Dominique Faudot

A New Robust Algorithm to Trace Curves

We are presenting here a new reliable algorithm to trace curves using interval arithmetic. We give several computable criteria which guarantee the convergence of the correction step of the classical predictor-corrector method. Our method avoids, for instance, to jump from a component of the curve to another one; this kind of mistake typically causes inconsistencies in the topology of intersecting surfaces in geometric modelers.

From A Medial Surface To A Mesh

Medial surfaces are well‐known and interesting surface skeletons. As such, they can describe the topology and the geometry of a 3D closed object. The link between an object and its medial surface is also intuitively understood by people. We want to exploit such skeletons to use them in applications like shape creation and shape deformation. For this purpose, we need to define medial surfaces as Shape Representation Models (SRMs). One of the very first task of a SRM is to offer a visualization of the shape it describes. However, achieving this with a medial surface remains a challenging problem.

INITIAL PARAMETRIC REPRESENTATION OF BLOBS

Blobs, developed by J.F. Blinn in 1982, are the implicit surfaces obtained by composition of a real numerical function and a distance function. Since, many authors (C. Murakami, H. Nishimura, G. Wyvill…) defined their own function of density, from these implicit surfaces are interesting from several points of view. In particular, their fusion makes it possible to easily obtain an implicit equation of resulting surface. However, these surfaces do not admit a parametric equation yet. In this article, we will establish the parametric equation of two blobs in fusion, defined by the function of density of C. Murakami, by using an algebraic method. Then, we will develop another method, based on the differential equations.

Volumetric reconstruction of unorganized set of points with implicit surfaces

Many solutions exist to rebuild a three-dimensional object represented by a set of points. The purpose of our work is to provide an automatic reconstruction from an unorganized cloud, describing an unknown shape, in the aim to compute its volume. The approach employed in this paper consists in filling the objects interior with isosurfaces of potential fields and to use their fusion property in order to find the full volume and the continuous shape of the sampled object. Thus, the first step of our reconstruction is to search a correct interior for the object described by the set of points. Then, comes the positioning of implicit primitives into the cloud, deep inside of it and close to the boundary. A controlled fusion of the isosurfaces guarantees that no holes are present, such that we obtain a complete shape filling.

Identifying and Structuring Skeletal Noise

A skeleton is a thin centered structure within an object, which describes its topology and its geometry. The medial surface is one of the most known and used skeleton formulation. As other formulations, it contains noise, which complexifies its structure with useless parts. The connectivity of a skeleton is then unpredictable due to these useless parts. It can be a problem to segment the skeleton into logical components for example. We present here a technique whose purpose is to identify and structure such skeletal noise. It only requires a skeleton as input, making this work independent from any skeletonization process used to obtain the skeleton. We show in this paper that we significantly reduce the skeletal noise and produce clean skeletons that still capture every aspects of a shape. Those clean skeletons have the same local topology as the input ones, but with a clearer connectivity.

Implicit and Parametric Representations of Three Blobs

We explain a new method to transform an implicit equation of three blobs in a parametric equation.

An alternative to medial axis for the 3d reconstruction of unorganized set of points using implicit surfaces

Rebuilding three-dimensional objects represented by a set of points is a classical problem in computer graphics. Multiple applications like medical imaging or industrial techniques require finding shape from scattered data. Therefore, the reconstruction of a set of points that represents a shape has been widely studied, depending on data source and reconstructions objectives. This purpose of this paper is to provide an automatic reconstruction from an unorganized cloud describing an unknown shape in order to provide a solution that will allow to compute the objects volume and to deform it with constant volume. The main idea in this paper consists in filling the objects interior with an equipotential surface resulting of the fusion of potential field primitives also called metaballs or blobs. Nevertheless, contrary to most of usual rebuilding methods based on implicit primitives blending, we do not compute any medial axis to set the primary objects. Indeed, a fast voxelization is used to find a summary contour from the discrete shape and to determine interior areas. Then, the positioning of implicit primitives rely on a multilayer system. Finally, a controlled fusion of the isosurfaces guarantees the lack of any holes and a respectful contour of the original object, such that we obtain a complete shape filling

STUDY OF VOLUME VARIATION OF IMPLICIT OBJECTS

We propose studying the variations of volume of implicit objects during an animation according to several points of view: choice of the function of density, variations of parameters such as the iso-value and the radius of influence for a given function, variations of the parameters inherent in a particular function. Modification of parameters of the function of density must be carried out with care. There are no rules concerning these variations. To avoid the non-monotonous variations, it is necessary to choose a function of density beforehand and study the intervals of variation of its parameters. A new discretization makes it possible to locate these variations for a later use in a process of control of these variations.

Skeletizing 3D-Objects by Projections

Skeletization is used to simplify an object and to give an idea of the global shape of an object. This paper concerns the continuous domain. While many methods already exist, they are mostly applied in 2D-space. We present a new method to skeletize the polygonal approximation of a 3D-object, based on projections and 2D-skeletization from binary trees.

B-Deformable Superquadrics for 3D Reconstruction

We propose a new model for 3D representation and reconstruction. It is based on deformable superquadrics and parametric B-Splines. The 3D object deformation method uses B-Splines, instead of a Finite Element Method (FEM). This new model exhibits advantages of B-Splines It is significantly faster than deformable superquadrics without loss of generality (no assumption is made on object shapes,).