Jean Serra

• Gustave Eiffel University

In obtaining a tractable solution to the problem of extracting a minimal partition from hierarchy or tree by dynamic programming, we introduce the braids of partition and h-increasing energies, the former extending the solution space from a hierarchy to a larger set, the latter describing the family of energies, for which one can obtain the solutio...

This theoretical paper provides a basis for the optimality of scale-sets by Guigues [6] and the optimal pruning of binary partition trees by Salembier-Garrido [11]. They extract constrained-optimal cuts from a hierarchy of partitions. Firstly, this paper extends their results to a larger family of partitions, namely the braid [9]. Secondly, the pap...

Cette étude concerne le partitionnement d'un ensemble de telle sorte que les séparations entre classes soient matérialisées. On le résoud, dans les cas continu et discret, au moyen de hiérarchies de tesselations dont les classes sont des ouverts réguliers. Dans le cas discret, le passage partition→tessellation s'exprime par des topologies d'Alexand...

This theoretical paper introduces a novel continuous representation of hierarchy of partitions, and generalizes the conditions of h-increasingness and scale increasingness [1] to obtain a global-local optimum on the hierarchy. It studies in particular the Lagrange optimization problem and gives the condition on the energy to achieve constrained opt...

This paper introduces a new fusion paradigm that reorders the contours of an input hierarchy of segmentations utilizing any external function. Ground truth partitions, markers, numerical functions, and other additional sources are used to obtain a new reordered hierarchy. This transformation aids in combining sets, functions, and partitions in diff...

A few theses developed in G. Matheron’s book “Estimating and Choosing” are commented. They are the concern, firstly of the notion of a range, or typical size of the phenomenon under study, secondly of the role of the infinity (why a variogram rather a covariance, since data sets are always finite?), and thirdly of the sense of stationarity (is it a...

In the current technical note we provide a topological generalization of hierarchy of partitions(HOP) structure, and the implications over the axioms of h-increasingness and scale increasingness [13]. Further in this study we will explicit the Lagrange optimization in the optimal cuts problem and the conditions necessary on the energy to obtain a g...

A hierarchy of segmentations(partitions) is a multiscale set representation of the image. This paper introduces a new set of scale space operators or transformations on the space of hierarchies of partitions. An ordering of hierarchies is proposed which is endowed by an ω-ordering based on a global energy over the classes of the hierarchy. A class...

In evaluating a hierarchy of segmentations H of an image by ground truth G, which can be partitions of the space or sets, we look for the optimal partition in H that “fits” G best. Two energies on partial partitions express the proximity from H to G, and G to H. They derive from a local version of the Hausdorff distance. Then the problem amounts to...

A new approach is proposed for finding optimal cuts in hierarchies of partitions by energy minimization. It rests on the notion of h-increasingness, allows to find best(optimal) cuts in one pass, and to obtain nice “climbing” scale space operators. The ways to construct h-increasing energies, and to combine them are studied, and illustrated by two...

We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph GG. To this end, we consider two dual adjunctions between the edge set and the vertex set of GG. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of GG and (ii) to extend it to sub...

This chapter presents the former approach, which was extended through the 1970s to numerical functions under the impulse of macro- and microscopic medical imaging. During the 1980s, when morphological operations were acting on increasingly varied objects, the need for unification led to the concept of complete lattice as a common framework. This al...

Hierarchical segmentation is a multi-scale analysis of an image and provides a series of simplifying nested partitions. Such a hierarchy is rarely an end by itself and requires external criteria or heuristics to solve problems of image segmentation, texture extraction and semantic image labelling. In this theoretical paper we introduce a novel fram...

Morphological operators may be taken up from the two entries of dilation and of connection. This tutorial focuses on the second entry, and leads to optimum partitionings of the images under study. Five notions, which derive from each other, are successively explored. The “set connection,” which generalizes the usual connectivities, opens the series...

The paper deals with global constraints for hierarchical segmentations. The proposed framework associates, with an input image, a hierarchy of segmentations and an energy, and the subsequent optimization problem. It is the first paper that compiles the different global constraints and unifies them as Climbing energies. The transition from global op...

A new approach is proposed for finding the ”best cut” in a hierarchy of partitions by energy minimization. Said energy must be ”climbing” i.e. it must be hierarchically and scale increasing. It encompasses separable energies [5], [9] and those which composed under supremum [14], [12]. It opens the door to multivariate data processing by providing l...

A new approach is proposed for finding the "best cut" in a hierarchy of partitions by energy minimization. Said energy must be "climbing" i.e. it must be hierarchically and scale increasing. It encompasses separable energies and those composed under supremum.

We begin by defining the setup and the framework of connective segmentation. Then we start from a theorem based on connective criteria, established for the power set of an arbitrary set. As the power set is an example of a complete lattice, we formulate and prove an analogue of the theorem for general complete lattices. Secondly, we consider partia...

Given a set E, the partitions of E are usually ordered by merging of classes. In segmentation procedures, this ordering often generates small parasite classes. A new ordering, called ”grain building ordering”, or GBO, is proposed. It requires a connection over E and states that \(A\preccurlyeq B\), with A,B ⊆ E, when each connected component of B c...

The paper consists in a survey of image analysis over the last fifty years, seen from the point of view of Mathematica Morphology.

This paper is devoted to hierarchies of partitions, on which all criteria are proved to be connective. Optimisations are addressed by minimizing energies that satisfy the condition of hierarchical increasingness. The optimal cuts through the hierarchies are found. It is shown that many of the classical techniques are variants of what is proposed.

This paper deals with the theory and applications of spatially-variant discrete mathematical morphology. We review and formalize the definition of spatially variant dilation/erosion and opening/closing for binary and gray-level images using exclusively the structuring function, without resorting to complement. This theoretical framework allows to b...

In this paper, a new multi-objective (MO) fuzzy-clustering technique is developed for unsupervised land-cover classification in remote-sensing images. This technique automatically partitions the pixels in the spectral domain into several clusters. The ...

This paper aims to study forest fires from risk maps and satellite data in order to predict the location of the burnt zones they provoke. The approach is based on the discrete stochastic model of a random spread, which describes both fire fronts and burnt areas. Under iteration, it provides the time evolution of the process. The probabilities of sp...

There are various ways to define digital convexity in Zn. The proposed approach focuses on structuring elements (and not the sets under study), whose digital versions should allow to construct hierarchies of operators satisfying Matheron semi-groups law γλγμ=γmax(λ,μ), where λ is a size factor. In Rn the convenient class is the Steiner one. Its ele...

This chapter first specifies some requirements on the measurements made on images so that they are in some way ‘well-behaved’ or reliable. It then formalizes this by defining a space of objects on which such measurements are possible, and what their properties are. The chapter also specifies how to construct these measurements and introduce the top...

This paper deals with the theory and applications of spatially-variant mathematical morphology. We formalize the definition of spatially variant dilation/erosion and opening/closing for gray-level images using exclusively the structuring function, without resorting to complement. This sound theoretical framework allows to build morphological operat...

The emergence of new data in multidimensional function lattices is studied. A typical example is the apparition of false colours when (R,G,B) images are processed. Two lattice models are specially analysed. Firstly, one considers a mixture of total and marginal orderings where the variations of some components are governed by other ones. This const...

We study some basic morphological operators acting on the lattice of all subgraphs of a (non-weighted) graph G. To this end, we consider two dual adjunctions between the edge set and the vertex set of G. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G....

The classical colour polar-based representations (HLS, HSV, etc.) lead to brightness and saturation with non consistent properties. The requirements for a correct quantitative colour polar representation are recalled. They lead to using norms, and in particular the L1 norm. Colour images are multivariable functions, and for segmenting them one must...

electronic version (10 pp.)

The watershed segmentation is a popular tool in image processing. Starting from an initial map, the border thinning transformation produces a map whose minima constitute the catchment basins of the watershed of the initial map. An interesting feature of the transformed map (called border kernel) is to convey not only the watershed partition but als...

These last five years, mathematical morphology evolved in a few directions, an important one being a theory and practice of segmentation. This theory is based on the search for maximal partitioning of the space. The central theorem of the paper (Theorem 7) identifies segmentation with some classes of connections. Just as connections, the segmentati...

The suitability of polar representation for quantitative image processing tasks is investigated. The classical colour polar-based representations (HLS, HSV, etc.) lead to brightness and saturation with nonconsistent properties. After a short critical analysis of the gamma correction, a new polar representation using the L1 norm is proposed. It sati...

After a formal definition of segmentation as the largest partition of the space according to a criterion σ and a function f, the notion of a morphological connection is reminded. It is used as an input to a central theorem of the paper (Theorem 8), that identifies segmentation with the connections that are based on connective criteria. Just as conn...

The choice of a suitable colour space representation is still a challenging task in the processing and analysis of colour images. Starting with the recently proposed family of polar coordinate systems LSH (improving the standard HLS) which have suitable properties for quantitative image processing, the derivation of two bivariate histograms: histr;...

Let E be an arbitrary space, and δ an extensive dilation of P(E) into itself, with an adjoint erosion ε. Then, the image δ[P(E)] of P(E) by δ is a complete lattice P where the sup is the union and the inf the opening of the intersection according to δ ε. The lattice L, named viscous, is not distributive, nor complemented. Any dilation α on P(E) adm...

Atherosclerotic and medial vascular calcifications are frequent in chronic renal failure patiens and predict their increased cardiovascular mortality. Experimental models for mice have been recently developed in order to study these disorders. The aim of this paper is to present the morphological image processing algorithms developed for the semi-a...

The paper deals with the use of the various color pieces of information for segmenting color images and sequences with mathematical morphology operators. It is divided in four parts. The first one is concerning the choice of the color space suitable for morphological processing. The choice of a connection which induces a specific segmentation is di...

The processing and analysis of colour images has become an important area of study and application.

The use of 3D-polar coordinate representations of the RGB colour space is widespread, although many of these representations, such as HLS and HSV, have deficiencies rendering them unsuitable for quantitative image analysis. Three prerequisites for 3D-polar coordinate colour spaces which do not suffer from these deficiencies are suggested, and the r...

This paper deals with the use of the vector levelings for coding color images. This class of morphological connected filters suppresses details but preserves the contours of the remaining objects. If the color images are filtered by independently leveling each color component, new colors may be introduced. In order to avoid this drawback, a total o...

Automatic analysis of cartographic images is an important task for the development of intelligent geographical information systems. Both geometrical features and color are powerful cues to extract spatial semantic objects. This contribution deals with the use of the various color pieces of information for partitioning color images and for extractin...

The use of mathematical morphology in the L*a*b* colour space is discussed. Initially, a description of the characteristics of the L*a*b* space and a comparison to the HLS space are given. This is followed by a theoretical demonstration of the use of weighting functions to impose a complete order on a vector space. Various colour weighting function...

The use of mathematical morphology in the CIE L*a*b* colour space is discussed. It is possible to impose a total order on the colour vectors in this space by using a weighting function and lexicographical order. An order analogous to one by colour saturation is suggested by making use of a weighting function based on an electrostatic potential. Thi...

Current content-based image retrieval techniques can typically per-form efficient and effective searches on heterogeneous image data-bases. This contribution deals with an approach based on the inte-gration of color and texture description which is applied to a very homogeneous database: a blood image bank. The content of images is very similar and...

Oriented textures are characterised by a dominant orientation at each point of the texture, and can be summarised by images encoding these dominant orientations. Because of the angular values in these images, standard morphological operators are not suited to their treatment. We discuss the application of the morphological circular centred operator...

The HLS colour space is widely used in image analysis as it is physically intuitive. As the hue component of this space is defined on the unit circle, standard greyscale image analysis operators, specifically morphological operators, are not applicable to it. A variation of the standard morphological operators which require the choice of an origin...

The processing and analysis of colour images has become an important area of study and appli- cation. The representation of the RGB colour space in 3D-polar coordinates (hue, saturation and brightness) can sometimes simplify this task by revealing characteristics not visible in the rect- angular coordinate representation. The literature describes m...

Foreword It may be useful to add a few lines in preamble of the document below, in order to explain the context. Following G. Matheron's retirement, in December 1996, one could hear the most fanciful tales concerning the birth of Mathematical Morphology. I told G. Matheron about them, which prompted our decision to clarify this question publicly be...

This paper generalizes the notion of symmetrical neighbourhoods, which have been used to define connectivity in the case of sets, to the wider framework of complete lattices having a sup-generating family. Two versions (weak and strong) of the notion of a symmetrical dilation are introduced, and they are applied to the generation of "connected comp...

Images encoding angular information are common in image analysis. Examples include the hue band of color images, or images encoding directional texture information. Applying mathematical morphology to image data distributed on the unit circle is not immediately possible, as the unit circle is not a lattice. Three approaches to solving this problem...

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. A nontopological axiomatics has been proposed by Matheron and Serra. The present paper extends it to complete sup-generated lattices. A connection turns out to be characterized by a family of openings labelled by the sup-generators, which partition each e...

This paper presents a segmentation method, the flat zone approach, that avoids some limitations of the watershed-plus-markers method. The watershed-plus-markers approach, which is the traditional segmentation technique in mathematical morphology, has two inherent problems: (1) the possible separation of a piecewise-constant region of the input imag...

This paper presents a historical overview on Mathematical Morphology during the last forty years, and some new developments and applications in different disciplines.

In this paper, we recall the development of image analysis techniques during this last century and give briefly some basements of image analysis and mathematical morphology. The development of these techniques today and a presentation of some applications from different disciplines are also presented.

This paper investigates morphological connected filters and, in particular, the so-called filters by reconstruction. A brief background is offered on the theory of morphological filtering. Then, the concept of connectivity is introduced within the morphological framework, which makes it possible to establish connected filters as those that do not i...

This paper is an overview on the concept of morphological filtering. Starting from openings and the associated granulometries, we discuss the notion and construction of morphological filters. Then the major differences between the ‘morphological’ and the ‘linear’ approaches are highlighted. Finally, the problem of optimal morphological filtering is...

This paper consists of a tutorial overview of morphological filtering, a theory introduced in 1988 in the context of mathematical morphology. Its first section is devoted to the presentation of the lattice framework. Emphasis is put on the lattices of numerical functions in digital and continuous spaces. The basic filters, namely the openings and t...

The notion of a connection is a non metric generalisation of connec-tivity proposed by Serra. Its axiomatics lies on the idea that the union of connected components that intersect is still connected. Such an approach allows a precise definition of image (or sequence) segmentation. It yields also powerful filters (levelings), and provides more class...

Abstract

1 Beaucoup de méthodes actuelles de segmentation d'image s'appuient sur des hiérarchies de transformées et visent à construire une partition optimale, au sens d'énergie minimale, qui emprunte ses classes aux divers niveaux. Pour pouvoir être e¢ caces, ces méthodes requièrent que l'énergie véri…e une certaine condition de croissance hiérarchique. Lo...

The Present study proposes a method for multidimensional images processing, which applies particularly when circular data (e.g. directions, color) are involved. At the first step each image is transformed into a partition of the space, independently of the others. Next, these partitions are grouped into a probability map of the borders to extract,...

The classical colour polar-based representations, of type (HSV, HLS, etc.), lead to brightness and saturation with non consistent properties. In this paper they are replaced by means of norms, in particular L1 and max-min. Two examples illustrate the suitable relevance of L1 norm with regard to segmentation issues. The first issue deals with the hi...

Reimpresión en el año 1992 Incluye bibliografía e índice v. 2. Theoretical advances