Laurent Najman

A 4D counter-example showing that DWCness does not imply CWCness in $n$-D

By Nicolas Boutry, Rocio Gonzalez-Diaz, Laurent Najman, Thierry Géraud

2020-07-21

In Combinatorial image analysis: Proceedings of the 20th international workshop, IWCIA 2020, novi sad, serbia, july 16–18, 2020

Abstract In this paper, we prove that the two flavours of well-composedness called Continuous Well-Composedness (shortly CWCness), stating that the boundary of the continuous analog of a discrete set is a manifold, and Digital Well-Composedness (shortly DWCness), stating that a discrete set does not contain any critical configuration, are not equivalent in dimension 4. To prove this, we exhibit the example of a configuration of 8 tesseracts (4D cubes) sharing a common corner (vertex), which is DWC but not CWC.

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An equivalence relation between morphological dynamics and persistent homology in 1D

By Nicolas Boutry, Thierry Géraud, Laurent Najman

2019-03-13

In Mathematical morphology and its application to signal and image processing – proceedings of the 14th international symposium on mathematical morphology (ISMM)

Abstract We state in this paper a strong relation existing between Mathematical Morphology and Discrete Morse Theory when we work with 1D Morse functions. Specifically, in Mathematical Morphology, a classic way to extract robust markers for segmentation purposes, is to use the dynamics. On the other hand, in Discrete Morse Theory, a well-known tool to simplify the Morse-Smale complexes representing the topological information of a Morse function is the persistence. We show that pairing by persistence is equivalent to pairing by dynamics.

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A tutorial on well-composedness

By Nicolas Boutry, Thierry Géraud, Laurent Najman

2017-10-12

In Journal of Mathematical Imaging and Vision

Abstract Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called well-composed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well-defined, and so on.

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Caractérisation des zones de mouvement périodiques pour applications bio-médicales

By Élodie Puybareau, Hugues Talbot, Laurent Najman

2017-06-28

In Actes du 26e colloque GRETSI

Abstract De nombreuses applications biomedicales impliquent l’analyse de séquences pour la caractérisation du mouvement. Dans cet article, nous considerons des séquences 2D+t où un mouvement particulier (par exemple un flux sanguin) est associé à une zone spécifique de l’image 2D (par exemple une artère). Mais de nombreux mouvements peuvent co-exister dans les séquences (par exemple, il peut y avoir plusieurs vaisseaux sanguins presents, chacun avec leur flux spécifique). La caractérisation de ce type de mouvement implique d’abord de trouver les zones où le mouvement est présent, puis d’analyser ces mouvements : vitesse, régularité, fréquence, etc.

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Well-composedness in Alexandrov spaces implies digital well-composedness in $Z^n$

By Nicolas Boutry, Laurent Najman, Thierry Géraud

2017-06-01

In Discrete geometry for computer imagery – proceedings of the 20th IAPR international conference on discrete geometry for computer imagery (DGCI)

Abstract In digital topology, it is well-known that, in 2D and in 3D, a digital set $X \subseteq Z^n$ is digitally well-composed (DWC), i.e., does not contain any critical configuration, if its immersion in the Khalimsky grids $H^n$ is well-composed in the sense of Alexandrov (AWC), i.e., its boundary is a disjoint union of discrete $(n-1)$-surfaces. We show that this is still true in $n$-D, $n \geq 2$, which is of prime importance since today 4D signals are more and more frequent.

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Periodic area-of-motion characterization for bio-medical applications

By Élodie Puybareau, Hugues Talbot, Laurent Najman

2017-02-20

In Proceedings of the IEEE international symposium on bio-medical imaging (ISBI)

Abstract Many bio-medical applications involve the analysis of sequences for motion characterization. In this article, we consider 2D+t sequences where a particular motion (e.g. a blood flow) is associated with a specific area of the 2D image (e.g. an artery) but multiple motions may exist simultaneously in the same sequences (e.g. there may be several blood vessels present, each with their specific flow). The characterization of this type of motion typically involves first finding the areas where motion is present, followed by an analysis of these motions: speed, regularity, frequency, etc.

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Hierarchical image simplification and segmentation based on Mumford-Shah-salient level line selection

By Yongchao Xu, Thierry Géraud, Laurent Najman

2016-05-20

In Pattern Recognition Letters

Abstract Hierarchies, such as the tree of shapes, are popular representations for image simplification and segmentation thanks to their multiscale structures. Selecting meaningful level lines (boundaries of shapes) yields to simplify image while preserving intact salient structures. Many image simplification and segmentation methods are driven by the optimization of an energy functional, for instance the celebrated Mumford-Shah functional. In this paper, we propose an efficient approach to hierarchical image simplification and segmentation based on the minimization of the piecewise-constant Mumford-Shah functional.

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Hierarchical segmentation using tree-based shape spaces

By Yongchao Xu, Edwin Carlinet, Thierry Géraud, Laurent Najman

2016-04-11

In IEEE Transactions on Pattern Analysis and Machine Intelligence

Abstract Current trends in image segmentation are to compute a hierarchy of image segmentations from fine to coarse. A classical approach to obtain a single meaningful image partition from a given hierarchy is to cut it in an optimal way, following the seminal approach of the scale-set theory. While interesting in many cases, the resulting segmentation, being a non-horizontal cut, is limited by the structure of the hierarchy. In this paper, we propose a novel approach that acts by transforming an input hierarchy into a new saliency map.

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Connected filtering on tree-based shape-spaces

By Yongchao Xu, Thierry Géraud, Laurent Najman

2015-06-05

In IEEE Transactions on Pattern Analysis and Machine Intelligence

Abstract Connected 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.

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How to make $n$D images well-composed without interpolation

By Nicolas Boutry, Thierry Géraud, Laurent Najman

2015-05-14

In Proceedings of the IEEE international conference on image processing (ICIP)

Abstract Latecki et al. have introduced the notion of well-composed images, i.e., a class of images free from the connectivities paradox of discrete topology. Unfortunately natural and synthetic images are not a priori well-composed, usually leading to topological issues. Making any $n$D image well-composed is interesting because, afterwards, the classical connectivities of components are equivalent, the component boundaries satisfy the Jordan separation theorem, and so on. In this paper, we propose an algorithm able to make $n$D images well-composed without any interpolation.

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