Inf-structuring functions and self-dual marked flattenings in bi-Heyting algebra

International audienceThis paper introduces a generalization of self-dual marked flattenings defined in the lattice of mappings. This definition provides a way to associate a self-dual operator to every mapping that decomposes an element into sub-elements (i.e. gives a cover). Contrary to classical flattenings whose definition relies on the complemented structure of the powerset lattices, our approach uses the pseudo relative complement and supplement of the bi-Heyting algebra and a new notion of \textit{inf-structuring functions} that provides a very general way to structure the space. We show that using an inf-structuring function based on connections allows to recover the original definition of marked flattenings and we provide, as an example, a simple inf-structuring function whose derived self-dual operator better preserves contrasts and does not introduce new pixel values

Connected Filtering on Tree-Based Shape-Spaces

International audienceConnected filters are well-known for their good contour preservation property. A popular implementation strategy relies on tree-based image representations: for example, one can compute an attribute characterizing the connected component represented by each node of the tree and keep only the nodes for which the attribute is sufficiently high. This operation can be seen as a thresholding of the tree, seen as a graph whose nodes are weighted by the attribute. Rather than being satisfied with a mere thresholding, we propose to expand on this idea, and to apply connected filters on this latest graph. Consequently, the filtering is performed not in the space of the image, but in the space of shapes built from the image. Such a processing of shape-space filtering is a generalization of the existing tree-based connected operators. Indeed, the framework includes the classical existing connected operators by attributes. It also allows us to propose a class of novel connected operators from the leveling family, based on non-increasing attributes. Finally, we also propose a new class of connected operators that we call morphological shapings. Some illustrations and quantitative evaluations demonstrate the usefulness and robustness of the proposed shape-space filters

Inf-structuring Functions: A Unifying Theory of Connections and Connected Operators

International audienceDuring the last decade, several theories have been proposed in order to extend the notion of set connections in mathematical morphology. These new theories were obtained by generalizing the definition to wider spaces (namely complete lattices) and/or by relaxing some hypothesis. Nevertheless, the links among those different theories are not always well understood, and this work aims at defining a unifying theoretical framework. The adopted approach relies on the notion of inf-structuring function which is simply a mapping that associates a set of sub-elements to each element of the space. The developed theory focuses on the properties of the decompositions given by an inf-structuring function rather than in trying to characterize the properties of the set of connected elements as a whole. We establish several sets of inf-structuring function properties that enable to recover the existing notions of connections, hyperconnections, and attribute space connections. Moreover, we also study the case of grey-scale connected operators that are obtained by stacking set connected operators and we show that they can be obtained using specific inf-structuring functions. This work allows us to better understand the existing theories, it facilitates the reuse of existing results among the different theories and it gives a better view on the unexplored areas of the connection theories

Flooding edge or node weighted graphs

Reconstruction closings have all properties of a physical flooding of a topographic surface. They are precious for simplifying gradient images or, filling unwanted catchment basins, on which a subsequent watershed transform extracts the targeted objects. Flooding a topographic surface may be modeled as flooding a node weighted graph (TG), with unweighted edges, the node weights representing the ground level. The progression of a flooding may also be modeled on the region adjacency graph (RAG) of a topographic surface. On a RAG each node represents a catchment basin and edges connect neighboring nodes. The edges are weighted by the altitude of the pass point between both adjacent regions. The graph is flooded from sources placed at the marker positions and each node is assigned to the source by which it has been flooded. The level of the flood is represented on the nodes on each type of graphs. The same flooding may thus be modeled on a TG or on a RAG. We characterize all valid floodings on both types of graphs, as they should verify the laws of hydrostatics. We then show that each flooding of a node weighted graph also is a flooding of an edge weighted graph with appropriate edge weights. The highest flooding under a ceiling function may be interpreted as the shortest distance to the root for the ultrametric flooding distance in an augmented graph. The ultrametric distance between two nodes is the minimal altitude of a flooding for which both nodes are flooded. This remark permits to flood edge or node weighted graphs by using shortest path algorithms. It appears that the collection of all lakes of a RAG has the structure of a dendrogram, on which the highest flooding under a ceiling function may be rapidly found.Comment: 165 pages, 21 figure

Un arbre des formes pour les images multivariées

Nowadays, the demand for multi-scale and region-based analysis in many computer vision and pattern recognition applications is obvious. No one would consider a pixel-based approach as a good candidate to solve such problems. To meet this need, the Mathematical Morphology (MM) framework has supplied region-based hierarchical representations of images such as the Tree of Shapes (ToS). The ToS represents the image in terms of a tree of the inclusion of its level-lines. The ToS is thus self-dual and contrast-change invariant which make it well-adapted for high-level image processing. Yet, it is only defined on grayscale images and most attempts to extend it on multivariate images - e.g. by imposing an “arbitrary” total ordering - are not satisfactory. In this dissertation, we present the Multivariate Tree of Shapes (MToS) as a novel approach to extend the grayscale ToS on multivariate images. This representation is a mix of the ToS's computed marginally on each channel of the image; it aims at merging the marginal shapes in a “sensible” way by preserving the maximum number of inclusion. The method proposed has theoretical foundations expressing the ToS in terms of a topographic map of the curvilinear total variation computed from the image border; which has allowed its extension on multivariate data. In addition, the MToS features similar properties as the grayscale ToS, the most important one being its invariance to any marginal change of contrast and any marginal inversion of contrast (a somewhat “self-duality” in the multidimensional case). As the need for efficient image processing techniques is obvious regarding the larger and larger amount of data to process, we propose an efficient algorithm that can be build the MToS in quasi-linear time w.r.t. the number of pixels and quadraticw.r.t. the number of channels. We also propose tree-based processing algorithms to demonstrate in practice, that the MToS is a versatile, easy-to-use, and efficient structure. Eventually, to validate the soundness of our approach, we propose some experiments testing the robustness of the structure to non-relevant components (e.g. with noise or with low dynamics) and we show that such defaults do not affect the overall structure of the MToS. In addition, we propose many real-case applications using the MToS. Many of them are just a slight modification of methods employing the “regular” ToS and adapted to our new structure. For example, we successfully use the MToS for image filtering, image simplification, image segmentation, image classification and object detection. From these applications, we show that the MToS generally outperforms its ToS-based counterpart, demonstrating the potential of our approachDe nombreuses applications issues de la vision par ordinateur et de la reconnaissance des formes requièrent une analyse de l'image multi-échelle basée sur ses régions. De nos jours, personne ne considérerait une approche orientée « pixel » comme une solution viable pour traiter ce genre de problèmes. Pour répondre à cette demande, la Morphologie Mathématique a fourni des représentations hiérarchiques des régions de l'image telles que l'Arbre des Formes (AdF). L'AdF représente l'image par un arbre d'inclusion de ses lignes de niveaux. L'AdF est ainsi auto-dual et invariant au changement de contraste, ce qui fait de lui une structure bien adaptée aux traitements d'images de haut niveau. Néanmoins, il est seulement défini aux images en niveaux de gris et la plupart des tentatives d'extension aux images multivariées (e.g. en imposant un ordre total «arbitraire ») ne sont pas satisfaisantes. Dans ce manuscrit, nous présentons une nouvelle approche pour étendre l'AdF scalaire aux images multivariées : l'Arbre des Formes Multivarié (AdFM). Cette représentation est une « fusion » des AdFs calculés marginalement sur chaque composante de l'image. On vise à fusionner les formes marginales de manière « sensée » en préservant un nombre maximal d'inclusion. La méthode proposée a des fondements théoriques qui consistent en l'expression de l'AdF par une carte topographique de la variation totale curvilinéaire depuis la bordure de l'image. C'est cette reformulation qui a permis l'extension de l'AdF aux données multivariées. De plus, l'AdFM partage des propriétés similaires avec l'AdF scalaire ; la plus importante étant son invariance à tout changement ou inversion de contraste marginal (une sorte d'auto-dualité dans le cas multidimensionnel). Puisqu'il est évident que, vis-à-vis du nombre sans cesse croissant de données à traiter, nous ayons besoin de techniques rapides de traitement d'images, nous proposons un algorithme efficace qui permet de construire l'AdF en temps quasi-linéaire vis-à-vis du nombre de pixels et quadratique vis-à-vis du nombre de composantes. Nous proposons également des algorithmes permettant de manipuler l'arbre, montrant ainsi que, en pratique, l'AdFM est une structure facile à manipuler, polyvalente, et efficace. Finalement, pour valider la pertinence de notre approche, nous proposons quelques expériences testant la robustesse de notre structure aux composantes non-pertinentes (e.g. avec du bruit ou à faible dynamique) et nous montrons que ces défauts n'affectent pas la structure globale de l'AdFM. De plus, nous proposons des applications concrètes utilisant l'AdFM. Certaines sont juste des modifications mineures aux méthodes employant d'ores et déjà l'AdF scalaire mais adaptées à notre nouvelle structure. Par exemple, nous utilisons l'AdFM à des fins de filtrage, segmentation, classification et de détection d'objet. De ces applications, nous montrons ainsi que les méthodes basées sur l'AdFM surpassent généralement leur analogue basé sur l'AdF, démontrant ainsi le potentiel de notre approch

Adjacency stable connected operators and set levelings

This paper studies morphological connected operators. Particularly, it focuses on an adjacency constraint, as well as on the so-called set levelings. Two important findings are reported in this work. First, the relationships between the so-called adjacency stable operators and set levelings are investigated, and an equivalence is established. This is an important result about how these concepts have been chronologically introduced, and it permits to apply some properties to the related operator class. Second, the implications and limits of a property about expressing certain connected operators as a sequential composition of an opening and a closing (and vice-versa) based on markers are discussed. Then, a commutative property for attribute alternated filters is presented.Pages: 215-22

1: Adjacency Stable Connected Operators and Set Levelings

Review of a work submitted to the International Symposium on Mathematical Morphology, 8 (ISMM)