Laurent Fuchs

Computation of homology groups and generators

Topological invariants are extremely useful in many applications related to digital imaging and geometric modelling, and homology is a classical one. We present an algorithm that computes the whole homology of an object of arbitrary dimension: Betti numbers, torsion coefficients and generators. Results on classical shapes in algebraic topology are presented and discussed.

Exact, robust and efficient full visibility computation in Plücker space

We present a set of new techniques to compute an exact polygon-to-polygon visibility in Plücker space. The contributions are based on the definition of the minimal representation of lines stabbing two convex polygons. The new algorithms are designed to indicate useless computations, which results in more compact visibility data, faster to exploit, and in a reduced computation time. We also define a simple robust and exact solution to handle degeneracies, where previous methods proposed aggressive solutions.

Computing homology for surfaces with generalized maps: application to 3d images

In this paper, we present an algorithm which allows to compute efficiently generators of the first homology group of a closed surface, orientable or not. Starting with an initial subdivision of a surface, we simplify it to its minimal form (minimal refers to the number of cells), while preserving its homology. Homology generators can thus be directly deduced from the minimal representation of the initial surface. Finally, we show how this algorithm can be used in a 3D labelled image in order to compute homology of each region described by its boundary.

An algorithm to decompose n -dimensional rotations into planar rotations

In this paper, we present an algorithm that decomposes an n-dimensional rotation into planar rotations. The input data are n points and their images by the rotation to be decomposed. An evaluation of the existing methods and numerical examples are also provided.

Insight in discrete geometry and computational content of a discrete model of the continuum

This article presents a synthetic and self contained presentation of the discrete model of the continuum introduced by Harthong and Reeb [J. Harthong, Elements pour une theorie du continu, Asterisque 109/110 (1983) 235-244.[1]; J. Harthong, Une theorie du continu, in: H. Barreau, J. Harthong (Eds.), La mathematiques non standard, Editions du CNRS, 1989, pp. 307-329.[2]] and the related arithmetization process which led Reveilles [J.-P. Reveilles, Geometrie discrete, calcul en nombres entiers et algorithmique, Ph.D. Thesis, Universite Louis Pasteur, Strasbourg, France, 1991.[3]; J.-P. Reveilles, D. Richard, Back and forth between continuous and discrete for the working computer scientist, Annals of Mathematics and Artificial Intelligence, Mathematics and Informatic 16(1-4) (1996) 89-152.[4]] to the definition of a discrete analytic line. We present then some basis on constructive mathematics [E. Bishop, D. Bridges, Constructive Analysis, Springer, Berlin, 1985.[5]], its link with programming [P. Martin-Lof, Constructive mathematics and computer programming, in: Logic, Methodology and Philosophy of Science, vol. VI, 1980, pp. 153-175.[6]; W.A. Howard, The formulae-as-types notion of construction, To H.B. Curry: Essays on Combinatory Logic, Lambda-calculus and Formalism, 1980, pp. 479-490.[7]] and we propose an analysis of the computational content of the so-called Harthong-Reeb line. More precisely, we show that a suitable version of this new model of the continuum partly fits with the constructive axiomatic of R proposed by Bridges [Constructive mathematics: a foundation for computable analysis, Theoretical Computer Science 219(1-2) (1999) 95-109.[8]]. This is the first step of a more general program on a constructive approach of the scaling transformation from discrete to continuous space.

Computing homology generators for volumes using minimal generalized maps

In this paper, we present an algorithm for computing efficiently homology generators of 3D subdivided orientable objects which can contain tunnels and cavities. Starting with an initial subdivision, represented with a generalized map where every cell is a topological ball, the number of cells is reduced using simplification operations (removal of cells), while preserving homology. We obtain a minimal representation which is homologous to the initial object. A set of homology generators is then directly deduced on the simplified 3D object.

A first look into a formal and constructive approach for discrete geometry using nonstandard analysis

In this paper, we recall the origins of discrete analytical geometry developed by J-P. Reveilles [1] in the nonstandard model of the continuum based on integers proposed by Harthong and Reeb [2,3]. We present some basis on constructive mathematics [4] and its link with programming [5,6]. We show that a suitable version of this new model of the continuum partly fits with the constructive axiomatic of R proposed by Bridges [7]. The aim of this paper is to take a first look at a possible formal and constructive approach to discrete geometry. This would open the way to better algorithmic definition of discrete differential concepts.

Simploidals sets: Definitions, operations and comparison with simplicial sets

The combinatorial structure of simploidal sets generalizes both simplicial complexes and cubical complexes. More precisely, cells of simploidal sets are cartesian product of simplices. This structure can be useful for geometric modeling (e.g. for handling hybrid meshes) or image analysis (e.g. for computing topological properties of parts of n-dimensional images). In this paper, definitions and basic constructions are detailed. The homology of simploidal sets is defined and it is shown to be equivalent to the classical homology. It is also shown that products of Bezier simplicial patches are well suited for the embedding of simploidal sets.

Designing a Topological Modeler Kernel: A Rule-Based Approach

In this article, we present a rule-based language dedicated to topological operations and based on graph transformations. Generalized maps are described as a particular class of graphs determined by consistency constraints. Hence, topological operations over generalized maps can be specified using graph transformations. The rules we define are provided with syntactic criteria which ensure that graphs computed by applying rules on generalized maps are also generalized maps. We have developed a static analyzer of transformation rules which checks the syntactic criteria in order to ensure the preservation of generalized map consistency constraints. Based on this static analyzer, we have designed a rule-based prototype of a kernel of a topology-based modeler that is generic in dimension. Since adding a new topological operation can be reduced to write a graph transformation rule, we directly obtain an extensible prototype where handled topological objects satisfy built-in consistency. Moreover, first benchmarks show that our prototype is reasonably efficient compared to a reference implementation of 3D generalized maps which use a classical implementation style.

Ω-Arithmetization: A Discrete Multi-resolution Representation of Real Functions

Multi-resolution analysis and numerical precision problems are very important subjects in fields like image analysis or geometrical modeling. In the continuation of previous works of the authors, we expose in this article a new method called the