On the reconstruction of convex sets from random normal measurements

We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error eta, we provide an upper bounds on the number of probes that one has to perform in order to obtain an eta-approximation of this convex set with high probability. Our result rely on the stability theory related to Minkowski's theorem

Surface discretisation with rectifying strips on Geodesics

The use of geodesic curves of surfaces has enormous potential in architecture due to their multiple properties and easy geometric control using digital graphic tools. Among their numerous properties, the geodesic curves of a surface are the paths along which straight strips can be placed tangentially to the surface. On this basis, a graphical method is proposed to discretize surfaces using straight strips, which optimizes material consumption since rectangular straight strips take advantage of 100% of the material in the cutting process. The contribution of the article consists of presenting the geometric constraints that characterize this type of panelling from the idea of “rectifying surface”, considering the material inextensible. Experimental prototypes that have been part of the research are also described and the final theoretical results are presented

Interpolating sparse scattered data using flow information

Scattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field - as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow - then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic

Feature-based Product Modelling in a Collaborative Environment

Ph.DDOCTOR OF PHILOSOPH

Surface Reconstruction via Geodesic Interpolation

International audienceThis paper is concerned with reconstruction of numerical or real surfaces based on the knowledge of some geodesic curves on the surface. So, considering two regular 3D-curves f0(t) and f1(t), our purpose is to construct a surface which interpolates these two curves in such a way that these two curves are geodesics on this surface. This will be accomplished using Hermite interpolation. For a real surface, it will be shown that geodesics can be acquired using a ribbon of micro-sensors