Posters
Posters presented on Monday October 21, 2024 - From 16h30 to 18h
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POSTER ABSTRACT
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Distance function and persistence diagrams of a generic submanifold and its samplings
Charles ARNAL
Inria
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Though the critical points of the distance function to a submanifold can be poorly behaved, we show that when the submanifold M is generic, the critical points and their projections on M are regular and stable. This has consequences on the Cech persistence diagram of M, as well as on the Cech persistence diagrams of randoms samplings of M; in particular, convergence results with respect to the Wasserstein distances on diagrams can be obtained. |
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Computing the volume of the Combinahedron
Marguerite BIN
Loria
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The combinahedron is a polytope similar to the permutahedron that can also be written as the Minkowski sum of several segments. Its zonotope structure allows us to compute its volume : in our case, the computation of the volume can be reduced to a combinatorial sum involving only trees. We can then use the Prüfer Encoding of a tree (and for another polytope, a new encoding of trees) to directly compute the sum. |
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Fast and bijective rigid digitized transformations
Stéphane BREUILS
Université Savoie Mont Blanc
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Preserving surfaces or volumes of digital objects is crucial when applying rigid transformations of 2D/3D digital objects in medical images and computer vision. To achieve this goal, the digital geometry community has focused on characterizing bijective digitized rotations and reflections. However, the angular distribution of these bijective rigid transformations is far from being dense. Other bijective approximations of rigid transformations have been proposed, but the state-of-the-art methods lack the experimental evaluations necessary to include them in real-life applications. This poster presents several new methods to approximate digitized rotations with bijective transformations, including the composition of bijective digitized reflections, bijective rotation by circles and bijective rotation through optimal transport. These new methods and several classical ones are compared in terms of accuracy with respect to Euclidean rotations, as well as computational complexity and practical speed in real-time applications and continuity. Finally, we show stability results after these bijective rigid transformations. |
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Optimisation in Geometric Reconstruction
Loïc DRIEU LA ROCHELLE
Université de Poitiers
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This work focuses on the polyhedralization of discrete surfaces. The purpose is to optimize the number of polygons present in the reconstruction while preserving the property of reversibility. Our approach consists in reducing the number of planes in our segmentation and achieve more efficient surface analysis algorithms leading to a final compact mesh representing the surface of our object. |
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Density of sphere packings
Thomas FERNIQUE
LIPN/CNRS
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We aim to show how to get density result on sphere packings beyond the celebrated Kepler conjecture. |
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Using Delaunay triangulation to enumerate saddle connections
Oscar FONTAINE
Université de Bordeaux
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A translation surface is a surface obtained by gluing finitely many euclidian polygons along their edges using translations. Outside the vertices of the polygons, the surface is locally isometric to the Euclidean plane. A saddle connection is a straight line segment in a translation surface whose two ends are conic singularities. It is a result of H. Masur that the number of saddle connections of length at most R is \Theta(R^2). We study the algorithmic problem of enumerating the saddle connections in a given translation surface. A naive enumeration algorithm allows to list saddle connections in \Theta(R^3). A more sophisticated method using Delaunay triangulations shows that one can achieve the optimal \Theta(R^2) for some specific surfaces. |
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On the Twin-Width of Smooth Manifolds
Kristóf HUSZÁR
Graz University of Technology
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Building on Whitney's classical method of triangulating smooth manifolds, we show that every compact d-dimensional smooth manifold admits a triangulation with dual graph of twin-width at most d^{O(d)}. In particular, it follows that every compact 3-manifold has a triangulation with dual graph of bounded twin-width. This is in sharp contrast to the case of treewidth, where for any natural number n there exists a closed 3-manifold such that every triangulation thereof has dual graph with treewidth at least n. To establish this result, we bound the twin-width of the incidence graph of the d-skeleton of the second barycentric subdivision of the 2d-dimensional hypercubic honeycomb. We also show that every compact, piecewise-linear (hence smooth) d-dimensional manifold has triangulations where the dual graph has an arbitrarily large twin-width. Joint work with Édouard Bonnet. |
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Hierarchical analysis of 3D X-ray CT images for granular materials
Lysandre MACKE
Université de Strasbourg
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We address the classical problem of image segmentation applied to 3D CT scans in order to allow topological analysis of granular materials such as sand grains. The current state of the art offers many efficient methods for this issue. However they are usually not applicable to images such as the CT scans that we use - and that are also very common in the medical field - that contain important artefacts caused by the acquisition process. Hence we propose a new segmentation method that is usable for our problematic, exploiting hierarchical structures called component trees. |
Posters presented on Tuesday October 22, 2024 - From 16h30 to 18h
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POSTER ABSTRACT
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Incremental Watershed Cuts: Interactive Segmentation Algorithm with Parallel Strategy
Quentin LEBON
ESIEE
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We present an incremental method for computing seeded watershed cuts for interactive image segmentation. We propose an algorithm based on the hierarchical image representation called the binary partition tree to compute a seeded watershed cut. Additionally, we leverage properties of minimum spanning forests to introduce a parallel method for labeling a connected component. We show that those algorithms fits perfectly in an interactive segmentation process by handling user interactions, seed addition or removal, in linear time with respect to the number of affected pixels. Run time comparisons with several state-of-the-art interactive and non-interactive watershed methods show that the proposed method can handle user interactions much faster than previous methods with a significant speedup ranging from 10 to 60 on both 2D and 3D images, thus improving the user experience on large images. |
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Persistent Intrinsic Volumes
Antoine COMMARET
Inria
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We address the problem of estimating the area, and more generally the intrinsic volumes, of a compact subset X of R^d from a set Y that is close in the Hausdorff distance. We introduce an estimator that enjoys a linear rate, of convergence as a function of the Hausdorff distance under mild regularity conditions on X. Our approach combines tools from both geometric measure theory and persistent homology, extending the noise filtering properties of persistent homology from the realm of topology to geometry. |
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Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect
Corentin LUNEL
Université Gustave Eiffel
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While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order. |
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A Parameter-Free Normal Vector Estimator on Digital Surfaces
Aude MARÊCHÉ
Université de Lorraine
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The processing of 3D digital objects often requires the computation and analysis of their geometrical features. The normal vectors of the object's surface in particular provide important information used in image processing applications. We present in this paper a new method for the estimation of normal vectors on the surface of a 3D digital object. It is both local and parameter-free. The proposed method involves the study of neighborhoods around points using planar sectors. Experimental evaluations using multi-grid approaches show that it is both faster and more robust than state-of-the-art methods in the field, while being of comparable accuracy. |
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How to Modify the Tree of Shapes of an Image: Connected Operators Without Gradient Inversion
Julien MENDES FORTE
Université Caen Normandie
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The tree of shapes is a hierarchical data structure that models a grey-level image via its level lines. It belongs to the family of morphological trees, which allow to design connected operators, i.e. non-linear filters that transform an image without creating new contours. Connected operators act by modifying the image-modeling tree, shifting the values of its nodes. This paradigm is frequently used with the component tree, another popular morphological tree. It is much less considered in the case of the tree of shapes despite its ability to model more finely the image. Indeed, shifting the values of the nodes of a tree of shapes is more complex, compared to other morphological trees. We investigate how to modify a tree of shapes by shifting the values of its nodes. We explain how to carry out this operation so that the modified / simplified tree remains the tree of shapes of the processed image. We propose algorithmic solutions and methodological schemes to reach that goal. We discuss on their properties and we illustrate their relevance by application examples of induced connected operators. |
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Mean curvature estimator on digital surfaces using a Varifold formulation
Romain NEGRO
Université Savoie Mont Blanc
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Estimating curvatures on digital surfaces is challenging: they possess very poor information (having locally only one of 6 different directions in 3D) and are non-differentiable just like triangle surfaces.
We propose to use the broad mathematical framework of varifolds, which encompasses all kind of surface representations (continuous, discrete, point clouds, ...). Varifolds decouple position and tangent plane information via two separate measurements. We show how to rectify the tangent measure in the case of digital surfaces using convergent geometric estimators of normals. With these modifications, the first variation of a varifold induces the mean curvature on these surfaces. |
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Geometrical Deep Learning with the help of mathematical morphology.
Santiago VELASCO-FORERO
Mines-PSL
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This poster presents recent results on the use of mathematical morphology in the context of deep learning to analyze point clouds, images and random processes. |
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