Abstract. The goal of this paper is to provide linear or quasi-linear algorithms for producing some of the various trees used in mathemetical morphology, in particular the trees corresponding to hierarchies of watershed cuts and hierarchies of constrained connectivity. A specific binary tree, corresponding to an ordered version of the edges of the minimum spanning tree, is the key structure in this study, and is computed thanks to variations around Kruskal algorithm for minimum spanning tree.

Abstract. In edge-weighted graphs, we provide a unified presentation of a family of popular morphological hierarchies such as component trees, quasi flat zones, binary partition trees, and hierarchical watersheds. For any hierarchy of this family, we show if (and how) it can be obtained from any other element of the family. In this sense, the main contribution of this paper is the study of all constructive links between these hierarchies.

The morphological Hit-or-Miss Transform (HMT) is a powerful tool for digital image analysis. Its recent extensions to grey level images have proven its ability to solve various template matching problems. In this paper we explore the capacity of various existing approaches to work in very noisy environments and discuss the generic methods used to improve their robustness to noise. We also propose a new formulation for a fuzzy morphological HMT which has been especially designed to deal with very noisy images. Our approach is validated through a pattern matching problem in astronomical images that consists of detecting very faint objects: low surface brightness galaxies. Despite their influence on the galactic evolution model, these objects remain mostly misunderstood by the astronomers. Due to their low signal to noise ratio, there is no automatic and reliable detection method yet. In this paper we introduce such a method based on the proposed hit-or-miss operator. The complete process is described starting from the building of a set of patterns until the reconstruction of a suitable map of detected objects. Implementation, running cost and optimisations are discussed. Outcomes have been examined by astronomers and compared to previous works. We have observed promising results in this difficult context for which Mathemat-

Abstract-This article aims to understand the practical features of hierarchies of morphological segmentations, namely the quasi-flat zones hierarchy and watershed hierarchies, and to evaluate their potential in the context of natural image analysis. We propose a novel evaluation framework for hierarchies of partitions designed to capture various aspects of those representations: precision of their regions and contours, possibility to extract high quality horizontal cuts and optimal non-horizontal cuts for image segmentation, and ease of finding a set of regions representing a semantic object. This framework is used to assess and to optimize hierarchies with respect to the possible pre-and post-processing steps. We show that, used in conjunction with a state-of-the-art contour detector, watershed hierarchies are competitive with complex state-of-the-art methods for hierarchy construction. In particular, the proposed framework allows us to identify a watershed hierarchy based on a novel extinction value, the number of parent nodes, that outperforms the other hierarchies of morphological segmentations. This coupled with the fact that watershed hierarchies satisfy clear global optimality properties and can be efficiently computed on large data, make them valuable candidates for various computer vision tasks.Index Terms-mathematical morphology, hierarchy of partitions, watershed segmentation, image analysis.

Abstract-Connected operators provide well-established solutions for digital image processing, typically in conjunction with hierarchical schemes. In graph-based frameworks, such operators basically rely on symmetric adjacency relations between pixels. In this article, we introduce a notion of directed connected operators for hierarchical image processing, by also considering non-symmetric adjacency relations. The induced image representation models are no longer partition hierarchies (i.e., trees), but directed acyclic graphs that generalize standard morphological tree structures such as component trees, binary partition trees or hierarchical watersheds. We describe how to efficiently build and handle these richer data structures, and we illustrate the versatility of the proposed framework in image filtering and image segmentation.

Connections in image processing are an important notion that describes 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 hyperconnection (h-connection) is a very promising notion. This paper addresses two major issues of this theory. First, we propose a new axiomatic that ensures that every h-connection generates decompositions that are consistent for image processing and, more precisely, for the design of h-connected filters. Second, we develop a general framework to represent the decomposition of an image into h-connections as a tree that corresponds to the generalization of the connected component tree. Such trees are indeed an efficient and intuitive way to design attribute filters or to perform detection tasks based on qualitative or quantitative attributes. These theoretical developments are applied to a particular fuzzy h-connection, and we test this new framework on several classical applications in image processing, i.e., segmentation, connected filtering, and document image binarization. The experiments confirm the suitability of the proposed approach: It is robust to noise, and it provides an efficient framework to design selective filters.

Abstract.We propose a quantitative evaluation of morphological hierarchies (quasi-flat zones, constraint connectivity, watersheds, observation scale) in a novel framework based on the marked segmentation problem. We created a set of automatically generated markers for the one object image datasets of Grabcut and Weizmann. In order to evaluate the hierarchies, we applied the same segmentation strategy by combining several parameters and markers. Our results, which shows important differences among the considered hierarchies, give clues to understand the behaviour of each method in order to choose the best one for a given application. The code and the marker datasets are available online.

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