Forcing nonperiodicity with a single tile

An aperiodic prototile is a shape for which infinitely many copies can be arranged to fill Euclidean space completely with no overlaps, but not in a periodic pattern. Tiling theorists refer to such a prototile as an "einstein" (a German pun on "one stone"). The possible existence of an einstein has been pondered ever since Berger's discovery of large set of prototiles that in combination can tile the plane only in a nonperiodic way. In this article we review and clarify some features of a prototile we recently introduced that is an einstein according to a reasonable definition. [This abstract does not appear in the published article.]Comment: 18 pages, 10 figures. This article has been substantially revised and accepted for publication in the Mathematical Intelligencer and is scheduled to appear in Vol 33. Citations to and quotations from this work should reference that publication. If you cite this work, please check that the published form contains precisely the material to which you intend to refe

Programmation et indécidabilités dans les systèmes complexes

N/AUn système complexe est un système constitué d'un ensemble d'entités quiinteragissent localement, engendrant des comportements globaux, émergeant dusystème, qu'on ne sait pas expliquer à partir du comportement local, connu, desentités qui le constituent. Nos travaux ont pour objet de mieux cerner lesliens entre certaines propriétés des systèmes complexes et le calcul. Parcalcul, il faut entendre l'objet d'étude de l'informatique, c'est-à-dire ledéplacement et la combinaison d'informations. À l'aide d'outils issus del'informatique, l'algorithmique et la programmation dans les systèmes complexessont abordées selon trois points de vue. Une première forme de programmation,dite externe, consiste à développer l'algorithmique qui permet de simuler lessystèmes étudiés. Une seconde forme de programmation, dite interne, consiste àdévelopper l'algorithmique propre à ces systèmes, qui permet de construire desreprésentants de ces systèmes qui exhibent des comportements programmés. Enfin,une troisième forme de programmation, de réduction, consiste à plonger despropriétés calculatoires complexes dans les représentants de ces systèmes pourétablir des résultats d'indécidabilité -- indice d'une grande complexitécalculatoire qui participe à l'explication de la complexité émergente. Afin demener à bien cette étude, les systèmes complexes sont modélisés par desautomates cellulaires. Le modèle des automates cellulaires offre une dualitépertinente pour établir des liens entre complexité des propriétés globales etcalcul. En effet, un automate cellulaire peut être décrit à la fois comme unréseau d'automates, offrant un point de vue familier de l'informatique, etcomme un système dynamique discret, une fonction définie sur un espacetopologique, offrant un point de vue familier de l'étude des systèmesdynamiques discrets.Une première partie de nos travaux concerne l'étude de l'objet automatecellulaire proprement dit. L'observation expérimentale des automatescellulaires distingue, dans la littérature, deux formes de dynamiques complexesdominantes. Certains automates cellulaires présentent une dynamique danslaquelle émergent des structures simples, sortes de particules qui évoluentdans un domaine régulier, se rencontrent lors de brèves collisions, avant degénérer d'autres particules. Cette forme de complexité, dans laquelletransparaît une notion de quanta d'information localisée en interaction, estl'objet de nos études. Un premier champ de nos investigations est d'établir uneclassification algébrique, le groupage, qui tend à rendre compte de ce type decomportement. Cette classification met à jour un type d'automate cellulaireparticulier : les automates cellulaires intrinsèquement universels. Un automatecellulaire intrinsèquement universel est capable de simuler le comportement detout automate cellulaire. C'est l'objet de notre second champ d'investigation.Nous caractérisons cette propriété et démontrons son indécidabilité. Enfin, untroisième champ d'investigation concerne l'algorithmique des automatescellulaires à particules et collisions. Étant donné un ensemble de particuleset de collisions d'un tel automate cellulaire, nous étudions l'ensemble desinteractions possibles et proposons des outils pour une meilleure programmationinterne à l'aide de ces collisions.Une seconde partie de nos travaux concerne la programmation par réduction. Afinde démontrer l'indécidabilité de propriétés dynamiques des automatescellulaires, nous étudions d'une part les problèmes de pavage du plan par desjeux de tuiles finis et d'autre part les problèmes de mortalité et depériodicité dans les systèmes dynamiques discrets à fonction partielle. Cetteétude nous amène à considérer des objets qui possèdent la même dualité entredescription combinatoire et topologique que les automates cellulaires. Unenotion d'apériodicité joue un rôle central dans l'indécidabilité des propriétésde ces objets

Shape Design and Optimization for 3D Printing

In recent years, the 3D printing technology has become increasingly popular, with wide-spread uses in rapid prototyping, design, art, education, medical applications, food and fashion industries. It enables distributed manufacturing, allowing users to easily produce customized 3D objects in office or at home. The investment in 3D printing technology continues to drive down the cost of 3D printers, making them more affordable to consumers. As 3D printing becomes more available, it also demands better computer algorithms to assist users in quickly and easily generating 3D content for printing. Creating 3D content often requires considerably more efforts and skills than creating 2D content. In this work, I will study several aspects of 3D shape design and optimization for 3D printing. I start by discussing my work in geometric puzzle design, which is a popular application of 3D printing in recreational math and art. Given user-provided input figures, the goal is to compute the minimum (or best) set of geometric shapes that can satisfy the given constraints (such as dissection constraints). The puzzle design also has to consider feasibility, such as avoiding interlocking pieces. I present two optimization-based algorithms to automatically generate customized 3D geometric puzzles, which can be directly printed for users to enjoy. They are also great tools for geometry education. Next, I discuss shape optimization for printing functional tools and parts. Although current 3D modeling software allows a novice user to easily design 3D shapes, the resulting shapes are not guaranteed to meet required physical strength. For example, a poorly designed stool may easily collapse when a person sits on the stool; a poorly designed wrench may easily break under force. I study new algorithms to help users strengthen functional shapes in order to meet specific physical properties. The algorithm uses an optimization-based framework — it performs geometric shape deformation and structural optimization iteratively to minimize mechanical stresses in the presence of forces assuming typical use scenarios. Physically-based simulation is performed at run-time to evaluate the functional properties of the shape (e.g., mechanical stresses based on finite element methods), and the optimizer makes use of this information to improve the shape. Experimental results show that my algorithm can successfully optimize various 3D shapes, such as chairs, tables, utility tools, to withstand higher forces, while preserving the original shape as much as possible. To improve the efficiency of physics simulation for general shapes, I also introduce a novel, SPH-based sampling algorithm, which can provide better tetrahedralization for use in the physics simulator. My new modeling algorithm can greatly reduce the design time, allowing users to quickly generate functional shapes that meet required physical standards

Fabrication-Aware Design with Performative Criteria

Artists and architects often need to handle multiple constraints during design of physical constructions. We define a performative constraint as any constraint on design that is tied to the performance of the model--either during fabrication, construction, daily use, or destruction. Even for small to medium scale models, there are functional criteria such as the ease of fabrication and the assembly process, or even the interplay of light with the material. Computational tools can greatly aid in this process, assisting with the lower-level performative constraints, while the designer handles the high-level artistic decisions. Additionally, using new fabrication methods, our tools can aid in lowering the difficulty of building complex constructions, making them accessible to hobbyists. In this thesis, we present three computational methods for designing with different approaches, each with a different material, fabrication method, and use case. The first method is a construction with intersecting planar pieces that can be laser cut or milled. These 3D forms are assembled by sliding pieces into each other along straight slits, and do not require other support such as glue or screws. We present a mathematical abstraction that formalizes the constraints between pieces as a graph, including fabrication and assembly constraints, and ensure global rigidity of the sculpture. We also propose an optimization algorithm to guide the user using automatic constraint satisfaction based on analysis of the constraint relation graph. We demonstrate our approach by creating several small- to medium-scale examples including functional furniture. The second method presents a solution to building a 3D sculpture out of existing building blocks that can be found in many homes. Starting from the voxelization of a 3D mesh we merge voxels to form larger bricks, and then analyze and repair structural problems based on a graph representation of the block connections. We then output layer-by-layer building instructions to allow a user to quickly and easily build the model. We also present extensions such as hollowing the models to use less bricks, limiting the number of bricks of each size, and including color constraints. We present both real and virtual brick constructions and associated timings, showing improvements over previous work. The final case presented tackles the inverse design problem of finding a surface to produce a target caustic on a receiver plane when light is refracted or reflected. This is an example where the performative constraint is the principal driver of the design. We introduce an optimal transport formulation to find a correspondence between the incoming light and the output target light distribution. We then show a 3D optimization that finds the surface that transports light based on the correspondence map. Our approach supports piecewise smooth surfaces that are as smooth as possible but allow for creases, to greatly reduce the amount of artifacts while allowing light to be completely diverted producing completely black regions. We show how this leads to a very large space of high-contrast, high-resolution caustic images, including point and line singularities of infinite light density as well as photo-realistic images. Our approach leads to surfaces that can be milled using standard CNC milling. We demonstrate the approach showing both simulated and fabricated examples

Compression, pose tracking, and halftoning

In this thesis, we discuss image compression, pose tracking, and halftoning. Although these areas seem to be unrelated at first glance, they can be connected through video coding as application scenario. Our first contribution is an image compression algorithm based on a rectangular subdivision scheme which stores only a small subsets of the image points. From these points, the remained of the image is reconstructed using partial differential equations. Afterwards, we present a pose tracking algorithm that is able to follow the 3-D position and orientation of multiple objects simultaneously. The algorithm can deal with noisy sequences, and naturally handles both occlusions between different objects, as well as occlusions occurring in kinematic chains. Our third contribution is a halftoning algorithm based on electrostatic principles, which can easily be adjusted to different settings through a number of extensions. Examples include modifications to handle varying dot sizes or hatching. In the final part of the thesis, we show how to combine our image compression, pose tracking, and halftoning algorithms to novel video compression codecs. In each of these four topics, our algorithms yield excellent results that outperform those of other state-of-the-art algorithms.In dieser Arbeit werden die auf den ersten Blick vollkommen voneinander unabhängig erscheinenden Bereiche Bildkompression, 3D-Posenschätzung und Halbtonverfahren behandelt und im Bereich der Videokompression sinnvoll zusammengeführt. Unser erster Beitrag ist ein Bildkompressionsalgorithmus, der auf einem rechteckigen Unterteilungsschema basiert. Dieser Algorithmus speichert nur eine kleine Teilmenge der im Bild vorhandenen Punkte, während die restlichen Punkte mittels partieller Differentialgleichungen rekonstruiert werden. Danach stellen wir ein Posenschätzverfahren vor, welches die 3D-Position und Ausrichtung von mehreren Objekten anhand von Bilddaten gleichzeitig verfolgen kann. Unser Verfahren funktioniert bei verrauschten Videos und im Falle von Objektüberlagerungen. Auch Verdeckungen innerhalb einer kinematischen Kette werden natürlich behandelt. Unser dritter Beitrag ist ein Halbtonverfahren, das auf elektrostatischen Prinzipien beruht. Durch eine Reihe von Erweiterungen kann dieses Verfahren flexibel an verschiedene Szenarien angepasst werden. So ist es beispielsweise möglich, verschiedene Punktgrößen zu verwenden oder Schraffuren zu erzeugen. Der letzte Teil der Arbeit zeigt, wie man unseren Bildkompressionsalgorithmus, unser Posenschätzverfahren und unser Halbtonverfahren zu neuen Videokompressionsalgorithmen kombinieren kann. Die für jeden der vier Themenbereiche entwickelten Verfahren erzielen hervorragende Resultate, welche die Ergebnisse anderer moderner Verfahren übertreffen

Hierarchical Variance Reduction Techniques for Monte Carlo Rendering

Ever since the first three-dimensional computer graphics appeared half a century ago, the goal has been to model and simulate how light interacts with materials and objects to form an image. The ultimate goal is photorealistic rendering, where the created images reach a level of accuracy that makes them indistinguishable from photographs of the real world. There are many applications ñ visualization of products and architectural designs yet to be built, special effects, computer-generated films, virtual reality, and video games, to name a few. However, the problem has proven tremendously complex; the illumination at any point is described by a recursive integral to which a closed-form solution seldom exists. Instead, computer simulation and Monte Carlo methods are commonly used to statistically estimate the result. This introduces undesirable noise, or variance, and a large body of research has been devoted to finding ways to reduce the variance. I continue along this line of research, and present several novel techniques for variance reduction in Monte Carlo rendering, as well as a few related tools. The research in this dissertation focuses on using importance sampling to pick a small set of well-distributed point samples. As the primary contribution, I have developed the first methods to explicitly draw samples from the product of distant high-frequency lighting and complex reflectance functions. By sampling the product, low noise results can be achieved using a very small number of samples, which is important to minimize the rendering times. Several different hierarchical representations are explored to allow efficient product sampling. In the first publication, the key idea is to work in a compressed wavelet basis, which allows fast evaluation of the product. Many of the initial restrictions of this technique were removed in follow-up work, allowing higher-resolution uncompressed lighting and avoiding precomputation of reflectance functions. My second main contribution is to present one of the first techniques to take the triple product of lighting, visibility and reflectance into account to further reduce the variance in Monte Carlo rendering. For this purpose, control variates are combined with importance sampling to solve the problem in a novel way. A large part of the technique also focuses on analysis and approximation of the visibility function. To further refine the above techniques, several useful tools are introduced. These include a fast, low-distortion map to represent (hemi)spherical functions, a method to create high-quality quasi-random points, and an optimizing compiler for analyzing shaders using interval arithmetic. The latter automatically extracts bounds for importance sampling of arbitrary shaders, as opposed to using a priori known reflectance functions. In summary, the work presented here takes the field of computer graphics one step further towards making photorealistic rendering practical for a wide range of uses. By introducing several novel Monte Carlo methods, more sophisticated lighting and materials can be used without increasing the computation times. The research is aimed at domain-specific solutions to the rendering problem, but I believe that much of the new theory is applicable in other parts of computer graphics, as well as in other fields