Polytopes, Feasible Regions and Occlusions in the n-view Reconstruction Problem

This paper assesses the question, given a arbitrary point in P3, can it be reconstructed by a given camera orbit? We show that a solution to this problem can be found by intersecting the frustrums of the cameras in the sequence creating a polyhedron that bounds the area in P3 observed by all cameras. For a projective set of cameras this can be considered as an expansion of the chetral inequalities. We also show an exception to this basic principle is encounted when the point in P3 is occluded. Thus giving a weak condition for occlusion of an arbitrary point in P3

Doctor of Philosophy

dissertationVolumetric parameterization is an emerging field in computer graphics, where volumetric representations that have a semi-regular tensor-product structure are desired in applications such as three-dimensional (3D) texture mapping and physically-based simulation. At the same time, volumetric parameterization is also needed in the Isogeometric Analysis (IA) paradigm, which uses the same parametric space for representing geometry, simulation attributes and solutions. One of the main advantages of the IA framework is that the user gets feedback directly as attributes of the NURBS model representation, which can represent geometry exactly, avoiding both the need to generate a finite element mesh and the need to reverse engineer the simulation results from the finite element mesh back into the model. Research in this area has largely been concerned with issues of the quality of the analysis and simulation results assuming the existence of a high quality volumetric NURBS model that is appropriate for simulation. However, there are currently no generally applicable approaches to generating such a model or visualizing the higher order smooth isosurfaces of the simulation attributes, either as a part of current Computer Aided Design or Reverse Engineering systems and methodologies. Furthermore, even though the mesh generation pipeline is circumvented in the concept of IA, the quality of the model still significantly influences the analysis result. This work presents a pipeline to create, analyze and visualize NURBS geometries. Based on the concept of analysis-aware modeling, this work focusses in particular on methodologies to decompose a volumetric domain into simpler pieces based on appropriate midstructures by respecting other relevant interior material attributes. The domain is decomposed such that a tensor-product style parameterization can be established on the subvolumes, where the parameterization matches along subvolume boundaries. The volumetric parameterization is optimized using gradient-based nonlinear optimization algorithms and datafitting methods are introduced to fit trivariate B-splines to the parameterized subvolumes with guaranteed order of accuracy. Then, a visualization method is proposed allowing to directly inspect isosurfaces of attributes, such as the results of analysis, embedded in the NURBS geometry. Finally, the various methodologies proposed in this work are demonstrated on complex representations arising in practice and research

Interactive ray tracing of arbitrary implicits with SIMD interval arithmetic

Journal ArticleWe present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions

Détection d’intersection via l’application de Gauss; revue et nouvelles techniques

This paper delves into the problem of detecting the intersection of two convexpolyhedra.It does so through the lens of Minkowski sums and Gauss maps, and with a biastowards applications in computer graphics and robotics.In the first part, we show how Minkowski sums and Gauss maps come into play,give a brief survey of techniques for pairs of simple shapes and describe alow-level optimization of a naive algorithm for convex polyhedra, which isapplied to tetrahedra.Novel applications to the ray casting problem are also given.In the second part, we take a more abstract approach to the problem anddescribe a new and very efficient and robust algorithm for detecting theintersection of two convex shapes.The new technique works directly on the unit sphere, interpreted as the sphereof directions.In particular, it is compared favourably to the ubiquitous algorithm ofGilbert, Johnson and Keerthi.Cet article discute du problème (décisionnel) de la détection de l'intersectionde deux polyèdres convexes. Il porte particulièrement sur les applications dece problème en informatique graphique et en robotique. La discussion s'y faitdu point de vue des sommes de Minkoswki et de l'application de Gauss.Dans la première partie, nous rappellons le rôle de ce point de vue dans lacompréhension de la géométrie du problème. Nous donnons un bref aperçu destechniques conçues pour certaines paires de formes simples, et nous proposonsun algorithme naïf mais optimisé, traitant des polyèdres convexes quelconques.Nous traitons en exemple une application aux paires de tétraèdres et uneapplication au problème du lancer de rayons.En deuxième partie, nous approchons le problème de manière plus abstraite etdécrivons un nouvel algorithme robuste et rapide pour la détection del'intersection de deux objets convexes (non nécessairement polyédrique).Ce nouvel algorithme travaille directement sur la sphère unité que nousinterprétons comme l'espace des directions. En particulier, notre nouvelletechnique est comparée favorablement à celle, fort répandue, de Gilbert,Johnson et Keerthi

Visualization of Industrial Structures with Implicit GPU Primitives

International audienceWe present a method to interactively visualize large industrial models by replacing most triangles with implicit GPU primitives: cylinders, cone and torus slices. After a reverse-engineering process that recovers these primitives from triangle meshes, we encode their implicit parameters in a texture that is sent to the GPU. In rendering time, the implicit primitives are visualized seamlessly with other triangles in the scene. The method was tested on two massive industrial models, achieving better performance and image quality while reducing memory use

Non-Euclidean Shells: A Study of Growth-Induced Fabrication and Mechanical Multi-Stability

Non-Euclidean shells are ubiquitous in both natural and man-made systems, yet fabrication at smaller length-scales (nanometer to micrometer) and the underlying mechanical behavior of these geometries is not well understood. In this dissertation, we develop a framework for improvements in fabrication and control over deformation pathways of non-Euclidean elastic shells. For programming a target non-Euclidean geometry, we study the non-uniform growth induced buckling in flat sheets. To deepen our understanding of this powerful mechanism, we present an experimental verification of its mathematical equivalence with a mechanism involving topological defects. We establish a framework for correlating topological defect-induced buckling, realized through a simple cut-and-glue construction, with growth induced buckling realized through non-uniform growth of patterned 2D hydrogel sheets. Validating the obtained mathematical results, we demonstrate fabrication of a cylindrical and conical dipole geometry through both mechanisms under consideration. In addition, upon applying a similar treatment on tetrahedron geometry we realize the limitations of growth-induced buckling mechanism for programming 3D geometry, and find that optimizing for pattern resolution and swelling range can lead to a higher fidelity in target geometries. Next, we turn towards the interplay between geometry and mechanics in non-Euclidean shells, to harness multi-stability between different geometric configurations. Under this theme, we study deformation of arbitrarily curved surfaces along pre-defined creases (curves with local weakening) finding that the continuity of this deformation can be predicted through a simple consideration of the curvature of the crease. We establish that simple geometric control over the crease curvature can be used to program on-demand snap-through instabilities between bi-stable states of developable, elliptic and hyperbolic surfaces. Using experiments, FEA and theoretical analysis of bending and stretching energies involved while indenting a hemispherical shell, we establish the geometric phase space in which the isometric state of a creased sphere is stable. Finally, we extend this notion by considering the ability to program multiple stable states in a tiled conical geometry, commonly found in bendable drinking straws (bendy straws) and other similar structures. These corrugated structures exhibit stability in axial (resulting in change in length), non-axial (change in ‘bent’ angle) and azimuthal direction (change in azimuthal angle) imparting a desirable high-degree of freedom in possible configurations. By analyzing the stability behavior of elastic double conical frusta shells, we study the necessary conditions for programming multi-stability, and find that axial stability depends on geometrical parameters. Interestingly, we conclude that a stable non-axial state requires a combination of appropriate geometry and presence of a pre-stress in azimuthal direction. The controlled multi-stability opens pathways toward a truly re-configurable shape programmable system, useful for manipulators and actuators in soft-robotic

Interactive ray tracing for volume visualization

Journal ArticleWe present a brute-force ray tracing system for interactive volume visualization, The system runs on a conventional (distributed) shared-memory multiprocessor machine. For each pixel we trace a ray through a volume to compute the color for that pixel. Although this method has high intrinsic computational cost, its simplicity and scalability make it ideal for large datasets on current high-end parallel systems