Sébastien Lefèvre
email: lefevre@lsiit.u-strasbg.fr

Extending Morphological Signatures for Visual Pattern Recognition

Keywords: Mathematical Morphology, Granulometry, Pattern spectrum, Morphological profile, Visual Pattern Recognition

Morphological signatures are powerful descriptions of the image content which are based on the framework of mathematical morphology. These signatures can be computed on a global or local scale: they are called pattern spectra (or granulometries and antigranulometries) when measured on the complete images and morphological profiles when related to single pixels. Their goal is to measure shape distribution instead of intensity distribution, thus they can be considered as a relevant alternative to classical intensity histograms, in the context of visual pattern recognition. A morphological signature (either a pattern spectrum or a morphological profile) is defined as a series of morphological operations (namely openings and closings) considering a predefined pattern called structuring element. Even if it can be used directly to solve various pattern recognition problems related to image data, the simple definitions given in the binary and grayscale cases limit its usefulness in many applications. In this paper, we introduce several 2-D extensions to the classical 1-D morphological signature. More precisely, we elaborate morphological signatures which try to gather more image information and do not only include a dimension related to the object size, but also consider on a second dimension a complementary information relative to size, intensity or spectral information. Each of the 2-D morphological signature proposed in this paper can be defined either on a global or local scale and for a particular kind of images among the most commonly ones (binary, grayscale or multispectral images). We also illustrate these signatures by several real-life applications related to object recognition and remote sensing.

Introduction

When applied on visual information such as digital images, pattern recognition and data mining techniques are very dependent on the accuracy of the image features they rely on. The image analysis and processing community has proposed from several decades a large variety of features to represent precisely the content of an image. The features which have received the most attention from the image scientists and engineers are context-independent, i.e. they can be used for a various panel of problems and a wide range of images. Among these features, the histogram is certainly the most plebiscited and represents the probability density function of the intensity values in the image. It can be analysed through various measures such as moments, entropy, uniformity, or other information criteria. However it does not take into account the spatial relationships between pixels.

On the contrary, approaches such as pattern spectra, granulometries, or morphological profiles are built from series of morphological filtering operations and thus involve a spatial information. They can be applied either on a local scale or a global scale. In the first case, the differential morphological profiles introduced in [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF] have shown their interest to deal with classification of remote sensing data. On a global scale, pattern spectra and granulometries are widely known in the image analysis community [START_REF] Soille | Morphological Image Analysis : Principles and Applications[END_REF]. However all these morphological measures are limited to a single evolution curve and so cannot consider simultaneously several dimensions. Some attempts have been made to generate vectorial granulometries [START_REF] Wilkinson | Generalized pattern spectra sensitive to spatial information[END_REF] or vectorial covariances [START_REF] Aptoula | Spatial morphological covariance applied to texture classification[END_REF], but they are more vectorial extensions than multidimensional extensions.

In this paper, we propose several extensions to build 2-D series of morphological measures. We first give the necessary definitions and show the similarity between the different morphological measures which have been proposed in the literature. We then introduce three different 2-D signatures, related to size-size, size-intensity, and sizespectrum information. The interest of our contribution is then illustrated by several applications related to object recognition and remote sensing. We believe however that these extensions can be used as appropriate image features to solve a larger panel of pattern recognition and image mining problems.

Preliminary definitions

Morphological signatures are tools provided by the framework of Mathematical Morphology (MM) and known as pattern spectra (and granulometries / antigranulometries) or morphological profiles when computed on a global or a local scale respectively. The theory of MM has been introduced by Matheron and Serra [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF] in the mid sixties and has been extensively used for 40 years particulary for spatial-based image analysis. The theoretical framework of complete lattices [START_REF] Ronse | Why mathematical morphology needs complete lattices[END_REF] is commonly used to elaborate morphological operators on monovalued (binary or grayscale) images. Multivalued (colour or multispectral) images require the choice of a specific vectorial ordering [START_REF] Aptoula | A comparative study on multivariate mathematical morphology[END_REF]. In this section, we will recall the definitions of the basic morphological operators from which we can then give the formulation of the morphological signature.

Fundamental operators

Let f : E → T represents the image to be morphologically processed, with E being the discrete coordinate grid (usually N 2 for a 2-D image) while T represents the set of possible image values. In the case of a binary image, T = {0, 1} where the objects and the background are respectively represented by values equal to 1 and 0. In the case of a grayscale image, T is generally a subset of Z, for instance [0, 255]. Finally, Z n is considered for T in case of multispectral images. Most of the available operators within the MM framework are applied on an input image f (either binary, grayscale or multispectral) and rely also on a predefined matching pattern B, called structuring element (SE). In this paper we assume that only flat structuring elements are involved, thus resulting in the definition of B as a subset of E.

Based on these notations, the definitions of the two basic morphological operators (namely dilation and erosion) are given by:

δ B (f )(x, y) = (r,s)∈B f (x -r, y -s), (x, y) ∈ E (1)
B (f )(x, y) = (r,s)∈B f (x + r, y + s), (x, y) ∈ E (2)
In other words, the dilation δ B (f ) results in an image where each pixel (x, y) is associated to the local maximum of f in the neighbourhood defined by the SE B. The erosion is the dual operator, considering the local minimum instead of the local maximum. We can then derive the opening and closing operators:

γ B (f ) = δ B ( B (f )) (3) φ B (f ) = B (δ B (f )) (4)
where the two fundamental operators are applied successively to filter the input image: erosion followed by dilation for the opening, dilation followed by erosion for the closing operator, thus resulting in a removal of either local maxima or minima. From these definitions we can observe that the strength of an opening or closing filter depends directly on the size of the SE B: considering an increasing size λ for the SE, the openings γ λ and closings φ λ will remove more and more details from the input image. Indeed the opening and closing operators ensure respectively the extensivity and anti-extensivity properties, in other words γ(f ) ⊂ f and f ⊂ φ(f ). For the sake of conciseness, we will use the operator ψ to represent either γ or φ, i.e. the opening or closing operation.

Morphological signatures

As indicated earlier in this paper, the morphological signature can be computed either on a local (i.e. pixel-related) or a global (i.e. image-related) scale, thus resulting in operators respectively known as morphological profile and pattern spectrum (or granulometry / antigranulometry). In order to give the formulation of these different operators, let us write Π ψ (f ) the series obtained from successive openings or closings:

Π ψ (f ) = {Π ψ λ (f ) | Π ψ λ (f ) = ψ λ (f ), ∀λ ∈ {0, . . . , n}} (5)
where ψ 0 (f ) = f and λ denotes the size of the SE B. We assume either ψ = γ or ψ = φ. More meaningful series ∆ ψ (f ) can be generated with differential measures:

∆ ψ (f ) = ∆ ψ λ (f ) | ∆ ψ λ (f ) = Π ψ λ (f ) -Π ψ λ-1 (f ), ∀λ ∈ {1, . . . , n} (6)
Finally we can group together opening and closing series to generate a single Π or ∆ series of length 2n:

Π(f ) = Π c (f ) | Π c (f ) = Π φ-c (f ), ∀c ∈ {-n, . . . , -1} Π γc (f ), ∀c ∈ {1, . . . , n} (7)
∆(f ) = ∆ c (f ) | ∆ c (f ) = ∆ φ-c (f ), ∀c ∈ {-n, . . . , -1} ∆ γc (f ), ∀c ∈ {1, . . . , n} (8)
The morphological signatures widely used in image analysis and visual pattern recognition can be defined from these different image series, either on a local or a global scale. In the former case, the morphological profile and the well-known differential morphological profile (DMP) are respectively defined as Π(f )(x, y) and ∆(f )(x, y) for a given pixel (x, y) in f . In the latter case, the pattern spectrum is built from an analysis of the image series Π(f ) gathering most frequently the image pixel values through the volume or sum operation, i.e. (x,y)∈E Π(f )(x, y). In the particular case of binary images, the image volume can either be computed as the sum of pixel values or as the amount of white pixels (or 1-pixels). Granulometries and antigranulometries are built similarly but are limited respectively to openings and closings, i.e.

(x,y)∈E Π γ (f )(x, y) and (x,y)∈E Π φ (f )(x, y). As for local signatures, the original series Π is often replaced in practice by its differential version ∆. Moreover, in order to obtain a shape probability distribution function, a normalisation may be involved, each term being then divided by the initial volume (x,y)∈E f (x, y).

We will now present several 2-D extensions to the principle of morphological signature, using the notations introduced in this section.

2-D morphological signatures

We have seen previously that the well-known (differential) morphological profile and pattern spectrum (or granulometry / antigranulometry) operators may be written as 1-D morphological signatures computed on a local or global scale. Here we will present several 2-D extensions to these signatures, in order to gather more information related either to size, intensity, or spectral information.

Size-size signature

In the Π and ∆ series, a unique parameter λ is considered for measuring the size evolution. In other words, the resulting series are generated by applying openings and closings with SE B λ of increasing size λ. The SE B λ for a given size λ can be obtained by dilating λ times the original SE B using a given SE S:

B λ = {B i | B i = δ S (B i-1 ), ∀i ∈ {2, . . . , λ}} λ (9)
with B 1 = B. The SE S can be chosen either as a simple SE such as a square, a cross, or a disc of small size (3 × 3 pixels for instance), or as B itself.

This definition assuming a single size parameter λ prevents us from performing accurate measurement. Indeed, it is not adequate to elliptical or rectangular shapes for instance, where the two independent axes should be taken into account.

So we introduce the 2-D size-size signature which is built using SE B α,β with two different size parameters α and β that vary independently. The SE B α,β for a given pair (α, β) of size parameters is obtained by dilating α and β times the original SE B with two SE S and T :

B α,β = {B i,j | B i,j = δ S (B i-1,j ), B i,j = δ T (B i,j-1 ), ∀i∀j ∈ {2, . . . , λ}} α,β (10) with B 1,1 = B.
A relevant choice for the two SE S and T consists in 1-D SE such as horizontal and vertical lines respectively, with a length of 3 pixels for instance.

It is then possible to generate the Π series from this 2-D set of SE B α,β :

Π ψ (f ) = Π ψ α,β (f ) | Π ψ α,β (f ) = ψ α,β (f ), ∀α∀β ∈ {0, . . . , n} (11)
where the application of ψ on f with a SE B α,β is noted ψ α,β (f ) and with the convention ψ 0,0 (f ) = f . The ∆ series measures the differential in both size dimensions:

∆ ψ (f ) = ∆ ψ α,β (f ) | ∆ ψ α,β (f ) = 2Π ψ α,β (f ) -Π ψ α-1,β (f ) -Π ψ α,β-1 (f ), ∀α∀β ∈ {1, . . . , n} (12)

Size-intensity signature

As we can notice in the definitions given in section 2, the pixel intensity values in grayscale images are used either directly (at local scale) or through the sum operator (at global scale). The original morphological signature does not take into account the distribution of intensity values in the image. This can be seen as a strong limitation as intensity distribution (usually measured by histogram) is a key feature when dealing with pattern recognition in image data. So we propose to build at the global scale a 2-D morphological signature which takes into account both the size and the intensity distributions. To do so, let us use the Kronecker delta function:

δ ij = 1 if i = j 0 if i = j (13)
and the histogram function h f : T → Z:

h f (θ) = (x,y)∈E δ θf (x,y) (14)
Alternatively, we can also use the normalised histogram function

h f : T → [0, 1] where h f (θ) = h f (θ)/ θ ∈T h f (θ ).
The formulation of the 2-D size-intensity signature is then given by the following Π series:

Π ψ (f ) = Π ψ λ,θ (f ) | Π ψ λ,θ (f ) = h ψ λ (f ) (θ), ∀θ ∈ T, ∀λ ∈ {0, . . . , n} (15)
and its derivative counterpart is given by the following ∆ series:

∆ ψ (f ) = ∆ ψ λ,θ (f ) | ∆ ψ λ,θ (f ) = h ψ λ (f )-ψ λ-1 (f ) (θ), ∀θ ∈ T, ∀λ ∈ {1, . . . , n} (16)
This 2-D size-intensity morphological signature gathers the properties of both the classical morphological signature and the intensity histogram by considering size and intensity information together.

Size-spectrum signature

We have considered until now that the input image was monovalued. In case of multispectral images where T is equal to Z k or N k , the spectral signature of the image pixels can be involved into the morphological signature. To do so, it is possible to first compute a morphological signature for each of the k spectral components (or bands) and then to combine these k signatures into a single one, either 1-D or 2-D.

In the case of a 2-D signature, the morphological series Π can be expressed as:

Π ψ (f ) = Π ψ λ,ω (f ) | Π ψ λ,ω (f ) = ψ λ (f ω ), ∀ω ∈ {1, . . . , k} , ∀λ ∈ {0, . . . , n} (17)
where f ω is a grayscale image representing the ω th spectral component of f . In this definition the marginal strategy is used, thus the correlation among the different spectral channels is completely ignored.

To avoid this limitation, it is possible to rather consider a vectorial ordering when applying the morphological operators γ and φ on the multispectral input image f . The purpose of a vectorial ordering is to give a way to order vectors and thus to compute vectorial extrema by means of the two operators sup v and inf v . Assuming a given vectorial ordering, the fundamental dilation and erosion operators are written:

δ v B (f )(x, y) = sup v (r,s)∈B f (x -r, y -s), (x, y) ∈ E (18) v B (f )(x, y) = inf v (r,s)∈B f (x + r, y + s), (x, y) ∈ E (19)
and from these operators it is possible to write the vectorial versions of the morphological opening γ v and closing φ v :

γ v B (f ) = δ v B ( v B (f )) (20) φ v B (f ) = v B (δ v B (f )) (21)
The morphological size-spectrum signature can finally be expressed as:

Π ψ (f ) = Π ψ λ,ω (f ) | Π ψ λ,ω (f ) = (ψ v λ (f )) ω , ∀ω ∈ {1, . . . , k} , ∀λ ∈ {0, . . . , n} (22)
where (ψ v λ (f )) ω = ψ λ (f ω ) in the specific case of a marginal ordering. For a comprehensive review of vectorial orderings and multivariate mathematical morphology, the reader can refer to [START_REF] Aptoula | A comparative study on multivariate mathematical morphology[END_REF].

Finally, we can also define the differential series ∆ as:

∆ ψ (f ) = ∆ ψ λ,ω (f ) | ∆ ψ λ,ω (f ) = Π ψ λ,ω (f ) -Π ψ λ-1,ω (f ), ∀ω ∈ {1, . . . , k} , ∀λ ∈ {1, . . . , n} (23)
We have presented here several 2-D extensions to the classical morphological signature. We will now show their potential interest when computed either on a local or a global scale through their use in various applications.

Applications

We have presented in the previous section several 2-D extensions to the morphological signature, which can be computed either on a local or a global scale. The potential interest of these extensions will be illustrated by a panel of applications made as various as possible and related to building detection / object recognition / pixel classification.

Building detection

The 2-D size-size signature has been used as a global scale to determine optimal attributes for image filtering in the context of building detection in panchromatic remote sensed images. Binary images are obtained by thresholding the initial graylevel images. The filter is a morphological opening whose goal is to remove all details smaller than the estimated size for buildings visible in the binary image.

Figure 1 illustrates the relevance of the proposed 2-D granulometry (using only the 2-D version of the ∆ γ series) to automatically determine the optimal SE to be used in the morphological filtering process among sizes (α, β) between (10, 10) and (50, 50). As we can observe, the proposed SE size (here 19 × 21 pixels, computed as the highest peak) helps the building detection process by greatly reducing the noisy areas which do not correspond to buildings. The overall process is given in [START_REF] Lefèvre | Automatic building extraction in vhr images using advanced morphological operators[END_REF].

Object recognition

In the second application we focus on object recognition in grayscale images. The feature used is the size-intensity signature which has proven to be more discrimant than histogram. The experiments were made on the COIL-20 dataset from Columbia University, which contains 1440 images representing 20 objects under 72 viewpoints.

Pixel classification

The third application illustrates the potential interest of the size-spectrum signature computed on a local scale to deal with multispectral image interpretation in remote sensing through pixel classification. The image used in this experiment is composed of 1100×900 pixels and 3 spectral components (green, red and near infra-red) and represents an urban area of Strasbourg, France (fig. 4, left). For each pixel (x, y) we consider here a 2-D morphological signature ∆ of size 10 × 3 with λ = 5 and ω = 3, i.e. 5 openings and 5 closings for 3 spectral bands. A bayesian classification was performed with 5 classes (buildings, roads, water, vegetation, and shadows). The learning set represents 1 % of the complete data and 10-fold cross-validation was involved. The overall accuracy was about 80-85 % with some classes (buildings, 75 %) been worse detected than others (water, vegetation, shadows, 95 %).

In figure 4 is given the original multispectral image in false colours and also an example of classification result using this 2-D size-spectrum local signature. More exhaustive results can be found in [START_REF] Aptoula | A comparative study on multivariate mathematical morphology[END_REF].

In this paper we have presented several 2-D extensions to the widely used 1-D morphological signature. This signature can be computed on a local or global scale, and is respectively known as (differential) morphological profile and pattern spectrum (or granulometry / antigranulometry). We propose to include another dimension related to size, intensity or spectral information in complement to the initial size distribution, thus obtaining 2-D size-size, size-intensity, and size-spectrum morphological signatures. We finally illustrate the potential interest of these extensions through their use in various applications: building detection, object recognition, and pixel classification.

Future work will include optimisation of the algorithms used to generate the 2-D signatures in order to reduce the computational cost. Another perspective is to exploit in a better way the 2-D signatures using matrix computation. We also consider to elaborate some n-D signatures to gather more information in a single signature. Finally, we will have to compare the proposed 2-D morphological signatures with well-known nonmorphological measures, such as wavelets or other textures descriptors.

Fig. 1 .

 1 Fig. 1. Relevance of the 2-D size-size global signature to determine optimal parameters for building detection (from left to right and top to bottom): input, unfiltered and filtered images, and curve visualisation (the highest peak determines the optimal parameter).

(Fig. 2 .

 2 Fig. 2. Comparison between the 2-D size-intensity global signature and intensity histogram for object recognition (the input is the top left image).

Fig. 3 .

 3 Fig. 3. Relevance of the 2-D size-intensity global signature to discriminate objects in graylevel images: three different objects (left) with relatively similar histograms and their respective 2-D signatures (right).

Fig. 4 .

 4 Fig. 4. Relevance of the 2-D size-spectrum local signature to classify pixels in multispectral remote sensed images: input (left) and classified (right) images.

Acknowledgements

The author wish to thank his colleagues Erchan Aptoula and Jonathan Weber for some of the experimental works made on remote sensing applications.


Benjamin Perret
email: bperret@unistra.fr

Sébastien Lefèvre
email: sebastien.lefevre@univ-ubs.fr

Christophe Collet
email: c.collet@unistra.fr

Toward a New Axiomatic for Hyper-connections

We propose an evolution of the hyper-connection axiomatic in order to improve the consistency of hyper-connected filters and to simplify their design. Our idea relies on the principle that the decomposition of an image into h-components must be necessary and sufficient. We propose a set of three equivalent axioms to achieve this goal. We show that an existing h-connection already fulfills these axioms and we propose a new h-connection based on flat functions that also fulfills these axioms. Finally we show that these new axioms bring several new interesting properties that simplify the use of h-connections and guarantee the consistency of h-connected filters as they ensure that: 1) every deletion of image components will effectively modify the filtered image 2) a deleted component can not re-appear in the filtered image.

Introduction

Connections in image processing describe how pixels can be grouped together according to their spatial relationships and/or their gray level values. In recent years, several works were devoted to the development of new theories of connections among which hyper-connection (h-connection) [START_REF] Serra | Connectivity on complete lattices[END_REF][START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF][START_REF] Wilkinson | An axiomatic approach to hyperconnectivity[END_REF] is a very promising notion. Contrary to traditional connections (like set connections), h-connections allow to decompose images into intersecting components.

In this paper, we investigate the problem of the consistency of connected filters (edge-preserving filters) based on h-connections. The usual way to produce a h-connected filter on an image is based on the following three steps: 1) consider the set of h-connected components of this image, 2) select a subset of theses components and 3) reconstruct an image using the select h-components. In this scheme, one expect that the result image contains the selected h-components and only those ones. But in the current theory, we can easily show that the result will contain the selected h-connected components but may also contain other components. These unwanted components are in fact h-components of the original image that have not been selected during the 2 nd step (see proposition 2 further). This undesirable effect can be circumvented by the addition of a new property that relies on the idea that the decomposition of an image into h-components must be necessary and sufficient to describe the image. We think that this property is fundamental for image processing as it implies that the decomposition into h-components completely describes the whole image (sufficient) and that none of its components is useless (necessary). These requirements enforce the consistency of the h-connected filters as they ensure that: 1) every deletion of image components will effectively modify the filtered image 2) a deleted component can not re-appear in the filtered image. From a more formal point of view, the sufficiency and necessity conditions can be formulated like this: the supremum of the h-components of an image must be equal to the whole image (sufficient) and the supremum of a family strictly included in the family of h-components must be strictly included in the image (necessary).

In this article we propose a set of three axioms to achieve this goal and we show that they are indeed equivalent. We show that an existing h-connection (the functions with a unique maximum) already fulfills these axioms and we propose a new h-connection based on flat functions, that may be seen as a generalization of the flat zones, that also fulfills these axioms. Finally we show that these new axioms bring several new interesting properties that simplify the use of h-connections and guarantee the consistency of h-connected filters.

This article is organized as follows: section 2 gives some preliminary definitions, section 3 presents the theory of h-connections and establishes some new results, section 4 presents our new theoretical developments and finally section 5 concludes this work.

Lattice and Set Connection

This section gives some preliminary definitions about lattices and set connections. In the sequel, sets and families are written in capital letters while elements of a set are written in lower-case letters. The logical conjunction (respectively disjunction) is noted ∧ (respectively ∨).

A lattice (L, ≤, , ) is composed of a non-empty set L with a partial order ≤ and two operators: a supremum and an infimum . In the following, we consider only complete lattices where every non empty family A ⊆ L has a supremum A = and an infimum A = ⊥ in L. A set S ⊆ L such that ⊥ / ∈ S is called a sup-generating family of L if every element of L can be written as the supremum of elements of S (∀a ∈ L, ∃B ⊆ S, a = B). The elements of S are called sup-generators. Important examples of lattices are:

the extended real line R = R {-∞, +∞}, which is a complete chain under the usual order, infimum, and supremum. It is sup-generated by R. the set of all subsets of a set E (noted P (E)), which is a complete lattice with the partial order defined by the inclusion relation, and the infimum and supremum given by the set intersection and union. It is sup-generated by the singletons of E. the set of functions from a set E into a lattice L (noted L E ), which is a lattice under the pointwise order ∀f,

g ∈ L E , f ≤ g ⇔ ∀x ∈ E, f (x) ≤ L g(x)
. The infimum and supremum are similarly defined by a pointwise application of the infimum L and supremum L of the underlying lattice. L E is supgenerated by the pulses: the functions δ t x , ∀x ∈ E, ∀t ∈ L, t = ⊥ defined by ∀y ∈ E, δ t x (y) = t if x = y, ⊥ otherwise. The properties of L E depend of the properties of the underlying lattice L.

A connection on P (E) is a family C ⊆ P (E) composed of the connected elements of E. Formally we say that C is a connection if [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF]:

1. {∅} ∈ C: the empty set is connected; 2. ∀x ∈ E, {x} ∈ C: the singletons (or points) are connected; 3. ∀A ⊆ C, A = ∅ ⇒ A ∈ C, the union of intersecting connected elements is connected.

Hyper-connection

We now present the theory of h-connections, we give the definition of a new h-connection and we establish two new properties. Being given a lattice L with a sup-generating family S, an h-connection C + on L is a subset of L verifying the following conditions [START_REF] Serra | Connectivity on complete lattices[END_REF]:

1. ⊥ ∈ C + : the least element is h-connected; 2. ∀s ∈ S, s ∈ C + : the points (i.e. the sup-generators) are h-connected;

3. ∀A ⊆ C, A ⇒ A ∈ C + , the supremum of overlapping h-connected elements is h-connected.
where is a predicate on P (L) called the overlap criterion. This predicate must be decreasing: ∀A ⊆ P (L), A ⇒ ∀b ∈ L, {b} A.

A simple example of h-connection is made of functions with a unique maximum [START_REF] Serra | Connectivity on complete lattices[END_REF][START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF]. This h-connection is defined on the lattice of functions L E and it is based on a primary set connection C o on E. Then, one says that a function has a unique maximum if it is connected at all levels with respect to the primary set connections. More formally, we define the set of connected functions by:

C + m = f ∈ L E | ∀t ∈ L, f ¬ t ∈ C o , with f ¬ t = {p ∈ E | t ≤ f (p)} the thresholding of f at level t.
The overlap criterion can be defined as:

∀ {f i } ⊆ L E , m {f i } ⇔ ∀t ∈ L, U t = ∅ or U t = ∅ with U t = f i ¬ t | f i ¬ t = ∅ .
Another new example of h-connection is given by the set of all flat functions. Let L E be the image space, let C o be a primary set connection on P (E), the set of flat functions is defined by:

C + p = f C,t ∈ L E | C ∈ C o , t ∈ L (1) with ∀x ∈ E, f C,t (x) = t if x ∈ C, ⊥ otherwise.
Then the overlap criterion p is defined by:

∀ {g i∈I } ⊆ C + p , p ({g i }) = true if U = ∅ and ∀i, j ∈ I, ∀x ∈ U, g i (x) = g j (x) false otherwise (2)
with U = i∈I supp (g i ) and supp (g i ) = {p ∈ E | g i (p) = ⊥} is the support of the function g i . Flat functions are overlapping if their supports intersect and if they have the same value on this intersection. 2). Proof. We must show that the three axioms of the h-connection are verified by (C + p , p ). First note that p is trivially decreasing. 1) By definition of set connections,

1. (C + p , p ) is an h-connection of L E (see Figure
∅ ∈ C o , ∀t ∈ L, f ∅,t = ⊥ ∈ C + p .
2) By definition of set connections, the singletons of E belong to C o , thus, ∀x ∈ E, ∀t ∈ L, the f {x},t = δ x,t ∈ C + p are the pulses of L E which are a sup-generating family S of L E .

3) Let {a i∈I } be a family of C + p , let also U = i∈I supp (a i ), and assume that U = ∅ and ∀i, j ∈ I, ∀x ∈ U, a i (x) = a j (x) then we must show that a i ∈ C + p . From property 3) of set connections, if U = ∅ then i∈I supp (a i ) = R ∈ C o , then from the definition of C + p and from the condition ∀i, j ∈ I, ∀x ∈ U, a i (x) = a j (x), all a i have the same value t on their supports, thus,

a i = f R,t ∈ C + p .
Hyper-connected operators and openings H-operators [START_REF] Wilkinson | An axiomatic approach to hyperconnectivity[END_REF] are applications from L into P (L) which extract the h-components marked by a sup-generator. Formally, being given a sup-generator s ∈ S, we define the h-operator by:

∀a ∈ L, γ * s (a) = h ∈ C + | s ≤ h ≤ a, ∀g ∈ C + , h ≤ g ≤ a ⇒ g = h (3)
The h-components are maximal h-connected elements and the h-components of a ∈ L are given by:

γ * (a) = s∈S γ * s (a) (4)
The h-opening marked by s is the supremum of the h-components of a above s:

∀a ∈ L, γ s (a) = γ * s (a) (5)
The two following properties hold [START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF]:

1. a = γ * (a): an element is the supremum of its h-components; 2. ∀b, c ∈ γ * (a), b = c ⇒ b c: two h-components do not overlap.

Z-operators

In [START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF], the authors propose the interesting notion of z-operators.

Let a ∈ L, the equivalence relation a ∼ on S is defined by:

∀b, c ∈ S, b a ∼ c ⇔ γ b (a) = γ c (a) (6)
Then, the z-operator is defined as the supremum of an equivalence class of a ∼:

∀s ∈ S, ζ s (a) = b ∈ S | b a ∼ s (7)
Finally, the set of all z-zones of a is noted:

ζ (a) = {ζ s (a) | ∀s ∈ S} (8)
The authors of [START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF] have chosen to base the z-operators on the h-connected openings, but we can also define them in terms of h-connected operators leading to a slightly different definition. Let a ∼ * be the equivalence relation defined by:

∀b, c ∈ S, b a ∼ * c ⇔ γ * b (a) = γ * c (a) (9)
The alternative z-operator * is the supremum of an equivalence class of a ∼ * :

∀s ∈ S, ζ * s (a) = b ∈ S | b a ∼ * s (10)
Generally, as h-components can be intersecting, ζ s (a) = ζ * s (a). Consider the set E = {a, b, c} and the h-connection {∅, {a} , {b} , {c} , {ab} , {bc} , {ac}} on P (E) with the overlap criterion defined as false (two elements are never overlapping, this is a valid criterion since it is trivially decreasing). Then, the h-components of E are {{ab} , {bc} , {ac}}. Thus, we have γ a

(E) = γ b (E) = γ c (E) = E and ζ a (E) = ζ b (E) = ζ c (E) = E. But γ * a (E) = {{ab} , {ac}}, γ * b (E) = {{ab} , {bc}} and γ * c (E) = {{ac} , {bc}} thus ζ * a (E) = {a} = ζ a (E).
In section 4, we show that under certain conditions, we have ∀a ∈ L, ∀s ∈ S, ζ s (a) = ζ * s (a). Figure 2 shows an example of function decomposition with the z-operators and the h-connection of functions having a unique maximum.

H-reconstruction

Being given a marker m ∈ L, the hyper-reconstruction (hreconstruction) of a ∈ L marked by m is defined by [START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF]:

a m = s∈S,s≤m γ s (a) (11) f a f f f f b c d e f f f f f g h i
Example of decomposition of the function f using the h-connection of flat functions. We propose another definition based on the alternative z-operators:

a ζ m = s∈S,s≤m ζ s (a) ( 12
)
The latter is more flexible, the two approaches are compared in Figure 3.

New properties

We establish here two properties that will help us in following proofs. Let a ∈ L:

Proposition 2. ∀ {h i } ⊆ γ * (a), {h i } ⊆ γ * ( {h i })
Proof. Let a in L, {h i } ⊆ γ * (a) and b = {h i }, for all i , we have h i ≤ b and b ≤ a. Suppose that there exists some h ∈ C + such that h i ≤ h ≤ b ≤ a. But by definition of γ * (a) and as h i belongs to γ * (a), we have

h i = h showing that h i ∈ γ * (b). Thus we have {h i } ⊆ γ * ( {h i }).
This property that was already known under a different form in [START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF] states that: being given a family of h-components, the h-components of its supremum is a superset of the given family. Those two families are generally not equal, take for example the h-connection given in section 3; the h-components of the element {abc} are {{ab} , {bc} , {ac}}. Now, consider the {{ab} , {bc}}, the supremum of this family is equal to {abc} and the h-components of this supremum is a strict superset of the family. If the h-opening of a marked by the sup-generator s is an h-component then there is no other h-component of a above s.

In the following, we assume the existence of a complete lattice L with the sup-generating family S and the h-connection C + such that the number of hcomponents of each element of L is finite.

Toward a new axiomatic

Despite the successful developments of h-connections in recent works [START_REF] Wilkinson | An axiomatic approach to hyperconnectivity[END_REF][START_REF] Ouzounis | Generalized Connected Morphological Operators for Robust Shape Extraction[END_REF][START_REF] Ouzounis | Hyperconnected attribute filters based on k-flat zones[END_REF][START_REF] Nempont | A new fuzzy connectivity measure for fuzzy sets[END_REF][START_REF] Wilkinson | Hyperconnectivity, attribute-space connectivity and path openings: Theoretical relationships[END_REF][START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF], the theory of h-connections is still not satisfactory. All of the successes obtained have been based on specific h-connections, whereas only a few general properties have been established. This lack of theoretical results comes from the very broad definition of h-connections. The third axiom (the overlap criterion) is especially problematic. On one hand, this axiom is satisfactory because it formalizes the intuitive union based approach. On the other hand, the definition of the overlap criterion is so wide that, in practice, this third axiom does not bring any meaningful property to the h-connection (for example, consider the overlap criterion which is always false, and the third axiom is never applicable). One can also observe that in all the h-connections used in practice [START_REF] Serra | Connectivity on complete lattices[END_REF][START_REF] Braga-Neto | A theoretical tour of connectivity in image processing and analysis[END_REF][START_REF] Nempont | A new fuzzy connectivity measure for fuzzy sets[END_REF][START_REF] Ouzounis | Generalized Connected Morphological Operators for Robust Shape Extraction[END_REF][START_REF] Ouzounis | Hyperconnected attribute filters based on k-flat zones[END_REF] the overlap criterion is defined as a tautology such that it correctly fits the set of connected elements. For example, instead of a constantly false overlap criterion, one can also define the following less trivial valid overlap tautologic criterion:

∀C + ∈ L, ∀A ⊆ C + , A ⇔ (∀B ⊆ A, B ∈ C + ).
To harden the definition of h-connections, we propose the fundamental property that an h-connection must provide for each element a decomposition in h-components that is necessary and sufficient. With the current theory it is clear that the decomposition in h-components is sufficient in the sense that an element is equal to the supremum of its h-components. But in general this decomposition is not necessary in the sense that only a subset of the h-components of an element can be necessary to retrieve the element by supremum.

More formally, one can define the following properties ∀a ∈ L:

P-1 -∀h ∈ γ * (a) , {h ∈ γ * (a) | h = h } < γ * (a): all h-components of an element are necessary to describe the element. P-2 -∀ {h i } ⊆ γ * (a) , ∀h ∈ γ * (a) , h ≤ {h i } ⇒ h ∈ {h i }: an h-component
cannot be covered by other h-components. P-3 -∀h ∈ γ * (a) , ∃s ∈ S, γ * s (a) = h: each h-component of an element can be individually retrieved through an h-opening.

Then, the following proposition holds: Proposition 4. Properties P-1, P-2, and P-3 are equivalent. In the following, we will say that a connection is accessible if the previous properties hold.

Proof. We first show P-2⇒P-1. Let {h i } ⊆ γ * (a), and assume that ∃h Now P-3⇒P-2: let {h i } ⊆ γ * (a) and h ∈ γ * (a). Assume that h ≤ {h i } then, by P-3, there exists s ∈ S such that γ s (a) = h and s ≤ h ≤ {h i }. By assumption, the family γ * (a) is finite, so there exists j such that s ≤ h j and from proposition 3 we can say that h = h j and thus h belongs to {h i }.

∈ {h i } such that {h ∈ γ * (a) | h = h } = γ * (a). Let F = {h ∈ γ * (a) | h = h }, one have h ≤ γ * (a) = F but
And finally, P-1⇒P-3: let h ∈ γ * (a), P-1 implies that B < a with B = {h ∈ γ * (a) | h = h}. Now, consider the smallest family {s i } ⊆ S such that (B ∪ {s i }) = a. Let s be an element of {s i }, we have for all h ∈ B, s h and s ≤ h thus γ s (a) = h. P-1 and P-2 are two direct formulations of the "necessity" condition, the first one from a global point of view, the second from a local point of view. P-3 shows that this condition naturally comes to the notion of special groups of sup-generators that give access to a unique h-component through an h-opening. From the image processing point of view, this means that every h-component of an image can be selected individually with a simple opening.

Proposition 5. The h-connection (C + m , m ) of functions with a unique maximum is accessible.

Proof. We show that C + m verifies the property P-3. Let a ∈ L E , h ∈ γ * (a) and x ∈ E such that h(x) = max y∈E h(y) (h(x) is in the maximum of h). Let the pulse δ x,h(x) and we show that γ * δ x,h(x) (a) = {h}. The inclusion {h} ⊆ γ * δ x,h(x) (a) is direct as δ x,h(x) ≤ h and h ∈ γ * (a). We now show the inverse inclusion

γ * δ x,h(x) (a) ⊆ {h}. Let h ∈ γ * δ x,h(x) (a). We start by showing that m ({h, h }) is true. Let t ∈ L, we have: if t ≤ h(x) then h ¬ t = ∅ and h ¬ t = ∅ (as δ x,h(x) ≤ h ), else if t > h(x) then h ¬ t = ∅ (h(x)
is the maximal height of h). And yet ∀t ∈ L, t ≤ h(x), we have

x ∈ h ¬ t ∩ h ¬ t = ∅. So m ({h, h }) is true. In consequence, {h, h } ∈ C + m , but h, h ≤ a so {h, h } ≤ a,
and as h and h are h-connected components of a, we have either h = h or h and h are not comparable. Assume that h and h are not comparable, then h < {h, h }, h < {h, h }, and as {h, h } ≤ a, h and h cannot be h-connected components of a. Thus we have h = h and γ * δ x,h(x) (a) ⊆ {h}. Thus, we have the double inclusion and γ * δ x,h(x) (a) = {h}. Every h-connected component of a can be obtained by an h-opening. The h-connection C + m verifies property P-3 and is accessible. Proposition 6. The h-connection (C + p , p ) of flat functions is accessible.

Proof. The demonstration is similar to the previous one using the opening γ * δ x,h(x) (a) = {h} with x ∈ E such that h(x) = a(y).

Accessible h-connections also have stronger properties:

Proposition 7. If C + is accessible, being given a family of h-components, the h-components of its supremum is the same family:

∀ {h i } ⊆ γ * (a), {h i } = γ * ( {h i })
Proof. We already have the first inclusion {h i } ⊆ γ * ( {h i }) by proposition 2.

The second inclusion γ * ( {h i }) ⊆ {h i } is directly given by P-2: let h ∈ γ * ( {h i }), as we have h ≤ {h i }, P-2 says that h belongs to {h i }. Finally, the double inclusion proves that {h i } = γ * ( {h i }).

Compared to proposition 2, this version is harder and it ensures that by selecting a family of h-components, its reconstruction by the supremum operator will not introduce new h-components. From the image processing point of view, this ensures that, when performing an h-connected filtering, a deleted h-component cannot re-appear after the reconstruction of the selected h-components. Another interesting property of accessible h-connections concerns z-operators: The reverse implication is given by: assume that we have s 1 This property removes the necessity of operating a choice between the two distinct approaches when dealing with z-operators. Thus it simplifies the process of creating image filters based on z-zones.

Conclusion

We have proposed an evolution of the notion of h-connection motivated by the desirable property that the decomposition of an image into connected components should be necessary and sufficient to describe this image. Due to this evolution, based on three equivalent axioms, we have set out several new important properties for image processing which ensure the consistency and simplify the use of h-connected filters. This theoretical work offers a strong basis for the development of hierarchical representations based on h-connections [START_REF] Perret | Hyperconnections and hierarchical representations for grayscale and multiband image processing[END_REF].

Figure 1 Fig. 1 .

 11 Fig. 1. The overlap criterion p. Functions a and b are not overlapping because they don't have the same value on their intersection. Functions c and d are not overlapping because their supports do not intersect. Functions e and f overlap.

Proposition

Fig. 3 .

 3 Fig. 3. Example of h-reconstruction with the h-connection of flat functions. Figure (a) shows the function f and the marker m. The figure (b) represents the result of the h-reconstruction f m of f by m proposed in [2]. Figure (c) represents the result f ζ m

Proposition 3 .

 3 ∀s ∈ S, ∀h ∈ γ * (a), if s ≤ h and γ s (a) = h ∈ C + then h = h . Proof. Let s in S, h ∈ C + such that γ s (a) = h and h ∈ γ * (a) such that s ≤ h . We have h ≤ {g ∈ γ * (a) | s ≤ g} = h, so s ≤ h ≤ h ≤ a and by definition of h-components h = h .

  that contradicts P-2 since h / ∈ F . Then, we show P-1⇒P-2: let {h i } ⊂ γ * (a) and assume that there exists h ∈ γ * (a) such that h / ∈ {h i } and h ≤ {h i }. Let b = {h i } = ({h i } ∪ {h}), we have by proposition 2 that h ∈ γ * (b) and that {h i } ⊆ γ * (b). And then we have {h ∈ γ * (b) | h = h} = {h i } = b which contradicts P-1.

Proposition 8 . 1 a∼ s 2 ⇔ s 1 a∼ * s 2 . 1 a∼ * s 2 ⇒ s 1 a∼ s 2 , 1 a∼ 1 a ∼ s 2

 81212112112 If C + is accessible, the equivalence relations a ∼ and a ∼ * are equivalent: ∀s 1 , s 2 ∈ S, sProof. First, we show that ∀s 1 , s 2 ∈ S, s this part does not need the accessibility property. We have:s * s 2 ⇔ γ * s1 (a) = γ * s2 (a) ⇒ γ * s1 (a) = γ * s2(a) ⇔ γ s1 (a) = γ s2 (a) ⇔ s

a ∼ s 2

 2 , which is equivalent to γ s1 (a) = γ s2 (a). Let h ∈ γ * s1 (a), we have h ≤ γ s1 (a) = γ * s1 (a) = γ * s2 (a). Then, P-2 implies that h ∈ γ * s2 (a) and thus γ * s1 (a) ⊆ γ * s2 (a). The same argument is used to prove that γ * s2 (a) ⊆ γ * s1 (a) showing that γ * s2 (a) = γ * s1 (a) which is equivalent to s 1 a ∼ * s 2 .