Dominique Attali

Computing and Simplifying 2D and 3D Continuous Skeletons

Skeletons provide a synthetic and thin representation of objects. Therefore, they are useful for shape description. Recent papers have proposed to approximate the skeleton of continuous shapes using the Voronoi graph of boundary points. An original formulation is presented here, using the notion of polyballs (we call polyball any finite union of balls). A preliminary work shows that their skeletons consist of simple components (line segments in 2D and polygons in 3D). An efficient method for simplifying 3D continuous skeletons is also presented. The originality of our approach consists in simplifying the shape without modifying its topology and in including these modifications on the skeleton. Depending on the desired result, we propose two strategies which lead to either surfacical skeletons or wireframe skeletons. Two angular criteria are proposed that allow us to build a size-invariant hierarchy of simplified skeletons.

r -regular shape reconstruction from unorganized points

In thw paper, the problem of reconstructing a surface, given a set of scattered data points is addressed. First, a precise formulation of the reconstruction problem is proposed. The solution is mathematically defined as a particular mesh of the surface called the normalized mesh. This solution has the property to be included inside the Delaunay graph. A criterion to select boundary faces inside the Delaunay graph is proposed. This criterion is proven to provide the exact solution in 2D for points sampling a r-regular shapes with a sampling path c < 0.38r. In 3D, this results cannot be extended and the criterion cannot retrieve every faces. Some heuristics are then proposed in order to complete the surface. the object [4, 5, 6]. More complex graphs have also been introduced like crhulls and a-shapes [7, 8]. a-shapes are a generalization of the convex hull of a point set. An a-shape is a polytope surrounding the set of points. The parameter a controls the maximum “curvature” of any cavity of the polytope. Several a-shapes with different values of a are presented in figure 1. The choice of the parameter a might be tricky. O .::.. ”.... ... . . . . . . . . . . -.. . -.. “.. . . ..“

Complexity of the delaunay triangulation of points on surfaces the smooth case

It is well known that the complexity of the Delaunay triangulation of N points in R 3, i.e. the number of its faces, can be O (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface.In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of R 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N).

A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

Abstract Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in R3 may be quadratic in the worst case, we show in this paper that it is only linear when the points are distributed on a fixed set of well-sampled facets of R3 (e.g. the planar polygons in a polyhedron). Our bound is deterministic and the constants are explicitly given.

Skeletal Reconstruction of Branching Shapes

We present a new method to reconstruct an implicit representation of a branching object from a set of data points scattered on its surface. The method is based on the computation of a geometric skeleton inside the data set. This skeleton is simplified in order to filter noise and converted into skeletal elements – a graph of interconnected curves – that generate an implicit surface. We use Bézier triangles as extra skeletal elements to perform bulge free blends between branches while controlling the blend extent. The result is a smooth reconstruction of the object, that can be computed whatever its topology. The skeleton offers compact storage, and provides an underlying structure for the reconstructed object, making it easier to edit in a modeling or animation environment.

Efficient data structure for representing and simplifying simplicial complexes in high dimensions

We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference Flag(G)\ K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.

Delaunay conforming iso-surface, skeleton extraction and noise removal

Iso-surfaces are routinely used for the visualization of volumetric structures. Further processing (such as quantitative analysis, morphometric measurements, shape description) requires volume representations. The skeleton representation matches these requirements by providing a concise description of the object. This paper has two parts. First, we exhibit an algorithm which locally builds an iso-surface with two significant properties: it is a 2-manifold and the surface is a subcomplex of the Delaunay tetrahedrization of its vertices. Secondly, because of the latter property, the skeleton can in turn be computed from the dual of the Delaunay tetrahedrization of the iso-surface vertices. The skeleton representation, although informative, is very sensitive to noise. This is why we associate a graph to each skeleton for two purposes: (i) the amount of noise can be identified and quantified on the graph and (ii) the selection of the graph subpart that does not correspond to noise induces a filtering on the skeleton. Finally, we show some results on synthetic and medical images. An application, measuring the thickness of objects (heart ventricles, bone samples) is also presented.

A new method for analyzing local shape in three-dimensional images based on medial axis transformation

In this paper, we propose a new approach based on three-dimensional (3-D) medial axis transformation for describing geometrical shapes in three-dimensional images. For 3-D-images, the medial axis, which is composed of both curves and medial surfaces, provides a simplified and reversible representation of structures. The purpose of this new method is to classify each voxel of the three-dimensional images in four classes: boundary, branching, regular and arc points. The classification is first performed on the voxels of the medial axis. It relies on the topological properties of a local region of interest around each voxel. The size of this region of interest is chosen as a function of the local thickness of the structure. Then, the reversibility of the medial axis is used to deduce a labeling of the whole object. The proposed method is evaluated on simulated images. Finally, we present an application of the method to the identification of bone structures from 3-D very high-resolution tomographic images.

Modeling noise for a better simplification of skeletons

The skeleton of an object is the locus of the centers of maximal discs included in the shape. The skeleton provides a compact representation of objects, useful for shape description and recognition. A well-known drawback of the skeleton transformation is its lack of continuity. This paper is concerned with the modeling of noise that may affect objects and the consequence of this noise on the skeleton. A graph (called the parameter graph) is introduced, on which branches due to noise are characterized. We deduce from this preliminary study a method to simplify skeletons. It depends on thresholds that can be chosen directly on the parameter graph associated to each skeleton.

Pruning Discrete and Semiocontinuous Skeletons

In this paper pruning techniques are illustrated, which allow us to suitably simplify the (discrete and semicontinuous) skeleton, by either deleting or shortening peripheral skeleton branches. To avoid excessive shortening, which might reduce the representative power of the skeleton, the relevance of the figure regions mapped in the skeleton branches is used to decide on pruning. Different definitions of relevance are introduced and features allowing the quantitative evaluation of the relevance are suggested.