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 optimization, in the lattice of cuts in the hierarchy.

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 build morphological operators whose structuring elements can locally adapt their shape and orientation across the dominant direction of the structures in the image. The shape and orientation of the structuring element at each pixel are extracted from the image under study: the orientation is given by means of a diffusion process of the average square gradient field, which regularizes and extends the orientation information from the edges of the objects to the homogeneous areas of the image; and the shape of the orientated structuring elements can be linear or it can be given by the distance to relevant edges of the objects. The proposed filters are used on binary and gray-level images for enhancement of anisotropic features such as coherent, flow-like structures. Results of spatially-variant erosions/dilations and openings/closings-based filters prove the validity of this theoretical sound and novel approach.

The aim of the study is to structure multivariate data in cylindrical polar representation, in order to segment them by composing the Rnluminance and hue partitions according to that of the saturation. Remote sensing applications are in mind. This goal requires to equip with a total ordering the unite sphere in Rn, where the hue is distributed. The proposed orders, and the subsequent lattices, do not depend on absolute coordinates, but are adapted either to the whole image under study, or to the neighbourhoods necessary for processing the image. They are obtained by choosing a few maxima on the distribution of the spherical coordinates of the image, and by taking as new variables the infima of the distances to these various maxima. The case of four dimensions, usual when dealing with the principal components in remote sensing, is treated in detail. A comparison of the results in various dimensions is performed on a multispectral aerial image of the university of Pavia.

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 also numeric information relative to the initial map. In this paper, we provide the space of all border kernels with a semi lattice and propose morphological operations (relative to this lattice) which allow for merging border kernels and building hierarchies of watersheds based in particular on connected filters.

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 segmentations can then be regrouped by suprema and infima, which permits serial and parallel combinations. The segmentation classes turn out to be independent of their location in the measuring field, assuming that a convenient neighborhood is experimentally accessible.

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 discussed. The authors then present the color segmentation approach which is based on a nonparametric pyramid of watersheds, with a comparative study of different color gradients. Another multiscale color segmentation algorithm is then introduced, relying on the merging of chromatic-achromatic partitions ordered by the saturation component.

Firstly, the paper proposes an axiomatic definition for the notion of "segmentation" in image processing, which is based on the idea of a maximal partition. Then a key theorem links segmentation with connection, on the one hand, and with connective criteria on the other one. A series of lattice properties are then developed. In a last part, two examples of segmentations are proposed.

MOTIVATION: DNA microarrays are an experimental technology which consists in arrays of thousands of discrete DNA sequences that are printed on glass microscope slides. Image analysis is an important aspect of microarray experiments. The aim of this step is to reduce an image of spots into a table with a measure of the intensity for each spot. Efficient, accurate and automatic analysis of DNA spot images is essential in order to use this technology in laboratory routines. RESULTS: We present an automatic non-supervised set of algorithms for a fast and accurate spot data extraction from DNA microarrays using morphological operators which are robust to both intensity variation and artefacts. The approach can be summarised as follows. Initially, a gridding algorithm yields the automatic segmentation of the microarray image into spot quadrants which are later individually analysed. Then the analysis of the spot quadrant images is achieved in five steps. First, a pre-quantification, the spot size distribution law is calculated. Second, the background noise extraction is performed using a morphological filtering by area. Third, an orthogonal grid provides the first approach to the spot locus. Fourth, the spot segmentation or spot boundaries definition is carried out using the watershed transformation. And fifth, the outline of detected spots allows the signal quantification or spot intensities extraction; in this respect, a noise model has been investigated. The performance of the algorithm has been compared with two packages: ScanAlyze and Genepix, showing its robustness and precision.

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 are discussed. In addition, some lexicographical vector orders, which allow mathematical morphology to be used in the HLS space, are presented. Included in these is a new saturationweighted hue order, which takes hue and saturation into account simultaneously. 1

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 are presented. First, difference-based operators are studied (e.g., gradient, top-hat). Second, a definition of grouped circular data is suggested, and "pseudo" morphological operators, which operate only on grouped data, are introduced. Finally, processing using pixel labeling is presented, leading to the development of a cyclic opening operator. Applications for treating the hue band of color images and for finding perturbations in wood texture are given.

This paper deals with the roles of assumption, mathematical models, and experimental control in pattern recognition. In order to base the discussion on actual examples, three studies are first described. The first one presents filtering of bird song via their sonograms. The second and the third ones deal with morphological operators on the unit circle, which are successively applied to hue gradient and top-hats in colour images, and to directional irregularities in wood textures. The discussion holds on the necessity to bring implicit assumptions to light, to evaluate the consistency of the mathematical formalism with respect to these assumptions, and finally to control experimentally the domain of pertinence of the proposed approach.

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. An algebraic definition, called connection, has been proposed by Serra to extend the notion of connectivity to complete sup-generated lattices. A connectio

This correspondence deals with the notion of connected operators. Starting from the definition for operator acting on sets, it is shown how to extend it to operators acting on function. Typically, a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions. It is shown that from any connected operator acting on sets, one can construct a connected operator for functions (however, it is not the unique way of generating connected operators for functions). Moreover, the concept of pyramid is introduced in a formal way. It is shown that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Filters by reconstruction are defined and their main properties are presented. Finally, some examples of application of connected operators and use of flat zones are described.

This paper presents a method to compute a partition intended to be used in a region-based coding system where the number of regions is a design constraint. A new flat zone approach is introduced that (a) selects a subset of those flat zones (regions) that are present in the output of a multiscale connected filter, and that (b) assigns the rest of the regions to one among those selected by using flat zone merging procedures. The filters used are connected sequential alternating filters and a new connected filter recently presented. These are flat operators, invariant under anamorphosis, that act on a function by extending its flat zones exclusively. A new theoretical proposition is stated. Our segmentation approach behaves like a connected operator and it preserves the flat zone inclusion property between the input and the output partition. Moreover, regions computed at the highest resolution level can be imposed in successive coarser levels. Among the flat zones which are present at the output of a connected filter, the most significant ones with respect to several criteria are distinguished. This region selection is based solely on an ordering of the regions. Absolute thresholding is avoided, and the proposed approach computes with linear dynamic changes.

In the lattice of the operators acting on a complete lattice, a link ? is the inf of an increasing and a decreasing mappings. The links generate a complete lattice, and are characterized, in case of complete distributivity, by the implication A ? X ? B ? ? (A) ^ ?(B) ? ?(X). They allow minimal representation of a large class of mappings by means of hit-or-miss. Special attention is devoted to the residuals, i.e. the set difference between the identity and increasing mappings. These particular links allow to provide a common form to skeletons, ultimate erosions and conditional bisectors.

Mathematical Morphology provides an alternative to classical filtering methods, such as convolution. To replace linearity by the increasing condition leads to the class of the M-filters. Although they are not invertible, they remove very accurately specific features from the images (peaks but not holes for ex.). The particular case of the opening is discussed and illustrated by an example drawn from astronomy.

A logical framework for digital 3-D treatment is given steming from the point of view of the set theory. Two different structures can equip the digital space, namely that of a module and that of a grid. The notions of size distributions, filterings, erosions, openings, thinnings, convexity, derive from the first structure, those of rotations, homotopy and connectivity from the second one. To digitalize, it is not enough merely to replace a Euclidean figure by a mosaic of pixels. We also have to replace Euclidean translations, rotations, convex hulls, connected particles, etc. by analogous versions on a grid of points. A number of umbrella notions result. The first is the notion of a module, classical in linear algebra, which governs increasing transformations, projections and convexity analyses. Then there is cur notion of a grid, which is the key to all the rotation and connectivity problems. Finally, one can also equip the grid with a metric, but we shall not do it here.

In Image processing, it is often implicitely admitted that an image is known when it has been segmented in various textures, in accordance with the judgement of the human eye. Two counter examples are presented, in biology and in petrology, showing that image analysis may also consist in synthetizing informations disseminated over a large number of pictures, and in predicting the relationships between physical properties and textures. Then, a formal definition of image understanding is given, which yields a particular approach based on structuring elements, and on a few logical properties. An illustration of the approach is provided by the analysis of a boolean structure.