Currents and finite elements as tools for shape space

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples

Efficient quadrature rules for subdivision surfaces in isogeometric analysis

We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions. The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of $(1\times n)$ quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration. Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull-Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including non-uniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull-Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis

Multiresolution curve editing with linear constraints

The use of multiresolution control toward the editing of freeform curves and surfaces has already been recognized as a valuable modeling too

Quadrupole moment of a magnetically confined mountain on an accreting neutron star: effect of the equation of state

Magnetically confined mountains on accreting neutron stars are promising sources of continuous-wave gravitational radiation and are currently the targets of directed searches with long-baseline detectors like the Laser Interferometer Gravitational Wave Observatory (LIGO). In this paper, previous ideal-magnetohydrodynamic models of isothermal mountains are generalized to a range of physically motivated, adiabatic equations of state. It is found that the mass ellipticity drops substantially, from \epsilon ~ 3e-4 (isothermal) to \epsilon ~ 9e-7 (non-relativistic degenerate neutrons), 6e-8 (relativistic degenerate electrons) and 1e-8 (non-relativistic degenerate electrons) (assuming a magnetic field of 3e12 G at birth). The characteristic mass M_{c} at which the magnetic dipole moment halves from its initial value is also modified, from M_{c}/M_{\sun} ~ 5e-4 (isothermal) to M_{c}/M_{\sun} ~ 2e-6, 1e-7, and 3e-8 for the above three equations of state, respectively. Similar results are obtained for a realistic, piecewise-polytropic nuclear equation of state. The adiabatic models are consistent with current LIGO upper limits, unlike the isothermal models. Updated estimates of gravitational-wave detectability are made. Monte Carlo simulations of the spin distribution of accreting millisecond pulsars including gravitational-wave stalling agree better with observations for certain adiabatic equations of state, implying that X-ray spin measurements can probe the equation of state when coupled with magnetic mountain models.Comment: 20 pages, 15 figures, to be published in MNRA

Part Description and Segmentation Using Contour, Surface and Volumetric Primitives

The problem of part definition, description, and decomposition is central to the shape recognition systems. The Ultimate goal of segmenting range images into meaningful parts and objects has proved to be very difficult to realize, mainly due to the isolation of the segmentation problem from the issue of representation. We propose a paradigm for part description and segmentation by integration of contour, surface, and volumetric primitives. Unlike previous approaches, we have used geometric properties derived from both boundary-based (surface contours and occluding contours), and primitive-based (quadric patches and superquadric models) representations to define and recover part-whole relationships, without a priori knowledge about the objects or object domain. The object shape is described at three levels of complexity, each contributing to the overall shape. Our approach can be summarized as answering the following question : Given that we have all three different modules for extracting volume, surface and boundary properties, how should they be invoked, evaluated and integrated? Volume and boundary fitting, and surface description are performed in parallel to incorporate the best of the coarse to fine and fine to coarse segmentation strategy. The process involves feedback between the segmentor (the Control Module) and individual shape description modules. The control module evaluates the intermediate descriptions and formulates hypotheses about parts. Hypotheses are further tested by the segmentor and the descriptors. The descriptions thus obtained are independent of position, orientation, scale, domain and domain properties, and are based purely on geometric considerations. They are extremely useful for the high level domain dependent symbolic reasoning processes, which need not deal with tremendous amount of data, but only with a rich description of data in terms of primitives recovered at various levels of complexity

Multilevel fast multipole algorithm for fields

An efficient implementation of the multilevel fast multipole algorithm is herein applied to accelerate the calculation of the electromagnetic near- and far-fields after the equivalent surface currents have been obtained. In spite of all the research efforts being drawn to the latter, the electric and/or magnetic fields (or other parameters derived from these) are ultimately the magnitudes of interest in most of the cases. Though straightforward, their calculation can be computationally demanding, and hence the importance of finding a sped-up accurate representation of the fields via a suitable setup of the method. A complete self-contained formulation for both near- and far-fields and for problems including multiple penetrable regions is shown in full detail. Through numerical examples we show that the efficiency and scalability of the implementation leads to a drastic reduction of the computation time.Ministerio de Economía y Competitividad | Ref. MAT2014-58201-C2-1-RMinisterio de Economía y Competitividad | Ref. MAT2014-58201-C2-2-RGobierno Regional de Extremadura | Ref. IB1318

Continuous collision detection for ellipsoids

We present an accurate and efficient algorithm for continuous collision detection between two moving ellipsoids. We start with a highly optimized implementation of interference testing between two stationary ellipsoids based on an algebraic condition described in terms of the signs of roots of the characteristic equation of two ellipsoids. Then we derive a time-dependent characteristic equation for two moving ellipsoids, which enables us to develop a real-time algorithm for computing the time intervals in which two moving ellipsoids collide. The effectiveness of our approach is demonstrated with several practical examples. © 2006 IEEE.published_or_final_versio

Parametric Paraglider Modeling

Dynamic simulations are invaluable for studying system behavior, developing control models, and running statistical analyses. For example, paraglider flight simulations could be used to analyze how a wing behaves when it encounters wind shear, or to reconstruct the wind field that was present during a flight. Unfortunately, creating dynamics models for commercial paraglider wings is difficult: not only are detailed specifications unavailable, but even if they were, a detailed model would be laborious to create. To address that difficulty, this project develops a paraglider flight dynamics model that uses parametric components to model commercial paraglider wings given only limited technical specifications and knowledge of typical wing design. To validate the model design and implementation, an aerodynamic simulation of a reference paraglider canopy is compared to wind tunnel measurements, and a dynamic simulation of a commercial paraglider system is compared to basic flight test data. The entirety of the models and example wings are available as an open source library built on the Python scientific computing stack