Correcting curvature-density effects in the Hamilton-Jacobi skeleton

The Hainilton-Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method

Time-dependent Orbifolds and String Cosmology

In these lectures, we review the physics of time-dependent orbifolds of string theory, with particular attention to orbifolds of three-dimensional Minkowski space. We discuss the propagation of free particles in the orbifold geometries, together with their interactions. We address the issue of stability of these string vacua and the difficulties in defining a consistent perturbation theory, pointing to possible solutions. In particular, it is shown that resumming part of the perturbative expansion gives finite amplitudes. Finally we discuss the duality of some orbifold models with the physics of orientifold planes, and we describe cosmological models based on the dynamics of these orientifolds.Comment: 74 pages, 20 figures. Updated version of lectures at RTN Winter School on Strings, Supergravity and Gauge Theory - Torino, January 2003. Contains many new results. Minor correction

Non-Euclidean geometry in nature

I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random graphs. For these systems I discuss the following interlinked questions: (i) the optimal embedding of plants leaves in the three-dimensional space, (ii) the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to chaotic Hamiltonian systems is adde

Application of the fast marching method for outdoor motion planning in robotics

In this paper, a new path planning method for robots used in outdoor environments is presented. The proposed method applies Fast Marching to a 3D surface represented by a triangular mesh to calculate a smooth trajectory from one point to another. The method uses a triangular mesh instead of a square one since this kind of grid adapts better to 3D surfaces. The novelty of this approach is that, before running the algorithm, the method calculates a weight matrix W based on the information extracted from the 3D surface characteristics. In the presented experiments these features are the height, the spherical variance, and the gradient of the surface. This matrix can be viewed as a difficulty map situated over the 3D surface and is used to limit the propagation speed of the Fast Marching wave in order to find the best path depending on the task requirements, e.g., the least energy consumption path, the fastest path, or the most plain terrain. The algorithm also gives the speed for the robot, which depends on the wave front propagation speed. The results presented in this paper show how, by varying this matrix W, the paths obtained are different. Moreover, as it is shown in the experimental part, this algorithm is also useful for calculating paths for climbing robots in much more complex environments. Finally, at the end of the paper, it is shown that this algorithm can also be used for robot avoidance when two robots approach each other, and they know each other's position.Comunidad de Madri

Integrable theories and loop spaces: fundamentals, applications and new developments

We review our proposal to generalize the standard two-dimensional flatness construction of Lax-Zakharov-Shabat to relativistic field theories in d+1 dimensions. The fundamentals from the theory of connections on loop spaces are presented and clarified. These ideas are exposed using mathematical tools familiar to physicists. We exhibit recent and new results that relate the locality of the loop space curvature to the diffeomorphism invariance of the loop space holonomy. These result are used to show that the holonomy is abelian if the holonomy is diffeomorphism invariant. These results justify in part and set the limitations of the local implementations of the approach which has been worked out in the last decade. We highlight very interesting applications like the construction and the solution of an integrable four dimensional field theory with Hopf solitons, and new integrability conditions which generalize BPS equations to systems such as Skyrme theories. Applications of these ideas leading to new constructions are implemented in theories that admit volume preserving diffeomorphisms of the target space as symmetries. Applications to physically relevant systems like Yang Mills theories are summarized. We also discuss other possibilities that have not yet been explored.Comment: 64 pages, 8 figure

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

Efficient image-based rendering

Recent advancements in real-time ray tracing and deep learning have significantly enhanced the realism of computer-generated images. However, conventional 3D computer graphics (CG) can still be time-consuming and resource-intensive, particularly when creating photo-realistic simulations of complex or animated scenes. Image-based rendering (IBR) has emerged as an alternative approach that utilizes pre-captured images from the real world to generate realistic images in real-time, eliminating the need for extensive modeling. Although IBR has its advantages, it faces challenges in providing the same level of control over scene attributes as traditional CG pipelines and accurately reproducing complex scenes and objects with different materials, such as transparent objects. This thesis endeavors to address these issues by harnessing the power of deep learning and incorporating the fundamental principles of graphics and physical-based rendering. It offers an efficient solution that enables interactive manipulation of real-world dynamic scenes captured from sparse views, lighting positions, and times, as well as a physically-based approach that facilitates accurate reproduction of the view dependency effect resulting from the interaction between transparent objects and their surrounding environment. Additionally, this thesis develops a visibility metric that can identify artifacts in the reconstructed IBR images without observing the reference image, thereby contributing to the design of an effective IBR acquisition pipeline. Lastly, a perception-driven rendering technique is developed to provide high-fidelity visual content in virtual reality displays while retaining computational efficiency.Jüngste Fortschritte im Bereich Echtzeit-Raytracing und Deep Learning haben den Realismus computergenerierter Bilder erheblich verbessert. Konventionelle 3DComputergrafik (CG) kann jedoch nach wie vor zeit- und ressourcenintensiv sein, insbesondere bei der Erstellung fotorealistischer Simulationen von komplexen oder animierten Szenen. Das bildbasierte Rendering (IBR) hat sich als alternativer Ansatz herauskristallisiert, bei dem vorab aufgenommene Bilder aus der realen Welt verwendet werden, um realistische Bilder in Echtzeit zu erzeugen, so dass keine umfangreiche Modellierung erforderlich ist. Obwohl IBR seine Vorteile hat, ist es eine Herausforderung, das gleiche Maß an Kontrolle über Szenenattribute zu bieten wie traditionelle CG-Pipelines und komplexe Szenen und Objekte mit unterschiedlichen Materialien, wie z.B. transparente Objekte, akkurat wiederzugeben. In dieser Arbeit wird versucht, diese Probleme zu lösen, indem die Möglichkeiten des Deep Learning genutzt und die grundlegenden Prinzipien der Grafik und des physikalisch basierten Renderings einbezogen werden. Sie bietet eine effiziente Lösung, die eine interaktive Manipulation von dynamischen Szenen aus der realen Welt ermöglicht, die aus spärlichen Ansichten, Beleuchtungspositionen und Zeiten erfasst wurden, sowie einen physikalisch basierten Ansatz, der eine genaue Reproduktion des Effekts der Sichtabhängigkeit ermöglicht, der sich aus der Interaktion zwischen transparenten Objekten und ihrer Umgebung ergibt. Darüber hinaus wird in dieser Arbeit eine Sichtbarkeitsmetrik entwickelt, mit der Artefakte in den rekonstruierten IBR-Bildern identifiziert werden können, ohne das Referenzbild zu betrachten, und die somit zur Entwicklung einer effektiven IBR-Erfassungspipeline beiträgt. Schließlich wird ein wahrnehmungsgesteuertes Rendering-Verfahren entwickelt, um visuelle Inhalte in Virtual-Reality-Displays mit hoherWiedergabetreue zu liefern und gleichzeitig die Rechenleistung zu erhalten

Screened poisson hyperfields for shape coding

We present a novel perspective on shape characterization using the screened Poisson equation. We discuss that the effect of the screening parameter is a change of measure of the underlying metric space. Screening also indicates a conditioned random walker biased by the choice of measure. A continuum of shape fields is created by varying the screening parameter or, equivalently, the bias of the random walker. In addition to creating a regional encoding of the diffusion with a different bias, we further break down the influence of boundary interactions by considering a number of independent random walks, each emanating from a certain boundary point, whose superposition yields the screened Poisson field. Probing the screened Poisson equation from these two complementary perspectives leads to a high-dimensional hyperfield: a rich characterization of the shape that encodes global, local, interior, and boundary interactions. To extract particular shape information as needed in a compact way from the hyperfield, we apply various decompositions either to unveil parts of a shape or parts of a boundary or to create consistent mappings. The latter technique involves lower-dimensional embeddings, which we call screened Poisson encoding maps (SPEM). The expressive power of the SPEM is demonstrated via illustrative experiments as well as a quantitative shape retrieval experiment over a public benchmark database on which the SPEM method shows a high-ranking performance among the existing state-of-the-art shape retrieval methods