximum, I j ′ is an open interval of [0, 1] (Proposition 13). If j ′ = r, that is f j ′ = ḃ, then ζ(γ) and ζ(γ ′ ) are included in f ↑ j so γ and γ ′ are included in a same 2×2 square of Z n (if dim( f j ) ≥ n -2) or in the same 2 × 2 × 2 cube of Z n (if dim( f j ) = n -3). Moreover, if n = 3, (α, β) = (6, 26) (hence, ε = -1) and dim( f j ) = 0, from Proposition 1, we derive that there is no facet with value 1x in f ↑ j . So, using one or the other version of the definition of digital equivalence, we can conclude that the arcs γ and γ ′ are equivalent. Otherwise ( j ′ < r), we set

I i + f i 0 +1 1 I ′ + r i=i 0 +2 f i 1 I i where I ′ = I ∪ I i 0 +1 .
I j ′ = ]y, z[ with 0 < y < z < 1. Let k ∈ [[0, s]] and k ′ ∈ [[0, t]] be the even integers defined by z ∈ J k ∪ J k+1 and z ∈ K k ′ ∪ K k ′ +1 . Since p p ′ , p p ′′ and g k , h k ′ ∈ F n n (in particular J k and K k ′ are open on the left), this definition of k and k ′ ensures that g k ⊇ f j ′ ⊆ h k ′ and thereby that g i ∈ f ↑ j for all i ≤ k and h i ∈ f ↑ j for all i ≤ k ′ . If f j ′ ∈ F n n , then necessarily q k = h k ′ = f j ′ and we set χ ′ 0 = ( f j ′ ). Otherwise (dim( f j ′ ) < n), we set χ ′ 0 = (g k , f j ′ , h k ′ ).
In both cases, χ ′ 0 is an arc in µ -1 ({x}) thus, from Proposition 16, we derive that there exists an ω-path γ

′ 0 in λ -1 ({x}) such that ζ(γ ′ 0 ) χ ′ 0 . We set ζ(γ ′ 0 ) = (e i ) u i=0
(u ≥ 0) and we have e 0 = g k and e

u = h k ′ . Moreover χ ′ 0 is an arc in f ↑ j ′ which is included in f ↑ j so ζ(γ ′ 0 ) is also an arc in f ↑ j . Let γ 0 , γ 1 and γ ′ 1 be the ω-paths in λ -1 ({x}) such that ζ(γ 0 ) = (g i ) k i=0 , ζ(γ 1 ) = (g i ) s i=k and ζ(γ ′ 1 ) = (h i ) t i=k ′ . We set γ ′′ = γ 0 .γ ′ 0 .γ ′ 1 . Since ζ(γ ′′ ) = ζ(γ 0 ).ζ(γ ′ 0 ).(h k ′ , . . . , h t ), it differs from ζ(γ ′ ) = (h 0 , . . . , h k ′ ).(h k ′ , . . . , h t ) only in f ↑ j .
So, whatever is the adjacency relation, we can conclude as above that the arcs γ ′′ and γ ′ are equivalent. In order to use the induction hypothesis, let us now define three new paths. (Of course, we still use the rule defined by Proposition 13 in order to get paths, this will not be indicated anymore.) Previously to the definition of the three paths, we need to choose a real

m ∈ ]y, z[ ∩ (J k ∪ J k+1 ) ∩ (K k ′ ∪ K k ′ +1 ). Such a real exists since J k ∪ J k+1 and K k ′ ∪ K k ′ +1
are open on the left and contain z. The first path is obtained from p by removing all faces before f j ′ and replacing them by the facet g k : ] where p |I means the restriction of the function p to the interval I. Since we remove at least two faces ( f 0 and f j ) and add just one, the length of the track of p 1 is strictly lower than r. We must beware that all new paths are regular. This implies to check the first interval in p [y,1] . This interval is [y, z[ (remember that ]y, z[ = I j ′ ) and is not a singleton. The second path is

p 1 = g k 1 [0,y[ + f ′ j 1 [y,z[ + p |[z,1
Proposition 18 Let (α, β) be a pair of adjacencies on Z n . Let x ∈ {0, 1}. Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n . Let a, b ∈ Z n . Let γ, γ ′ be two ω-paths from a to b in λ -1 ({x}). If the arc ζ(γ ′ ) is a deformation in µ -1 ({x}) of the arc ζ(γ), then γ and γ ′ are equivalent in λ -1 ({x}).
Proof Let γ, γ ′ be two ω-paths from a to b in λ -1 ({x}) such that ζ(γ ′ ) is a deformation of ζ(γ). By definition, there exist an integer r ≥ 0 and a sequence S of arcs in µ -1 ({x}) (with µ = ζ ε (λ) where ε is the connectivity function associated to (α, β)), S = (χ 0 = ζ(γ), . . . , χ i , . . . , χ r = ζ(γ ′ )), such that, for each i ∈ [ [1, r]], either χ i is an elementary stretching of χ i-1 or χ i-1 is an elementary stretching of χ i . From Lemma 4, we derive that there exists a sequence S p of ω-paths in λ -1 ({x}), S p = (γ 1 , . . . , γ i , . . . , γ r ) such that for all

i ∈ [[1, r]], ζ(γ i ) χ i-1 and ζ(γ i ) χ i . Then, from Lemma 5, we deduce the sequence of equivalence γ ∼ γ 1 ∼ . . . ∼ γ r ∼ γ ′ . So he have γ ∼ γ ′ .
Let a ∈ λ -1 ({x}). Let π D (λ -1 ({x}), a) be the digital fundamental group of λ -1 ({x}) with basepoint a and ρ(µ -1 ({x}), ȧ) be the group of arcs in µ -1 (x) from ȧ to ȧ, up to deformations.

From Propositions 14 and 17, we know that the function ζ defined by

ζ : π D (λ -1 (x), a) → ρ(µ -1 (x), ȧ) [γ] → [ζ(γ)]
where [y] denotes the equivalence class of y (for the equivalence relation on digital paths of Z n on the left side and for the deformation on arcs of F n on the right side), is welldefined. Proposition 15 then states that ζ is a morphism. Propositions 16 and 18 give the surjectivity and the injectivity of ζ, respectively. We conclude that the two groups π D (λ -1 ({x}), a) and ρ(µ -1 ({x}), ȧ) are isomorphic and, since ρ(µ -1 ({x}), ȧ) and π(µ -1 ({x}), ȧ) are isomorphic (Theorem 2), the following theorem holds.

Theorem 4 Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β). Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . For any a ∈ λ -1 ({x}), the digital fundamental group of λ -1 ({x}) with basepoint a is isomorphic to the fundamental group of the poset (µ -1 ({x}), ⊆) with basepoint ȧ.

Conclusion

In this article, a modus operandi has been proposed to embed digital binary images, equipped with a pair of standard adjacencies, in the space of cubical complexes. In particular, it has been proved that it preserves the connected components of both object and background and preserves also the (digital) fundamental groups. These results, associated to those proposed in [START_REF] Mazo | Paths, homotopy and reduction in digital images[END_REF], justify the soundness of all contributions previously devoted to design homotopy type-preserving binary image processing methods, especially concerning the correctness of their behaviour with respect to the "continuous" topology of the handled digital objects. They also permit to establish links between image processing/analysis methods developed either in classical digital spaces (Z n ) or cubical complexes (F n ), and to potentially unify some of them.

Processing binary digital images without topology alteration has been an active research field for several years. Nowadays, an increasing interest for the extension of this field to the case of colour/label images (i.e., not only binary images, but more generally p-ary ones) can be observed. Some applicative methods devoted to p-ary images have already been proposed [START_REF] Poupon | Multi-object deformable templates dedicated to the segmentation of brain deep structures[END_REF][START_REF] Bazin | Topology-preserving tissue classification of magnetic resonance brain images[END_REF][START_REF] Miri | Topology-preserving discrete deformable model: Application to multi-segmentation of brain MRI[END_REF]. However, the putative preservation of topological properties by such methods generally relies on ad hoc hypotheses or incompletely proved theoretical bases.

From a theoretical point of view, the issue of topology and topology preservation in digital p-ary images has not been intensively considered yet [START_REF]Latecki, 3D well-composed pictures[END_REF][START_REF] Ségonne | A novel level set framework for the segmentation of medical images under topology control[END_REF][START_REF] Bazin | Digital homeomorphisms in deformable registration[END_REF][START_REF] Siqueira | Topological repairing of 3D digital images[END_REF][START_REF] Damiand | Topological model for 3D image representation: Definition and incremental extraction algorithm[END_REF][START_REF] Dupas | Multi-label simple points definition for 3D images digital deformable model[END_REF], and the proposed contributions still do not provide solutions in the most general cases in Z n and F n . In this context, further works will now consist in extending the approach proposed in this article by defining a mapping enabling to embed p-ary images from Z n to F n in a sound and tractable fashion, in particular to allow the development of homotopy-type-preserving image processing procedures.

A Auxiliary properties on complexes and posets

Lemma 6 Let f, g be two facets of F n . If f ∩ g ∅ then, for all h ∈ F n , we have f ∩ g ≺ h ⇒ (h ⊆ f or h ⊆ g).

Proof Set f = Π n i=1 f i , g = Π n i=1 g i and f ∩ g = Π n i=1 k i with f i , g i ∈ F 1 1 and k i ∈ F 1 1 ∪ F 1 0 for all i ∈ [ [1, n]]. Let h = Π n i=1 h i , with h i ∈ F 1 1 ∪ F 1 0 , be a face which covers f ∩ g. Then, there exists i 0 ∈ [ [1, n]] such that h i0 ∈ F 1 1 , k i0 ∈ F 1 0 , f i0 ∩g i0 = k i0 ⊂ h i0 and, for all i i 0 , k i = h i = f i ∩g i . But, readily, there exist only two faces in F 1 1 which include a given face of F 1 0 . Thus, h i0 = f i0 or h i0 = g i0 and, straightforwardly, h ⊆ f or h ⊆ g.

Lemma 7

Let 0 ≤ m < n. Let f ∈ F n m . Then: (i) Card( f ↑+ ) = 2 n-m . (ii) For {g 1 , g 2 } ∈ opp( f ). f ↑+ = g ↑+ 1 ⊔g ↑+ 2 (where ⊔ denotes the disjoint union).

Proof (i) We can assume, without loss of generality, that f = n i=1 I i where I i ∈ F 1 1 if i ≤ m and I i ∈ F 1 0 otherwise. Obviously, for each I i (m + 1 ≤ i ≤ n) there are two sets in F 1 1 including I i and therefore, there are 2 n-m products m i=1 I i × n i=m+1 J i where

I i ⊂ J i ∈ F 1 1 for each i ∈ [[m + 1, n]].
(ii) We apply the first part of the lemma to the three faces f, g 1 , g 2 : there are 2 n-m facets in f ↑ and 2 n-m-1 facets in g ↑ 1 as in g ↑ 2 . Since, by definition of opp, g ↑ 1 ∩ g ↑ 2 = ∅, we conclude that f ↑+ = g ↑+ 1 ⊔ g ↑+ 2 .

The following two lemmas use the definitions of Card -and Card + given in Section 3.4.

Lemma 8

Let f be a k-face in F n (with n -3 ≤ k ≤ n -1). Let r ∈ [[1, 2 n-k-1 ]]. Let E be a set of facets in f ↑ such that Card -(E) ≥ 2r. Then, there exist two faces g, h ∈ F n , opposite with respect to f such that Card(E ∩ g ↑ ) ≥ r and Card(E ∩ h ↑ ) ≥ r.

Proof We set m = ndim( f ) = nk. Thanks to the duality on the ordering in F n , the statement of this lemma is a consequence of the following one that we will prove: "if f is a facet of F m and E is a set of 2r 0-faces in f ↓ , not in the configuration depicted in Figure 14(k), then there exists a facet g of f ↓⋆ which contains exactly r elements of E". Since the result is obvious if m = 1 and we have restricted our lemma to m ≤ 3, we can check all configurations, up to rotations, symmetries and duality (e.g., choosing six points among eight amounts to choose two of them). These configurations, in 2-D and 3-D, are depicted in Figure 14 and the only one, numbered (k), in which no facet of the boundary of f ↓ contains half of the points corresponds to the configuration of a trihedron. Proof As in the proof of Lemma 8, thanks to duality, we can replace the statement of this lemma by the following one that we will prove: "if c is a facet of in F n (1 ≤ n ≤ 3), E a set of 0-faces in c ↓ such that there exist n facets of c ↓⋆ which contain, each, at least r faces of E (1 ≤ r ≤ 2 n-1 ), then Card(E) ≥ 2r -1 or r = 3 and E is in the configuration depicted in Figure 15(b) (in which case, Card(E) ≥ 2r -2)". If r = 1, the property is obvious, as is the case where two of the n facets of c ↓⋆ which contain r faces of E are parallel (since two parallel faces do not share 0-faces, in this case Card(E) ≥ 2r). Hence, in the sequel we assume that all these n facets share a same 0-face. Moreover, it suffices to study minimal configurations, that is, sets E which satisfy the hypothesis while none of their proper subsets does. If n = 2 and r = 2, there is only one such minimal configuration (up to rotations and symmetries), depicted in Figure 15(a). When n = 3, let b, f, l be the three facets of c ↓⋆ that contain r faces of E, t be the 0-face contained in these three facets, and x, y, z be the 0-faces contained in exactly two of these three facets (in Figure 15(b-e), b, f, l are located at the bottom, front and left positions, respectively). If r = 2 the first two 0-faces of E contained in b cannot be shared with f and l (but can be shared with either f or l). Thus, there must exist at least a third face of E in f or l, implying that Card(E) ≥ 3. If r = 3, the only minimal configuration with {t, x, y, z} ⊆ E is depicted in Figure 15(b) and the only minimal configuration, up to rotations, with t ∈ E and Card({x, y, z} ∩ E) = 2 is depicted on Figure 15(c). The case Card({x, y, z} ∩ E) = 1 is not possible. The only minimal configuration with t E is depicted on Figure 15(d). If r = 4, the only minimal configuration is depicted on Figure 15(e). The reader can then easily check from Figure 15(b-e) that the statement of the lemma is true.

Lemma 10 Let (T, ) be a poset. The path product is compatible with the order relation: for any a, b, c ∈ T and any paths p p ′ from a to b and q q ′ from b to c, we have p.q p ′ .q ′ . Proof Let t ∈ [0, 1]. If t ≤ 1 2 , then, p.q(t) = p(2t) p ′ (2t) = p ′ .q ′ (t). If t ≥ 1 2 , then p.q(t) = q(2t -1) q ′ (2t -1) = p ′ .q ′ (t).

  (i) ∅, X are open sets; (ii) any finite intersection of open sets is an open set; (iii) any union of open sets is an open set. The complement in X of an open set is called a closed set. From the above definition, any finite union of closed sets is a closed set and any intersection of closed sets is a closed set. A set of open sets is a basis for a topology if any open set is a union of open sets of this basis. A neighbourhood of a point x ∈ X is a subset of X including an open set containing x.

  is also the union of all open sets included in Y). Closure and interior are dual notions since ¬(Y • ) = (¬Y) ↓ and ¬(Y ↓ ) = (¬Y) • where ¬Y = X \ Y. An open set Y is a regular open set if Y = (Y ↓ ) • and a closed set is a regular closed set if Y = (Y • ) ↓ . The complement of a regular open set is a regular closed set.

Fig. 1

 1 Fig. 1 Two representations of a set of faces F = { f, g, h} in Z 2 with, for instance, f = {0} × {1}, g = {0, 1} × {0, 1} and h = {1} × {0, 1}. The face g is a facet of F (and also of F 2 ).

  x∈X µ(x)) for the maximum (resp. minimum) of the set {µ(x) | x ∈ X} and we also write µ(a)∨µ(b) (resp. µ(a)∧µ(b)) for x∈{a,b} µ(x) (resp. x∈{a,b} µ(x)).

Fig. 2

 2 Fig. 2 Two opposite faces in F 3 with respect to (a) a 2-face, (b) a 1face, (c) a 0-face. The dashed boxes in (b) and (c) show other pairs of opposite faces.

Fig. 3

 3 Fig. 3 Two ε-regular images (a) on F 2 with ε(2) = -1, ε(1) = 1, (b) on F 3 with ε(3) = -1, ε(2) = ε(1) = 1.

Fig. 4

 4 Fig. 4 Images ζ ε (λ) : F 2 → {0, 1} for some given λ : Z 2 → {0, 1}.

Fig. 5

 5 Fig. 5 Images ζ ε (λ) : F 3 → {0, 1} for some given λ : Z 3 → {0, 1}. (a) With ε(3) = -1, we obtain the 6-adjacency in Z 3 . (b) With ε(3) = 1 and ε(2) = -1, we obtain the 18-adjacency in Z 3 . (c, d) With ε(3) = ε(2) = 1, we obtain the 26-adjacency in Z 3 .

Fig. 7

 7 Fig. 7 (a,b) A torus built with six 3-faces illustrates how the two 6-adjacencies can be obtained. There are two 3-faces of the background which intersect the six 3-faces of the torus. These two white-faces are 26-adjacent but in (a) they do not intersect in the background (for the torus is a horn-torus) while, in (b), they do (the torus is a ring-torus). Hence, in (a), we model a (6, 18)-adjacency relation while in (b), we model a (6, 26)adjacency relation.(c,d) An object built from three facets with two connectivity functions which could a priori be used to model the 18-adjacency (see Figure5(b)). In (c) we can see a red 0-face between the three cubes. This is what is expected for the background must have a 6-adjacency. In (d), there is a hole instead of the red 0-face, which is not correct in 18-adjacency.

  .

Fig. 8 (

 8 Fig. 8 (a) A (-1, 1)-regular image µ : F 2 → {0, 1}. (b) The (1, -1)regular image -µ. (c) The (1, -1)-regular image ¬µ. (d) The (-1, 1)regular image -(¬µ) = ¬(-µ).

Fig. 9

 9 Fig. 9 (a) A digital image λ : Z 2 → {0, 1}. (b) The negative image ¬λ associated to λ. (c) The ε-regular image ζ ε (λ) associated to λ with ε = (1, -1, -1). (d) The (-ε)-regular image ζ -ε (¬λ) associated to ¬λ.Note that (b,d) have been rotated of -π/2 along the (nearly) vertical axis in order to ease the visualisation of (d).

  we derive that a g ↓ or b g ↓ . Therefore, µ(a) = 0 or µ(b) = 0 and µ(a) ∧ µ(b) = 0. Since this last equality is true for each pair in opp( f ), we can state that {a,b}∈opp( f ) µ(a) ∧ µ(b) = 0. So, ε(m) cannot be equal to -1. Hence, ε(m) = 1, and then ε(k) = 1 for all k ≥ m.

(

Fig. 10 (

 10 Fig.10(a) If ε(3) = -1, the integer k defined in Proposition 9 is equal to n -1 = 2. Then the proposition states that, since the intersection (in red) of the two facets of µ -1 ({1}) (in blue) is in µ -1 ({1}) there must exist a path composed of facets and of faces of dimension higher than 2 between these two facets, as the one depicted in (b). (c) If ε(3) = 1, ε(2) = -1, the integer k defined in Proposition 9 is equal to n -2 = 1. So, there must exist a path composed of facets and of faces of dimension higher than 1 between these two facets as the one depicted in (d).

Fig. 11 (

 11 Fig. 11 (a) Symbolic representation of a trihedron related to a face f ∈ F n such that dim( f ) = n -3. Black dots: f ↑+ 1 ; white dots f ↑+ \ f ↑+ 1 . The dash lines represent the existence of a face of dimension n -1 forming the intersection between two facets of f ↑+ . (b-d) Examples of trihedra, with three connectivity functions ε (one of the blue facets is hidden).

  and sup(J 2 ) = min(J 3 ). As previously observed, it may happen that, for y = g ′ or y = g, dim(y) ≤ nk where k = 2 if α = 2n, k = 3 if α = 18. Let us write ḋ for the facets following y in ζ(γ 0 ) (d = b or d = c). From Proposition 9, we derive that there exists an arc χ = ζ(γ 1 ) from ȧ to ḋ in y ↑ where γ 1 is an ω-path from a to d. By introducing, if necessary, this path γ 1 between a and b if γ 0 = (a, b), or between a and c if γ 0 =

  Fig.[START_REF] Cointepas | Cellular complexes: A tool for 3D homotopic segmentation in brain images[END_REF] We use the dual order in F n to represent the unit square/cube: the facets are depicted by points, the (n -1)-faces by edges and so on (the face m is then seen as the square {a, b, c, d} or the cube {a, b, c, d, e, f, g, h}). The arc ζ(γ) is drawn like a solid black curve. Since ζ ε (λ) is a regular image, whatever the connectivity function ε, when two 4/6-adjacent vertices (which are actually facets) of the depicted polyhedron are on ζ(γ), the edge between them belongs to λ -1 ({1}) and when the four edges of a square are on ζ(γ) then the square between them belongs to λ -1 ({1}). (a) All the vertices, and thus m, are in λ -1 ({1}). (b, c, d) These three figures are only concerned by (α, β) = (6, 18), i.e., ε = (-1, 1, -1). In each figure, the length of γ is greater than or equal to 6, so, from Table2m ∈ λ -1 ({1}).

Lemma 5

 5 Then p is a path whose track is χ and p p ′ q ′ . So, ζ(γ ′ ) χ and ζ(γ ′ ) χ ′ . Let (α, β) be a pair of adjacencies on Z n . Let ε be the connectivity function associated to (α, β).Let x ∈ {0, 1}. Let ω = α if x = 1 and ω = β if x = 0. Let λ : Z n → {0, 1} be an image in Z n and µ = ζ ε (λ) be the corresponding image in F n . Let a, b ∈ Z n .Let χ be an arc from the facet ȧ to the facet ḃ in µ -1 ({x}) and γ, γ ′ be two ω-paths from a to b such that ζ(γ) χ and ζ(γ ′ ) χ. Then γ and γ ′ are equivalent in λ -1 ({x}).

Fig. 14

 14 Fig. 14 Configurations (up to rotations, symmetries and duality) used in the proof of Lemma 8 (see text). (a-c) 2-D configurations; the full line indicate the 1-dimensional facet which satisfies the property. (d-m) 3-D configurations; the full lines indicate the 2-dimensional facet which satisfies the property.

Lemma 9

 9 Let f be a k-face in F n (with n -3 ≤ k ≤ n -1). Let r ∈ [[1, 2 n-k-1 ]]. Let E be a set of facets in f ↑ . If there exist nk faces which cover f and whose stars contain at least r facets of E, then Card + (E) ≥ 2r -1.

Fig. 15

 15 Fig. 15 Configurations used in the proof of Lemma 9 (see text). Black points: faces of E. Black edges: edges of three facets of the cube that contain at least r faces of E. (a) n = 2 and r = 2. (b-d) n = 3 and r = 3. (e) n = 3 and r = 4.

In the literature devoted to cubical complexes, it is generally assumed that F ⊆ F n is a finite set of faces. This constraint can however be relaxed in the context of this work.

[START_REF] Kong | A digital fundamental group[END_REF] We keep the usual designation though it would be more coherent in this work to define such a sequence of points as an α-arc in order to reserve the word path for the functional point of view.

Note that each one of the 2 n connectivity functions ε : [[1, n]] → {-1, 1} may enable to retrieve a standard adjacency for the object (possibly redundantly, see Figure5(c,d)), but does not necessarily induces a standard pair of adjacencies, as exemplified inFigures 6

and 7.

The notation χ χ ′ does not mean that is an order (it is actually only a pre-order on regular finite paths). For instance, if a b c are three distinct faces of F n , it can readily be proved that (a, c) (a, b, c) (a, c).

In this article, the terminology "complex image" denotes a subset of F n .

We could also invoke the existence of a smallest element to derive the contractibility of C ∩ λ -1 ({1}) (see, e.g., Corollary 8 in[START_REF] Mazo | Paths, homotopy and reduction in digital images[END_REF]) and thus its simple connectivity.

If ζ(γ) = ( f i ) r i=0 (r ≥ 2 and f 0 = f r = a) we use the sequence of elementary stretchings ( f 0 ) → ( f 0 , m, f r ) → ( f 0 , m, f r-1 , f r ) → . . . → ( f 0 , m, f 1 , . . . , f r ) → ( f 0 , f 1 , . . . , f r ).

obtained in a similar way from p ′ by removing all faces before g k , so its track is ζ(γ 1 ): [START_REF] Rosenfeld | Sequential operations in digital picture processing[END_REF] is either associated to g k (and then has to be merged with [0, m[), or is associated to g k+1 but in this case, by the definition of k, z is also in this interval so the first interval can never be a singleton. The third path p ′′ 1 is obtained from p ′′ by removing all faces before h k ′ and putting in front of h k ′ the faces of the arc χ ′ 0 with, for the choice of the intervals, the constraint that p ′′ 1 must be equal to p ′ 1 and p 1 on [0, y[:

where the intervals K ′ i are supposed to be well-defined. The track of

. The first interval in p ′′ [m,1] is either associated to h k ′ = e u (and then is to be merged with K ′ u ), or is associated to h k ′ +1 but in this case, as above, z is also in this interval so the first interval can never be a singleton. Hence, we have three regular paths p 1 , p ′ 1 , p ′′ 1 equal to p, p ′ , p ′′ on [m, 1] so we have p ′ We can now apply the induction hypothesis: the ωpaths γ 1 and γ ′ 0 .γ ′ 1 are equivalent. Furthermore, the ω-paths γ = γ 0 .γ 1 and γ ′′ = γ 0 .γ ′ 0 .γ ′ 1 are equivalent. Since, we have proved before that γ ′′ and γ ′ are equivalent, we can eventually conclude that γ and γ ′ are equivalent.


Alain Lascoux
email: alain.lascoux@univ-mlv.fr

Addition of ±1 : Application to Arithmetic

We show that some classical identities about multiplicative functions, and the Riemann zeta-function, may conveniently be interpreted in terms of the addition of ±1 to some alphabets.

Résumé

Nous montrons que des identités classiques concernant des fonctions multiplicatives de la théorie des nombres, et la fonction zeta de Riemann, s'interprètent commodément en terme de l'opération : "Ajouter ±1 à un alphabet".

Introduction

In good courses of classical analysis, like the one of Jean-Yves Thibon for the Marne-La-Vallée students, one finds exercises of the following type:

Exercise 99. Show the following two identities:

ζ(s) 3 /ζ(2s) = τ (n 2 ) n -s ζ(s) 4 /ζ(2s) = τ (n) 2 n -s ,
where ζ(s) is the Riemann zeta function, and τ (n) is the number of divisors of n.

Such identities seem to be totally out of reach of my own students 2 , who have learnt from me only symmetric functions, division by 1, and divided differences.

I shall however show that addition of ±1 is sufficient to generate such identities, and the link with symmetric function theory will be provided by the notion of multiplicative function:

f : N → commutative ring is multiplicative if and only if f (mn) = f (m)f (n) whenever m, n are relatively prime.

Equivalently, the prime decomposition of n determines the value of f (n):

n = p prime p kp ⇒ f (n) = p prime f p kp .
But instead of considering the sequence

[f (p 0 ), f (p 1 ), f (p 2 ), f (p 3 ), . . .] ,
or the generating function ∞ k=0 f (p k )z k , as does Lagrange, Symmetric Function Theory [START_REF] Lascoux | Symmetric functions and combinatorial operators on polynomials[END_REF] tells us that we better introduce an alphabet A p , such that, for k ∈ N, f (p k ) is the k-th complete function in this alphabet, denoted by S k (A p ), i.e., such that

∞ k=0 f (p k )z k = σ z (A p ) := ∞ k=0 z k S k (A p ) = a∈Ap (1-za) -1 .
In other words, each generating function is formally factorized and gives rise to an alphabet.

In summary, a multiplicative function is determined by a collection of alphabets A = {A p , p prime}. In this sutation, this multiplicative function will be denoted by µ A (.).

It may be appropriate, before going any further, to recall a few facts about symmetric functions. An alphabet A = {a 1 , a 2 , . . .} is written as the sum of its elements: A = a 1 + a 2 + • • • , and the generating function of complete functions in A is

σ z (A) = ∞ k=0 z k S k (A) = i (1 -za i ) -1 .
Adding a letter x to A is denoted by A → A + x. Adding three copies of the letter 1 is denoted by A → A + 3.

More generally, given alphabets A, B, and complex numbers k, r, the generating function σ z (kA ± B + rx) is equal to

σ z (kA ± B + rx) = α∈A (1-zα) -k β∈B (1-zβ) ∓1 (1 -zx) -r . (1)
We may need to specialize a letter to 3, but this must not be confused with taking three copies of 1. To allow nevertheless specializing a letter to a complex number r inside a symmetric function, without introducing intermediate variables, we write r for this specialization. Boxes have to be treated as single variables.

In other words, given a letter x, and complex numbers r, α, β, then

σ z (α r + β x) = (1 -z r) -α (1 -x) -β .
When all alphabets A p are equal to the same alphabet A, we write A for the collection of A = {A, A, . . .}. We shall also need the collection where A p = p , that we denote by A = p . It corresponds to the multiplicative function f (n) = n.

We shall now formulate properties of multiplicative functions in terms of another classical object in number theory.

Given a function f from N in a commutative ring, one can associate to it a Dirichlet series3

F (f ; s) = n≥1 f (n)n -s ,
the most notable being the Riemann zeta function ζ(s) := n≥1 n -s . The product of Dirichlet series corresponds to the convolution of functions on N:

F (f ; s)F (g; s) = F (f g; s) = n d|n f (d)g(n/d) n -s .
When f is a multiplicative function, say f = µ A , we write ζ(A; s) for the corresponding Dirichlet series. In this notation, the Riemann zeta function is

ζ(1; s), that is, each alphabet A p is equal to 1.
The product of two series ζ(A; s), ζ(B; s) is the series ζ(A + B; s), because for every p, σ z (A p )σ z (B p ) = σ z (A p + B p ). Thus addition of alphabets corresponds to products of Dirichlet series.

Notice that the convolution of multiplicative functions with the classical Möbius function µ -1 or its inverse µ 1 (cf. [START_REF] Apostol | Introduction to Analytic Number Theory[END_REF]), or the product of a Dirichlet series and ζ(s) -1 or ζ(s), are now interpreted as the addition to each alphabet A p of -1 or 1 respectively:

µ A-1 (n) = d: d|n µ(n/d) µ A (d) ⇔ ζ(A -1, s) = ζ(A, s)/ζ(s) (2)
µ A+1 (n) = d: d|n µ A (d) ⇔ ζ(A + 1, s) = ζ(A, s) ζ(s) . (3)
2 Adding 1

We are now in the position to solve the exercise stated at the beginning of this text.

For the alphabets 2 --1 and 3 --1 we have

σ z 2 --1 = (1 -(-1)z)(1 -z) -2 = 1 + 3z + 5z 2 + 7z 3 + • • • σ z 3 --1 = (1 -(-1)z)(1 -z) -3 = 1 + 4z + 9z 2 + 16z 3 + • • • . Since τ (p 2k ) = 2k + 1, it follows that ζ(s) 3 /ζ(2s) = ζ 2 --1 , s = n≥1 τ (n 2 ) n -s . Similarly, τ (p k ) 2 = (k+1) 2 implies ζ(s) 4 /ζ(2s) = ζ 3 --1 , s = n≥1 τ (n) 2 n -s ,
and this proves the two required formulas.

In fact, many of the classical arithmetical multiplicative functions occur in products of zeta functions, as shows the following table. We need only identify the generating functions σ z (A p ) for all alphabets appearing in the left column, to ensure the validity of the expansions given in the right column.

alphabet series σz(Ap) expansion -1 1/ζ(s) 1 -z µ -1 (n) n -s 2 ζ(s) 2 (1-z) -2 τ (n) n -s r ζ(s) r (1-z) -r τ r (n) n -s --1 ζ(s)/ζ(2s) 1 -(-z) |µ -1 (n)| n -s 1 --1 ζ(s) 2 /ζ(2s) (1+z)/(1-z) 2 ω(n) n -s 2 --1 ζ(s) 3 /ζ(2s) (1+z)/(1-z) 2 τ (n 2 ) n -s 3 --1 ζ(s) 4 /ζ(2s) (1+z)/(1-z) 3 τ (n) 2 n -s 1 + p α ζ(s) ζ(s -α) (1-z) -1 (1-zp α ) -1 d|n d α n -s p -1 ζ(s -1)/ζ(s) (1-z)/(1-zp) φ(n) n -s
In the table, we use the function τ (n), which, as before, is the number of divisors of n, and, more generally, τ r (n), r > 2, is the number of integral vectors [n 1 , . . . , n r ] such that n = n 1 • • • n r . In addition, we denote by ω(n) the number of prime divisors of n and by φ(n) the number of integers less than n and prime to n.

Let us elaborate in some more detail about the relation between the third column (generating function for the alphabet) and the fourth column (Dirichlet series), in the order the alphabets are listed in the table:

• 1 -z, i.e. µ(p) = -1, µ(p k ) = 0, k ≥ 2.

This is this the classical

Möbius function;

• (1 -z) -2 , µ(p k ) = k + 1 = τ (p k ) is the number of divisors of p k ; • (1 -z) -r , µ(p k ) = k+r-1 k-1 is the number of r-tuple [n 1 , . . . , n r ] such that n 1 • • • n r = p k ; • 1 -(-z), µ(p) = 1, µ(p k ) = 0, k ≥ 2; • (1 + z)/(1 -z), µ(p k ) = 2, k ≥ 1; • (1 + z)/(1 -z) 2 , µ(p k ) = 2k + 1 is the number of divisors of (p k ) 2 ; • (1 + z)/(1 -z) 3 , µ(p k ) = (k + 1) 2 is the square of the number τ (p k ) of divisors of p k ; • (1 -z) -1 (1 -zp α ) -1 , µ(p k ) = 1 + p α + • • • + p kα is the sum of the α-th powers of the divisors {1, p, . . . , p k } of p k ; • (1 -z)/(1 -zp), µ(p k ) = p(p k-1 -1)
is the number of integers less than p k and prime to p.

A natural generalization of the classical Möbius function proceeds by taking some complex number α, instead of -1, and setting all A p = α. Recall that

S k (α) = α(α + 1) • • • (α + k -1)/ k! = α + k -1 k .
Hence,

µ α p p k = p α + k -1 k . ( 4
)
These Möbius functions of higher order have been considered by Fleck [START_REF] Fleck | Über gewisse allgemeine zahlentheoretische Funktionen, insbesondere eine der Funktion µ(m) verwandte Funktion µ k (m), Sitzungsber[END_REF][START_REF] Dickson | History of the theory of numbers[END_REF] 4 , who proved the convolution formula

µ α µ -1 = µ α-1 .
Taking Dirichlet series instead of alphabets, Fleck's results translate into the seemingly more elaborate formula

∞ n=1 µ α (n)n -s = ζ(s) ∞ n=1 µ α-1 (n)n -s , (5)
which is nothing else but

α = 1 + (α -1) . (6)

Hadamard Products

One can use other operations on alphabets than just adding ±1. For example, the "pointwise" product of two multiplicative functions is still a multiplicative function. It corresponds to the Hadamard product of generating series:

i≥0 z i S i (A) , i≥0 z i S i (B) -→ i≥0 z i S i (A)S i (B) ,
or the Hadamard product of Dirichlet series, and induces an "Hadamard product" on pair of alphabets : (A, B) → A H B. Indeed, given two finite alphabets A and B, the Hadamard product of σ z (A) and σ z (B) is a rational function, with denominator

σ z (-AB) := a∈A b∈B (1 -zab) -1 .
However, in general the numerator will be too complicated to be applied to the theory of the Riemann zeta function. I shall nevertheless give three explicit cases that I found in the literature.

The first one is due to Ramanujan:

ζ(s)ζ(s -α)ζ(s -β)ζ(s -α -β) ζ(2s -α -β) = n≥1 d|n d α d|n d β n s . (7)
Indeed,

1 -z 2 xy (1 -z)(1 -zx)(1 -zy)(1 -zxy) = i≥0 (1 + x + • • • + x i )(1 + y + • • • + y i )z i ,
which means that the Hadamard product of

(1 -z) -1 (1 -zx) -1 and (1 -z) -1 (1 -zy) -1
is the left-hand side. This left-hand side can be written σ z (1+x+y +xy -Ω),

with Ω an alphabet of cardinality 2 such that S 1 (-Ω) = 0 and S 2 (-Ω) = -xy. When x = p α , y = p β , the generating function σ z (1 + x + y + xy -Ω) specializes to the component depending on

p of ζ(s)ζ(s-α)ζ(s-β)ζ(s-α-β) ζ(2s-α-β)
, and, thus eventually, Ramanujan's identity is indeed just (1+x) H (1+y) = (1+x+y+xy) -Ω .

(8)

The second example uses the Hadamard powers of the simplest5 series, (1 -z) -2 .

By definition the k-th Hadamard power of 1

+ 2z + 3z 2 + • • • is 1 + 2 k z + 3 k z 2 + 4 k z 3 + • • • = E k (z) (1 -z) k+1 ,
where E k (z) is the k-th Eulerian polynomial, dear to combinatorialists [START_REF] Riordan | Combinatorial identities[END_REF]Ch. 7]. Define now, for any positive integer k, the function

η k (s) = 1 1 -2 -s p prime p≡1 (mod 4) E k (p -s ) 1 -p -s r prime r≡3 (mod 4) 1 1 -r -2s . (9)
Therefore, thanks to our computations of the Hadamard powers of (1 -z) -2 and (1 -z 2 ) -1 , we see that η k (s) is the k-th Hadamard power of η 1 (s).

The alphabets corresponding to η 1 (s) are

A 2 = 1 , A p = 2 , A r = 1 + -1 , since σ z (1) = 1/(1 -z), σ z (2) = 1/(1 -z) 2 , σ z 1 + -1 = 1/(1 -z 2 ).
Subtracting 1 from all the alphabets, one gets a simpler function:

L(s) := η 1 (s)/ζ(s) = p≡1 1 1 -p -s r≡3 1 1 + r -s = ∞ n=0 (-1) n (2n + 1) s , (10)
corresponding to the alphabets

A 2 = 0 , A p = 1 , A r = -1 .
Since the Hadamard square of (1 -z) -2 is equal to

(1 + z) (1 -z) -3 = σ z 3 --1 ,
the function η 2 (s) is associated to the alphabets

A 2 = 1 , A p = 3 --1 , A r = 1 + -1 .
The Hadamard cube of (1 -z) -2 being

(1 + 4z + z 2 ) (1 -z) -4 = σ z (4 -α -β) , with α = -2 + √ 3 , β = -2 -√ 3 , the function η 3 (s) is associated to the alphabets

A 2 = 1 , A p = 4 -α -β , A r = 1 + -1 .
Finally, the fourth Hadamard power being

(1 + 11z + 11z 2 + z 3 ) (1 -z) -5 = (1 + z)(1 + 10z + z 2 ) (1 -z) -5
= σ z 5 --1 -α-β , with α = -5 + 2 √ 6 , β = -5 -2 √ 6 , the function η 4 (s) is associated to the alphabets

A 2 = 1 , A p = 5 --1 -α -β , A r = 1 + -1 .
Let us close this text by considering higher powers (1 -z) -r , with r > 2. I shall restrict myself to their Hadamard squares.

Since 1 (1 -z) r H 1 (1 -z) r = P r-1

1 + z 1 -z 1 (1 -z) r ,
where P j is the Legendre polynomial [START_REF] Riordan | Combinatorial identities[END_REF] of degree j, one has, using the function τ r (n) defined above,

Partially supported by the EC's IHRP Program, within the Research Training Network "Algebraic Combinatorics in Europe", grant HPRN-CT-2001-00272.

and also of me, because J.-Y. Thibon does not write the solution.

We shall not address convergence issues, but shall treat the Dirichlet series as formal series. However, for all identities that we consider here, the formal treatment is sufficient to also deduce their validity as identities for analytic series. For, all our series converge in some domain, which must in fact be a (right) half plane (see[START_REF] Apostol | Introduction to Analytic Number Theory[END_REF] Theorem 11.1]), and then it follows from uniqueness of coefficients (see[START_REF] Apostol | Introduction to Analytic Number Theory[END_REF] Theorem 11.3]) that the formal identity is at the same time an analytic identity.

and still reappear in recent literature[START_REF] Brown | On a certain kind of generalized number-theoretical Möbius function[END_REF].

The Hadamard product of f (z) with (1 -z) -1 being f (z), one has to argue between (1 -z) -2 and (1 -2z) -1 for the title of being the simplest non-trivial series. Notice that not only 1/(1 -z) is stable under Hadamard powers, but so is 1/(1 -z α ) for any positive integer α.