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 the equivalence between hierarchical segmentations and ultrametric watersheds

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.Comment: 19 pages, double-colum

Transformations topologiques discrètes

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Convolutional Nets and Watershed Cuts for Real-Time Semantic Labeling of RGBD Videos

International audienceThis work addresses multi-class segmentation of indoor scenes with RGB-D inputs. While this area of research has gained much attention recently, most works still rely on handcrafted features. In contrast, we apply a multiscale convolutional network to learn features directly from the images and the depth information. Using a frame by frame labeling, we obtain nearly state-of-the-art performance on the NYU-v2 depth dataset with an accuracy of 64.5%. We then show that the labeling can be further improved by exploiting the temporal consistency in the video sequence of the scene. To that goal, we present a method producing temporally consistent superpixels from a streaming video. Among the di erent methods producing superpixel segmentations of an image, the graph-based approach of Felzenszwalb and Huttenlocher is broadly employed. One of its interesting properties is that the regions are computed in a greedy manner in quasi-linear time by using a minimum spanning tree. In a framework exploiting minimum spanning trees all along, we propose an efficient video segmentation approach that computes temporally consistent pixels in a causal manner, filling the need for causal and real-time applications. We illustrate the labeling of indoor scenes in video sequences that could be processed in real-time using appropriate hardware such as an FPGA

Weighted fusion graphs: merging properties and watersheds

International audienc

Minimal simple pairs in the cubic grid

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs

Topological properties of thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. In contrast to the situation in Z^n , n>=3, it was proved in the 80s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds

Discrete euclidean skeletons in increased resolution

Orientadores: Roberto de Alencar Lotufo, Michel CouprieTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: A extração de esqueletos Euclidianos é uma tema de grande importância na área de processamento de imagens e tem sido discutido pela comunidade científica já há mais de 20 anos. Hoje é consenso que os esqueletos Euclidianos devem ter as seguintes características: ï¬?nos, centrados, homotópicos e reversíveis, i.e., suficientes para a reconstrução do objeto original. Neste trabalho, introduzimos o Eixo Mediano Euclidiano Exato em Resolução Aumentada -HMA, com o objetivo de obter um eixo mediano mais ï¬?no do que o obtido pela definição clássica. Combinando o HMA com um eï¬?ciente algoritmo de afinamento paralelo homotópico, propomos um esqueleto Euclidiano que é centrado, homotópico, reversível e mais ï¬?no que os já existentes na literatura. O esqueleto proposto tem a particularidade adicional de ser único e independente de decisões arbitrárias. São dados algoritmos e provas, assim como exemplos de aplicações dos esqueletos propostos em imagens reais, mostrando as vantagens da proposta. O texto inclui também uma revisão bibliográfica sobre algoritmos de transformada de distância, eixo mediano e esqueletos homotópicosAbstract: The extraction of Euclidean skeletons is a subject of great importance in the domain of image processing and it has been discussed by the scientiï¬?c community since more than 20 years.Today it is a consensus that Euclidean skeletons should present the following characteristics: thin, centered, homotopic and reversible, i.e., sufï¬?cient for the reconstruction of the original object. In this work, we introduce the Exact Euclidean Medial Axis in Higher Resolution -HMA, with the objective of obtaining a medial axis which is thinner than the one obtained by the classical medial axis deï¬?nition. By combining the HMA with an efï¬?cient parallel homotopic thinning algorithm we propose an Euclidean skeleton which is centered, homotopic, reversible and thinner than the existing similars in the literature. The proposed skeleton has the additional particularity of being unique and independent of arbitrary choices. Algorithms and proofs are given, as well as applicative examples of the proposed skeletons in real images, showing the advantages of the proposal. The text also includes an overview on algorithms for the Euclidean distance transform algorithms, the medial axis extraction, as well as homotopic skeletonsDoutoradoEngenharia de ComputaçãoDoutor em Engenharia Elétric

A unified topological framework for digital imaging

International audienceIn this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on Z^n and equipped with a standard pair of adjacencies) as an image defined in the space of cubical complexes (F^n). In particular, it is shown that all the standard pairs of adjacencies in Z^n can then be correctly modelled in F^n. Moreover, it is established that the digital fundamental group of a digital image in Z^n is isomorphic to the fundamental group of its corresponding image in F^n, thus proving the topological correctness of the proposed approach. From these results, it becomes possible to establish links between topology-oriented methods developed either in classical digital spaces (Z^n) or cubical complexes (F^n)