## About

127

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Introduction

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October 2004 - August 2021

April 2003 - September 2004

April 1998 - March 2003

## Publications

Publications (127)

In this article, we enrich the framework of morphological hierarchies with new acyclic graphs and trees. These structures lie at the convergence of hierarchical models and topological descriptors. We define them in the context of digital grey-level imaging. We discuss their links with component-trees, trees of shapes and adjacency trees. This analy...

In this article, we propose algorithms for pixelwise deformations of digital convex sets preserving their convexity using the combinatorics on words to identify digital convex sets via their boundary words, namely Lyndon and Christoffel words. The notion of removable and insertable points are used with a geometric strategy for choosing one of those...

Topology preservation is a property of affine transformations in R2, but not in Z2. In this article, given a binary object X⊂Z2 and an affine transformation A, we propose a method for building a binary object X^⊂Z2 resulting from the application of A on X. Our purpose is, in particular, to preserve the homotopy type between X and X^. To this end, w...

A statistical appearance model of blood vessels based on variational autoencoder (VAE) is well adapted to image intensity variations. However, images reconstructed with such a statistical model may have topological defects, such as loss of bifurcation and creation of undesired hole. In order to build a 3D anatomical model of blood vessels, we incor...

Geometric algebra has become popularly used in applications dealing with geometry. This framework allows us to reformulate and redefine problems involving geometric transformations in a more intuitive and general way. In this paper, we focus on 2D bijective digitized reflections and rotations. After defining the digitization through geometric algeb...

Topology preservation is a property of rigid motions in \({\mathbb R^2}\), but not in \({\mathbb {Z}}^2\). In this article, given a binary object \({\mathsf {X}} \subset {\mathbb {Z}}^2\) and a rational rigid motion \({\mathcal R}\), we propose a method for building a binary object \(\mathsf X_{\mathcal R}\subset \mathbb Z^2\) resulting from the ap...

Convexity is one of the useful geometric properties of digital sets in digital image processing. There are various applications which require deforming digital convex sets while preserving their convexity. In this article, we consider the contraction of such digital sets by removing digital points one by one. For this aim, we use some tools of comb...

Convexity is one of the useful geometric properties of digital sets in digital image processing. There are various applications which require deforming digital convex sets while preserving their convexity. In this article, we consider the contraction of such digital sets by removing digital points one by one. For this aim, we use some tools of comb...

The original version of the article was published in Mathematical Morphology - Theory and Applications 2 (2017) 55–75. Unfortunately, the original version contains a mistake: in the definition of Dif ( C1 , C2 ) in Section 3.6, max should be replaced by min. In this erratum we correct the formula defining Dif ( C1 , C2 ).

Hierarchical image segmentation provides region-oriented scale-spaces: sets of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Guimarães et al. proposed a hierarchical graph-based image segmentation (HGB) method based on the Felzenszwalb-Huttenlocher dissi...

We propose an efficient algorithm that removes unimportant regions from a hierarchical partition tree, while preserving the hierarchical partition structure. Various experiments demonstrate that applying this algorithm on various classification or segmentation problems does indeed improve the results by a large margin. Code is available online at h...

Higra — Hierarchical Graph Analysis is a C++/Python library for efficient sparse graph analysis with a special focus on hierarchical methods capable of handling large amount of data. The main aspects of hierarchical graph analysis addressed in Higra are the construction of hierarchical representations (agglomerative clustering, mathematical morphol...

Recently, a sufficient condition, namely quasi-regularity, has been proposed for preserving the connectivity during the process of digitization of a continuous object whose boundary is not necessarily differentiable. Under this condition, a rigid motion scheme for digital objects of \(\mathbb {Z}^2\) is proposed to guarantee that a well-composed ob...

Rigid motions (i.e. transformations based on translations and rotations) are simple, yet important, transformations in image processing. In \(\mathbb {R}^n\), they are both topology and geometry preserving. Unfortunately, these properties are generally lost in \(\mathbb {Z}^n\). In particular, when applying a rigid motion on a digital object, one g...

This book constitutes the thoroughly refereed proceedings of the 21st IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI 2019, held in Marne-la-Vallée, France, in March 2019.
The 38 full papers were carefully selected from 50 submissions. The papers are organized in topical sections on discrete geometric models and transf...

Hierarchical image segmentation provides a region-oriented scale-space, i.e. a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. However, most image segmentation algorithms, among which a graph-based image segmentation method relying on a region mergi...

Digitized rotations on discrete spaces are usually defined as the composition of a Euclidean rotation and a rounding operator; they are in general not bijective. Nevertheless, it is well known that digitized rotations defined on the square grid are bijective for some specific angles. This infinite family of angles has been characterized by Nouvel a...

Hierarchies of partitions are generally represented by dendrograms (direct representation). They can also be represented by saliency maps or minimum spanning trees. In this article, we precisely study the links between these three representations. In particular, we provide a new bijection between saliency maps and hierarchies based on quasi-flat zo...

This article is a first attempt towards a general theory for hierarchizing non-hierarchical image segmentation method depending on a region-dissimilarity parameter which controls the desired level of simpli fication: each level of the hierarchy is “as close as possible” to the result that one would obtain with the non-hierarchical method using the...

Purpose: We aim at studying intrinsic structures of the intracranial subarachnoid space.
Material and methods: Magnetic resonance images were obtained using the SPACE sequence, and the segmentation of the superior intracranial subarachnoid space was performed using geometrical features and a topological assumption of the shapes. Given such segmenta...

The goal of video cosegmentation is to jointly extract the common foreground regions and/or objects from a set of videos. In this paper, we present an approach for video cosegmentation that uses graph-based hierarchical clustering as its basic component. Actually, in this work, video cosegmentation problem is transformed into a graph-based clusteri...

Rigid motions in \(\mathbb {R}^2\) are fundamental operations in 2D image processing. They satisfy many properties: in particular, they are isometric and therefore bijective. Digitized rigid motions, however, lose these two properties. To investigate the lack of injectivity or surjectivity and more generally their local behavior, we extend the fram...

Rigid motions on \(\mathbb {R}^2\) are isometric and thus preserve the geometry and topology of objects. However, this important property is generally lost when considering digital objects defined on \(\mathbb {Z}^2\), due to the digitization process from \(\mathbb {R}^2\) to \(\mathbb {Z}^2\). In this article, we focus on the convexity property of...

Euclidean rotations in \(\mathbb {R}^2\) are bijective and isometric maps, but they generally lose these properties when digitized in discrete spaces. In particular, the topological and geometric defects of digitized rigid motions on the square grid have been studied. This problem is related to the incompatibility between the square grid and rotati...

In the field of digital geometry, numerous advances have been recently made to efficiently represent a simple polygonal shape; from dominant points of a curvature-based representation, a binary shape is efficiently represented even in presence of noise. In this article, we exploit recent results of such digital contour representations and propose a...

Action classification in videos has been a very active field of research over the past years. Human action classification is a research field with application to various areas such as video indexing, surveillance, human-computer interfaces, among others. In this paper, we propose a strategy based on decreasing the number of features in order to imp...

Rigid motions are fundamental operations in image processing. While bijective and isometric in \(\mathbb {R}^3\), they lose these properties when digitized in \(\mathbb {Z}^3\). To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on...

Euclidean rotations in \(\mathbb {R}^n\) are bijective and isometric maps. Nevertheless, they lose these properties when digitized in \(\mathbb {Z}^n\). For \(n=2\), the subset of bijective digitized rotations has been described explicitly by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In the case of 3D digitized rotations, the...

Image registration has become a crucial step in a wide range of imaging domains, from computer vision to computer graphics. The core of image registration consists of determining the transformation that induces the best mapping between two images. This problem is ill-posed; it is also difficult to handle, due to the high size of the images and the...

Rigid motions are fundamental operations in image processing. While they are bijective and isometric in \(\mathbb {R}^2\), they lose these properties when digitized in \(\mathbb {Z}^2\). To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on \(\mathbb {Z}^2\), initially proposed by Nouvel and R...

Hierarchical image segmentation provides a set of image segmentations at different detail levels in which coarser details levels can be produced by simple merges of regions from segmentations at finer detail levels. However, many image segmentation algorithms relying on similarity measures lead to no hierarchy. One of interesting similarity measure...

Cerebrospinal fluid imaging plays a significant role in the clinical diagnosis
of brain disorders, such as hydrocephalus and Alzheimer's disease. While
three-dimensional images of cerebrospinal fluid are very detailed, the complex
structures they contain can be time-consuming and laborious to interpret.
This paper presents a simple technique tha...

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 min...

Curvature is a continuous and infinitesimal notion. These properties induce geometrical difficulties in digital frameworks, and the following question is naturally asked: "How to define and compute curvatures of digital shapes?" In fact, not only geometrical but also topological difficulties are also induced in digital frameworks. The deeper questi...

Rigid transformations are involved in a wide variety of image processing applications, including image registration. In this context, we recently proposed to deal with the associated optimization problem from a purely discrete point of view, using the notion of discrete rigid transformation (DRT) graph. In particular, a local search scheme within t...

In the continuous domain, rigid transformations are topology-preserving operations. Due to digitization, this is not the case when considering digital images, i.e., images defined on Z^n. In this article, we begin to investigate this problem by studying conditions for digital images to preserve their topological properties under all rigid transform...

Rigid transformations are involved in a wide range of digital image processing applications. In such a context, they are generally considered as continuous processes, followed by a digitization of the results. Recently, rigid transformations on ℤ
2 have been alternatively formulated as a fully discrete process. Following this paradigm, we investiga...

We provide conditions under which 2D digital images preserve their topological properties under rigid transformations. We consider the two most common digital topology models, namely dual adjacency and well-composedness. This paper leads to the proposal of optimal preprocessing strategies that ensure the topological invariance of images under arbit...

Rigid image registration is an essential image processing task, with a large body of applications. This problem is usually formulated in the continuous domain, often in the context of an optimization framework. This approach leads to sometimes unwanted artifacts, e.g. due to interpolation. In the case of purely discrete applications, e.g., for temp...

We study the conditions under which the topological properties of a 2D well-composed binary image are preserved under arbitrary rigid transformations. This work initiates a more global study of digital image topological properties under such transformations, which is a crucial but under-considered problem in the context of image processing, e.g., f...

Purpose: Medical image visualization is an important step in the medical diagnosis of hydrocephalus. In this paper, we present planar representations called volumetric relief maps that are generated from three-dimensional images of the cerebrospinal fluid within the cortical subarachnoid space. Such maps are visually interpreted at once and allow t...

Hierarchical video segmentation provides region-oriented scale-space, i.e., a set of video segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Hierarchical methods have the interesting property of preserving spatial and neighboring information among segmented region...

Hydrocephalus is a neurological disorder that usually results from obstruction of the cerebrospinal fluid outflow in the ventricles or in the subarachnoid space. Magnetic resonance imaging offers a great deal of information to specialists in the clinical diagnosis and treatment processes of hydrocephalus. Recently we have proposed a new magnetic re...

Rigid transformations are involved in a wide range of digital image processing applications. When applied on discrete images, rigid transformations are usually performed in their associated continuous space, requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as fully discr...

A digital annulus is defined as a set of grid points lying between two circles sharing an identical center and separated by a given width. This paper deals with the problem of fitting a digital annulus to a given set of points in a 2D bounded grid. More precisely, we tackle the problem of finding a digital annulus that contains the largest number o...

In ℝ2, rigid transformations are topology-preserving operations. However, this property is generally no longer true when considering digital images instead of continuous ones, due to digitization effects. In this article, we investigate this issue by studying discrete rigid transformations (DRTs) on ℤ2. More precisely, we define conditions under wh...

Rigid transformations are useful in a wide range of digital image processing applications. In this context, they are generally considered as continuous processes, followed by discretization of the results. In recent works, rigid transformations on ℤ 2 have been formulated as a fully discrete process. Following this paradigm, we investigate – from a...

In this paper, we present a new three-dimensional (3D) tangent estimator by extending the two-dimensional (2D) λ-maximal segment tangent (λ-MST) estimator, which has very good theoretical and practical behaviors. We show that our proposed estimator keeps the same time complexity, accuracy and experimental asymptotic behaviors as the original 2D one...

This paper exploits the problem of fitting special forms of annuli that correspond to 4-connected digital circles to a given set of points in 2D images in the presence of noise by maximizing the number of inliers, namely the consensus set. We prove that the optimal solutions can be described by solutions with three points on the annulus boundary. T...

Hierarchical image segmentation provides a region-oriented scale-space, i.e., a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Most image segmentation algorithms, such as region merging algorithms, rely on a criterion for merging that does not lead...

Hierarchical image segmentation provides region-oriented scalespace, i.e., a
set of image segmentations at different detail levels in which the
segmentations at finer levels are nested with respect to those at coarser
levels. Most image segmentation algorithms, such as region merging algorithms,
rely on a criterion for merging that does not lead to...

We consider the following fitting problem: given an arbitrary set of N points in a bounded grid in dimension d, find a digital hyperplane that contains the largest possible number of points. We first observe that the problem is 3SUM-hard
in the plane, so that it probably cannot be solved exactly with computational complexity better than O(N
2), and...

In this paper, we study 3D rotations on grid points computed by using only integers. For that purpose, we investigate the intersection between the 3D half-grid and the rotation plane. From this intersection, we define 3D hinge angles which determine a transit of a grid point from a voxel to its adjacent voxel during the rotation. Then, we give a me...

This article presents a new method for fitting a digital line or plane to a given set of points in a 2D or 3D image in the presence of noise by maximizing the number of inliers, namely the consensus set. By using a digital model instead of a continuous one, we show that we can generate all possible consensus sets for model fitting. We present a det...

Given a set of discrete points in a 2D digital image containing noise, we formulate our problem as robust digital line fitting. More precisely, we seek the maximum subset whose points are included in a digital line, called the optimal consensus. The paper presents an efficient method for exactly computing the optimal consensus by using the topologi...

In this paper, we focus on 3D rotations on grid points computed by using only integers. For that purpose, we study the intersection
between the 3D half-grid and the rotation plane. From this intersection, we define 3D hinge angles which determine a transit
of a grid point from a voxel to its adjacent voxel during the rotation. Then, we give a metho...

This paper presents a method for fitting a digital plane to a given set of points in a 3D image in the presence of outliers. We present a new method that uses a digital plane model rather than the conventional continuous model. We show that such a digital model allows us to efficiently examine all possible consensus sets and to guarantee the soluti...

Rotations in the discrete plane are important for many applications such as image matching or construction of mosaic images. We suppose that a digital image A is transformed to another digital image B by a rotation. In the discrete plane, there are many angles giving the rotation from A to B, which we call admissible rotation angles from A to B. Fo...

This paper presents a method for segmenting a 3D point cloud into planar surfaces using recently obtained discrete-geometry results. In discrete geometry, a discrete plane is defined as a set of grid points lying between two parallel planes with a small distance, called thickness. In contrast to the continuous case, there exist a finite number of l...

Rotations in the discrete plane are important for many applications such as image matching or construction of mosaic images.
In this paper, we propose a method for estimating a rotation angle such that the rotation transforms a digital image A into another digital image B. In the discrete plane, there are many angles that can give the rotation from...