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The center of mass of a point set lying on a manifold generalizes the celebrated Euclidean centroid, and is ubiquitous in statistical analysis in non Euclidean spaces. In this work, we give a complete characterization of the weighted... more
The center of mass of a point set lying on a manifold generalizes the celebrated Euclidean centroid, and is ubiquitous in statistical analysis in non Euclidean spaces. In this work, we give a complete characterization of the weighted p-mean of a finite set of angular values on S1, based on a decomposition of S1 such that the functional of interest has at most one local minimum per cell. This characterization is used to show that the problem is decidable for rational angular values –a consequence of Lindemann’s theorem on the transcendence of π, and to develop an effective algorithm parameterized by exact predicates. A robust implementation of this algorithm based on multi-precision interval arithmetic is also presented, and is shown to be effective for large values of n and p. We use it as building block to implement the k-means and k-means++ clustering algorithms on the flat torus, with applications to clustering protein molecular conformations. These algorithms are available in th...
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Research Interests: Computer Science, Chemistry, Water, Protein Science, Macromolecular X-Ray Crystallography, and 10 moreCurvature, Protein Sequence Analysis, Molecular Computers, Structure activity Relationship, Protein Binding, Protein Data Bank, Biochemistry and cell biology, Voronoi Diagram, Solvents, and binding sites
The understanding of surfaces embedded in E3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects... more
The understanding of surfaces embedded in E3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of core concepts from differential topology applied to smooth embedded surfaces. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of poi...
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Research Interests: Engineering, Mathematics, Computer Science, Numerical Analysis, Differential Geometry, and 14 moreTaylor Expansion, Graphics, Mathematical Sciences, Computer Aided Geometric Design, Multivariate Interpolation, Surface, Key words, Convergence Rate, Approximation, Interpolation, Computer Graphic, Point Clouds, Point Cloud, and Polynomial
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Summary: Intervor is a software computing a parameter-free representation of macro–molecular interfaces, based on the α-complex of the atoms. Given two interacting partners, possibly with water molecules squeezed in-between them,... more
Summary: Intervor is a software computing a parameter-free representation of macro–molecular interfaces, based on the α-complex of the atoms. Given two interacting partners, possibly with water molecules squeezed in-between them, Intervor computes an interface model which has the following characteristics: (i) it identifies the atoms of the partners which are in direct contact and those whose interaction is water mediated, (ii) it defines a geometric complex separating the partners, the Voronoi interface, whose geometric and topological descriptions are straightforward (surface area, number of patches, curvature), (iii) it allows the definition of the depth of atoms at the interface, thus going beyond the traditional dissection of an interface into a core and a rim. These features can be used to investigate correlations between structural parameters and key properties such as the conservation of residues, their polarity, the water dynamics at the interface, mutagenesis data, etc. A...
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Summary: The ever increasing number of structural biological data calls for robust and efficient software for analysis. Easy Structural Biology Template Library (ESBTL) is a lightweight C++ library that allows the handling of PDB data and... more
Summary: The ever increasing number of structural biological data calls for robust and efficient software for analysis. Easy Structural Biology Template Library (ESBTL) is a lightweight C++ library that allows the handling of PDB data and provides a data structure suitable for geometric constructions and analyses. The parser and data model provided by this ready-to-use include-only library allows adequate treatment of usually discarded information (insertion code, atom occupancy, etc.) while still being able to detect badly formatted files. The template-based structure allows rapid design of new computational structural biology applications and is fully compatible with the new remediated PDB archive format. It also allows the code to be easy-to-use while being versatile enough to allow advanced user developments. Availability: ESBTL is freely available under the GNU General Public License from http://esbtl.sf.net. The web site provides the source code, examples, code snippets and do...
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Balls and spheres are amongst the simplest 3 D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several... more
Balls and spheres are amongst the simplest 3 D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several communities, we are not aware of any robust algorithm, and present the first such algorithm. Our calculation relies on the decomposition of the volume of the union into convex regions, namely the restrictions of the balls to their regions in the power diagram. Theoretically, we establish a formula for the volume of a restriction, based on Gauss' divergence theorem. The proof being constructive, we develop the associated algorithm. On the implementation side, we carefully analyse the predicates and constructions involved in the volume calculation, and present a certified implementation relying on interval arithmetic. The result is certified in the sense that the exact volume belongs to the interval computed. Experimental results are presented on hand-craf...
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Surfaces of R 3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in computer graphics, computer aided... more
Surfaces of R 3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in computer graphics, computer aided design, computational geometry, and computer vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpolation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, and extraction of the differential properties. This article presents a generic C++ software package solving these problems. On the theoretical side and as established in a companion paper, the interpolation and approximation methods provided achieve the best asymptotic error bounds known to date. On the implementation side and following state-of-the-art coding rules in computational geometry, genericity of the package is achieved thanks to four template ...
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Abstract: Given a finite sampling P R, of an unknown surface S, surface reconstruction is concerned with the calculation of a model of S from P. The model can be represented as a smooth or a triangulated surface, and is expected to match... more
Abstract: Given a finite sampling P R, of an unknown surface S, surface reconstruction is concerned with the calculation of a model of S from P. The model can be represented as a smooth or a triangulated surface, and is expected to match S from a topological and geometric standpoints. In this survey, we focus on the recent developments of Delaunay based surface reconstruction methods, which were the first methods (and in a sense still the only ones) for which one can precisely state properties of the reconstructed surface. We outline the foundations of these methods from a geometric and algorithmic standpoints. In particular, a careful presentation of the hypothesis used by these algorithms sheds light on the intrinsic diculties of the surface reconstruction problem —faced by any method, Delaunay based or not. Key-words: Reverse engineering, Shape approximation, Surface reconstruction, Delaunay,
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Conformal alpha shapes are a new filtration of the Delaunay triangulation of a finite set of points in ℝd. In contrast to (ordinary) alpha shapes the new filtration is parameterized by a local scale parameter instead of the global scale... more
Conformal alpha shapes are a new filtration of the Delaunay triangulation of a finite set of points in ℝd. In contrast to (ordinary) alpha shapes the new filtration is parameterized by a local scale parameter instead of the global scale parameter in alpha shapes. The local scale parameter conforms to the local geometry and is motivated from applications and previous algorithms in surface reconstruction. We show how conformal alpha shapes can be used for surface reconstruction of non-uniformly sampled surfaces, which is not possible with alpha shapes.