Interactive Hausdorff distance computation for general polygonal models

Figure 1: Interactive Hausdorff Distance Computation. Our algorithm can compute Hausdorff distance between complicated models at interactive rates (the first three figures). Here, the green line denotes the Hausdorff distance. This algorithm can also be used to find penetration depth (PD) for physically-based animation (the last two figures). It takes only a few milli-seconds to run on average. We present a simple algorithm to compute the Hausdorff distance between complicated, polygonal models at interactive rates. The algorithm requires no assumptions about the underlying topology and geometry. To avoid the high computational and implementa-tion complexity of exact Hausdorff distance calculation, we approx-imate the Hausdorff distance within a user-specified error bound. The main ingredient of our approximation algorithm is a novel polygon subdivision scheme, called Voronoi subdivision, combined with culling between the models based on bounding volume hier-archy (BVH). This cross-culling method relies on tight yet simple computation of bounds on the Hausdorff distance, and it discards unnecessary polygon pairs from each of the input models alterna-tively based on the distance bounds. This algorithm can approxi-mate the Hausdorff distance between polygonal models consisting of tens of thousands triangles with a small error bound in real-time, and outperforms the existing algorithm by more than an order of magnitude. We apply our Hausdorff distance algorithm to the mea-surement of shape similarity, and the computation of penetration depth for physically-based animation. In particular, the penetration depth computation using Hausdorff distance runs at highly interac-tive rates for complicated dynamics scene

Discrete Geometry

[no abstract available

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Adaptive Sampling for Geometric Approximation

Geometric approximation of multi-dimensional data sets is an essential algorithmic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an epsilon-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1+epsilon) times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. Combined with a careful regularization of the distance minimizers, we obtain a generalized data structure that essentially matches state-of-the-art results specific to the Euclidean distance. Finally, we investigate the generation of Voronoi meshes, where a given domain is decomposed into Voronoi cells as desired for a number of important solvers in computational fluid dynamics. The challenge is to arrange the cells near the boundary to yield an accurate surface approximation without sacrificing quality. We propose a sampling algorithm for the placement of seeds to induce a boundary-conforming Voronoi mesh of the correct topology, with a careful treatment of sharp and non-manifold features. The proposed algorithm achieves significant quality improvements over state-of-the-art polyhedral meshing based on clipped Voronoi cells

Discrete quantum geometries and their effective dimension

In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension $d$ on low energy scales to a real number $0<\alpha<d$ on high energy scales. In the particular case of $\alpha=1$ these results allow to understand the quantum geometry as effectively fractal.Comment: PhD thesis, Humboldt-Universit\"at zu Berlin; urn:nbn:de:kobv:11-100232371; http://edoc.hu-berlin.de/docviews/abstract.php?id=4204

Chopper: Partitioning models into 3D-printable parts

3D printing technology is rapidly maturing and becoming ubiquitous. One of the remaining obstacles to wide-scale adoption is that the object to be printed must fit into the working volume of the 3D printer. We propose a framework, called Chopper, to decompose a large 3D object into smaller parts so that each part fits into the printing volume. These parts can then be assembled to form the original object. We formulate a number of desirable criteria for the partition, including assemblability, having few components, unobtrusiveness of the seams, and structural soundness. Chopper optimizes these criteria and generates a partition either automatically or with user guidance. Our prototype outputs the final decomposed parts with customized connectors on the interfaces. We demonstrate the effectiveness of Chopper on a variety of non-trivial real-world objects.National Science Foundation (U.S.) (Grant CCF-1012147)National Science Foundation (U.S.) (Grant IIS-1116296)Intel Corporation (Science and Technology Center for Visual Computing

Multi-Black-Hole Configurations as Models for Inhomogeneous Cosmologies

On the largest scales, the Universe is homogeneous and isotropic, whereas on smaller scales, various structures immediately begin to emerge. The transition from an inhomogeneous spacetime to the homogeneous and isotropic Friedmann universe is not sufficiently understood yet. Modern cosmology rests on the hypothesis that the LambdaCDM-model applies and, indeed, this model is very successful. On the other hand, as the precision of observations steadily increases, it is more than likely that inhomogeneities will no longer be negligible in the future. For this reason, the study of inhomogeneous cosmological models is reasonable. In this thesis, we consider the question which Friedmann universe is the best fit to a particular given inhomogeneous spacetime, which is known as the fitting problem. We consider models in which matter is replaced by a discrete configuration of black holes, that is, we concentrate on vacuum solutions to Einstein's equations. Since the full system of the field equations is too complicated to find an exact time-dependent solution for the whole spacetime, we restrict ourselves to approximative models as well as solutions to the initial value problem. In the former case, we reconsider Swiss-cheese and Lindquist-Wheeler models. In both models, the spacetime around a mass is described by the Schwarzschild metric. In the latter case, we determine the spatial metric of a space-like hypersurface. We limit our attention to time-symmetric initial data characterised by the vanishing of the extrinsic curvature. In this case, we are able to find a solution for an arbitrary number of black holes using the conformal method. Clearly, it is not reasonable to assume that every configuration of black holes leads to a spacetime which may be approximated well by a Friedmann solution. Such an approximation should be possible if the masses are distributed somehow uniformly. The aim of this thesis is to clarify this statement and to provide criteria which allow quantitative statements about the degree of uniformity. We determine the parameters of the fitted dust universe, in particular the scale factor. Our considerations are supported by several example configurations. In particular, we provide a new method based on Lie sphere geometry to construct various configurations with a high degree of uniformity in a surprisingly simple fashion. Moreover, we provide a generalisation to an approximative inhomogeneous model given by Lindquist and Wheeler. In this case, it is possible to determine the parameters of the fitted Friedmann universe even if we do not know the exact solution. Under certain conditions, this model becomes similar to a Swiss-cheese model, allowing us to formulate first expectations on the time evolution, which is otherwise mostly disregarded within the framework of this thesis