Laurent Najman

Topological properties of the first non-local digitally well-composed interpolation on $n$-D cubical grids

By Nicolas Boutry, Laurent Najman, Thierry Géraud

2020-09-03

In Journal of Mathematical Imaging and Vision

Abstract

In discrete topology, we like digitally well-composed (shortly DWC) interpolations because they remove pinches in cubical images. Usual well-composed interpolations are local and sometimes self-dual (they treat in a same way dark and bright components in the image). In our case, we are particularly interested in $n$-D self-dual DWC interpolations to obtain a purely self-dual tree of shapes. However, it has been proved that we cannot have an $n$-D interpolation which is at the same time local, self-dual, and well-composed. By removing the locality constraint, we have obtained an $n$-D interpolation with many properties in practice: it is self-dual, DWC, and in-between (this last property means that it preserves the contours). Since we did not published the proofs of these results before, we propose to provide in a first time the proofs of the two last properties here (DWCness and in-betweeness) and a sketch of the proof of self-duality (the complete proof of self-duality requires more material and will come later). Some theoretical and practical results are given.

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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)

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. This result is surprising since we know that CWCness and DWCness are equivalent in 2D and 3D. To reach our goal, we use local homology.

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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. Furthermore, self-duality and injectivity of these pairings are proved.

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How to make $n$-D plain maps Alexandrov-well-composed in a self-dual way

By Nicolas Boutry, Thierry Géraud, Laurent Najman

2019-02-04

In Journal of Mathematical Imaging and Vision

Abstract

In 2013, Najman and Géraud proved that by working on a well-composed discrete representation of a gray-level image, we can compute what is called its tree of shapes, a hierarchical representation of the shapes in this image. This way, we can proceed to morphological filtering and to image segmentation. However, the authors did not provide such a representation for the non-cubical case. We propose in this paper a way to compute a well-composed representation of any gray-level image defined on a discrete surface, which is a more general framework than the usual cubical grid. Furthermore, the proposed representation is self-dual in the sense that it treats bright and dark components in the image the same way. This paper can be seen as an extension to gray-level images of the works of Daragon et al. on discrete surfaces.

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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. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state-of-the-art of well-composedness, summarizing its different flavours, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

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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. Dans cet article, nous proposons une méthode appropriée pour détecter et caractériser simultanément les zones où le mouvement est présent dans une séquence. Nous pouvons ensuite classer ce mouvement en zones cohérentes en utilisant un apprentissage non supervisé et produire des métriques directement utilisables pour diverses applications. Nous illustrons et validons cette même méthode sur l’analyse du flux sanguin chez l’embryon de poisson.

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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. In this article, we propose a methodology called “area-of-motion characterization” suitable for simultaneously detecting and characterizing areas where motion is present in a sequence. We can then classify this motion into consistent areas using unsupervised learning and produce directly usable metrics for various ap- plications. We illustrate this methodology for the analysis of cilia motion on ex-vivo human samples, and we apply and validate the same methodology for blood flow analysis in fish embryo.

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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. This method conforms to the current trend that consists in producing hierarchical results rather than a unique partition. Contrary to classical approaches which compute optimal hierarchical segmentations from an input hierarchy of segmentations, we rely on the tree of shapes, a unique and well-defined representation equivalent to the image. Simply put, we compute for each level line of the image an attribute function that characterizes its persistence under the energy minimization. Then we stack the level lines from meaningless ones to salient ones through a saliency map based on extinction values defined on the tree-based shape space. Qualitative illustrations and quantitative evaluation on Weizmann segmentation evaluation database demonstrate the state-of-the-art performance of our method.

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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. It relies on the notion of shape space: a graph representation of a set of regions extracted from the image. Each region is characterized with an attribute describing it. We weigh the boundaries of a subset of meaningful regions (local minima) in the shape space by extinction values based on the attribute. This extinction-based saliency map represents a new hierarchy of segmentations highlighting regions having some specific characteristics. Each threshold of this map represents a segmentation which is generally different from any cut of the original hierarchy. This new approach thus enlarges the set of possible partition results that can be extracted from a given hierarchy. Qualitative and quantitative illustrations demonstrate the usefulness of the proposed method.

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