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

From hyperconnections to hypercomponent tree: Application to document image binarization

International audienceIn this paper, we propose an extension of the component tree based on at zones to hyperconnections (h-connections). The tree is dened by a special order on the h-connection and allows non at nodes. We apply this method to a particular fuzzy h-connection and we give an ecient algorithm to transform the component tree into the new fuzzy h-component tree. Finally, we propose a method to binarize document images based on the h-component tree and we evaluate it on the DIBCO 2009 benchmarking dataset: our novel method places rst or second according to the dierent evaluation measures. Hierarchical and tree based representations have become very topical in image processing. In particular, the component tree (or Max-Tree) has been the subject of many studies and practical works. Nevertheless, the component tree inherits the weaknesses of the at zone approach, namely its high sensitivity to noise, gradients and diculty to manage disconnected objects. Even if some solutions have been proposed to preserve the component tree [5, 4], it seems that a more general framework for grayscale component tree [1] based on non at zones becomes necessary. In this article, we propose a method to design grayscale component tree based on h-connections. The h-connection theory has been proposed in [7] and developed in [1, 3, 4, 8, 9]. It provides a general denition of the notion of connected component in arbitrary lattices. In Sec. 2, we present the h-connection theory and a method to generate a related hierarchical representation. This method is applied to a fuzzy h-connection in Sec. 3 where an algorithm is given to transform a Max-Tree into the new grayscale component tree. In Sec. 4, we illustrate the interest of this tree with an application on document image binarization. 2 H-component Tree This section presents the basis of the h-connection theory [7, 1] and gives a denition of the h-component tree. The construction of the tree is based on the z-zones [1] of the h-connection, together with a special partial ordering. Z-zones are particular regions where all points generate the same set of hyperconnected (h-connected) components and the entire image can be divided into such zones. Under a given condition, the Hasse diagram obtained in this way is acyclic and provides a tree representation. Let L be a complete lattice furnished with the partial ordering ≤, the inmum , the supremum. The least element of L is denoted by ⊥ = L. We assume the existence of a sup-generatin

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

Toward a new axiomatic for hyper-connections

International audienceWe 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 reappear in the filtered image

Generalized Morphology using Sponges

Mathematical morphology has traditionally been grounded in lattice theory. For non-scalar data lattices often prove too restrictive, however. In this paper we present a more general alternative, sponges, that still allows useful definitions of various properties and concepts from morphological theory. It turns out that some of the existing work on “pseudo-morphology” for non-scalar data can in fact be considered “proper” mathematical morphology in this new framework, while other work cannot, and that this correlates with how useful/intuitive some of the resulting operators are

Connected image processing with multivariate attributes: an unsupervised Markovian classification approach

International audienceThis article presents a new approach for constructing connected operators for image processing and analysis. It relies on a hierarchical Markovian unsupervised algorithm in order to classify the nodes of the traditional Max-Tree. This approach enables to naturally handle multivariate attributes in a robust non-local way. The technique is demonstrated on several image analysis tasks: filtering, segmentation, and source detection, on astronomical and biomedical images. The obtained results show that the method is competitive despite its general formulation. This article provides also a new insight in the field of hierarchical Markovian image processing showing that morphological trees can advantageously replace traditional quadtrees