A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing

New characterizations of minimum spanning trees and of saliency maps based on quasi-flat zones

We study three representations of hierarchies of partitions: dendrograms (direct representations), saliency maps, and minimum spanning trees. We provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as used in image processing and characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results form a toolkit for new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies

Connected morphological operators for binary images

This paper presents a comprehensive discussion on connected morphological operators for binary images. Introducing a connectivity on the underlying space, every image induces a partition of the space in foreground and background components. A connected operator is an operator that coarsens this partition for every input image. A connected operator is called a grain operator if it has the following `local property': the value of the output image at a given point $x$ is exclusively determined by the zone of the partition of the input image that contains $x$. Every grain operator is uniquely specified by two grain criteria, one for the foreground and one for the background components. A well-known criterion is the area criterion demanding that the area of a zone is not below a given threshold. The second part of the paper is devoted to connected filters and grain filters. It is shown that alternating sequential filters resulting from grain openings and closings are strong filters and obey a strong absorption property, two properties that do not hold in the classical non-connected case

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

Ship detection in SAR images based on Maxtree representation and graph signal processing

© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.This paper discusses an image processing architecture and tools to address the problem of ship detection in synthetic-aperture radar images. The detection strategy relies on a tree-based representation of images, here a Maxtree, and graph signal processing tools. Radiometric as well as geometric attributes are evaluated and associated with the Maxtree nodes. They form graph attribute signals which are processed with graph filters. The goal of this filtering step is to exploit the correlation existing between attribute values on neighboring tree nodes. Considering that trees are specific graphs where the connectivity toward ancestors and descendants may have a different meaning, we analyze several linear, nonlinear, and morphological filtering strategies. Beside graph filters, two new filtering notions emerge from this analysis: tree and branch filters. Finally, we discuss a ship detection architecture that involves graph signal filters and machine learning tools. This architecture demonstrates the interest of applying graph signal processing tools on the tree-based representation of images and of going beyond classical graph filters. The resulting approach significantly outperforms state-of-the-art algorithms. Finally, a MATLAB toolbox allowing users to experiment with the tools discussed in this paper on Maxtree or Mintree has been created and made public.Peer ReviewedPostprint (author's final draft