Pierre-Alain Fayolle

Procedural function-based modelling of volumetric microstructures

We propose a new approach to modelling heterogeneous objects containing internal volumetric structures with size of details orders of magnitude smaller than the overall size of the object. The proposed function-based procedural representation provides compact, precise, and arbitrarily parametrized models of coherent microstructures, which can undergo blending, deformations, and other geometric operations, and can be directly rendered and fabricated without generating any auxiliary representations (such as polygonal meshes and voxel arrays). In particular, modelling of regular lattices and cellular microstructures as well as irregular porous media is discussed and illustrated. We also present a method to estimate parameters of the given model by fitting it to microstructure data obtained with magnetic resonance imaging and other measurements of natural and artificial objects. Examples of rendering and digital fabrication of microstructure models are presented.

Cultural Heritage Preservation Using Constructive Shape Modeling

Issues of digital preservation of shapes and internal structures of historical cultural objects are discussed. An overview of existing approaches to digital preservation related to shape modeling is presented and corresponding problems are considered. We propose a new digital preservation paradigm based on both constructive modeling reflecting the logical structure of the objects and open standards and procedures. Constructive Solid Geometry (CSG) and Function Representation (FRep) are examined and practically applied as mathematical representations producing compressed yet precise data structures, thus providing inter‐operability between current and future computer platforms crucial to archiving. Examples of CSG reconstruction of historical temples and FRep modeling of traditional lacquer ware are given. We examine the application of fitting of a parameterized FRep model to a cloud of data points as a step towards automation of the modeling process. Virtual venues for public access to cultural heritage objects including real time interactive simulation of cultural heritage sites over the Web are discussed and illustrated.

Constructive Heterogeneous Object Modeling Using Signed Approximate Real Distance Functions

We introduce a smooth approximation of the min/max operations, called signed approximate real distance function (SARDF), for maintaining an approximate signed distance function in constructive shape modeling. We apply constructive distance-based shape modeling to design objects with heterogeneous material distribution in the constructive hypervolume model framework. The introduced distance approximation helps intuitively model material distributions parametrized by distances to so-called material features. The smoothness of the material functions, provided here by the smoothness of the defining function for the shape, helps to avoid undesirable singularities in the material distribution, like stress or concentrations. We illustrate application of the SARDF operations by two- and three-dimensional heterogeneous object modeling case studies.

On Variational and PDE-Based Distance Function Approximations

In this paper, we deal with the problem of computing the distance to a surface (a curve in two dimensional) and consider several distance function approximation methods which are based on solving partial differential equations (PDEs) and finding solutions to variational problems. In particular, we deal with distance function estimation methods related to the Poisson‐like equations and generalized double‐layer potentials. Our numerical experiments are backed by novel theoretical results and demonstrate efficiency of the considered PDE‐based distance function approximations.

Segmentation of discrete point clouds using an extensible set of templates

We present an algorithm for segmenting a discrete three-dimensional point-set (i.e., partitioning an input discrete point-set into appropriate subsets). The algorithm consists in the iteration of two main steps which are: fitting the parameters of template primitives from a user-specified list of primitives and extracting the points from the input point-set matching the best fitted primitive. We illustrate the results of applying our algorithm to several examples of three-dimensional point-sets.

3D shape reconstruction of template models using genetic algorithms

We present in this communication a method, which enables to fit a 3D object defined by a functional representation (FRep) to a dataset of 3D points on its surface. A parametric FRep model sketching the point-set is fitted to the point-set. The best fined parameters of the model are obtained by using a genetic algorithm, well known for its interesting properties in non-linear optimization. The efficiency of the approach is illustrated for reverse engineering applications.

Shape recovery using functionally represented constructive models

We propose a method which helps to fit existing parameterized function representation (FRep) models to a given dataset of 3D surface points. Best fitted parameters of the model are obtained by using a hybrid algorithm combining simulated annealing and Levenberg-Marquardt methods. The efficiency of the approach is shown for recovery of two test items. We show through the CAD model processing an application of the proposed approach to the shape recovery followed by finite element mesh generation and adaptation.

An evolutionary approach to the extraction of object construction trees from 3D point clouds

In order to extract a construction tree from a finite set of points sampled on the surface of an object, we present an evolutionary algorithm that evolves set-theoretic expressions made of primitives fitted to the input point-set and modeling operations. To keep relatively simple trees, we use a penalty term in the objective function optimized by the evolutionary algorithm. We show with experiments successes but also limitations of this approach.

Technical note: Signed Lp-distance fields

Abstract We introduce and study a family of generalized double-layer potentials which are used to build smooth and accurate approximants for the signed distance function. Given a surface, the value of an approximant at a given point is a power mean of distances from the point to the surface points parameterized by the angle they are viewed from the given point. We analyze mathematical properties of the potentials and corresponding approximants. In particular, approximation accuracy estimates are derived. Our theoretical results are supported by numerical experiments which reveal the high practical potential of our approach.

Web-based Constructive Shape Modeling Using Real Distance Functions

We present an approach and a web-based system implementation for modeling shapes using real distance functions. The system consists of an applet supporting the HyperFun modeling language. The applet is extended with primitives defined by Euclidean distance from a point to the surface of the shape. Set-theoretic operations (union, intersection, difference) that provide an approximation of the Euclidean distance to a shape built in a constructive way are introduced. Such operations have a controllable error of the exact Euclidean distance to the shape and preserve C1 continuity of the overall function, which is an important condition for further operations and applications. The proposed system should help model various shapes, store them in a concise form, and exchange them easily between different entities on a network. The applet offers also the possibility to export the models defined in the HyperFun language to other formats for raytracing or rapid prototyping.