338 research outputs found
B\'ezier representation of the constrained dual Bernstein polynomials
Explicit formulae for the B\'ezier coefficients of the constrained dual
Bernstein basis polynomials are derived in terms of the Hahn orthogonal
polynomials. Using difference properties of the latter polynomials, efficient
recursive scheme is obtained to compute these coefficients. Applications of
this result to some problems of CAGD is discussed.Comment: 10 page
Generating Surfaces of Variable Eccentricity Within a Ray Tracer
Polynomial surfaces used in ray tracing have recently been improved upon allowing for three dimensional applications. Among these are surfaces that have a varying eccentricity. This paper will discuss a method for finding real roots of polynomials [allowing us to create these surfaces]. First, we will give the reader a basic comprehension of the workings of a ray tracer, a general understanding of three dimensional polynomial surfaces, how this newly implemented root finder functions, and how these concepts enable us to create surfaces of variable eccentricity. Then, examples will be provided to demonstrate the capabilities of the program
MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves
This paper studies the problem of finding the smallest -simplex enclosing
a given -degree polynomial curve. Although the Bernstein and
B-Spline polynomial bases provide feasible solutions to this problem, the
simplexes obtained by these bases are not the smallest possible, which leads to
undesirably conservative results in many applications. We first prove that the
polynomial basis that solves this problem (MINVO basis) also solves for the
-degree polynomial curve with largest convex hull enclosed in a
given -simplex. Then, we present a formulation that is \emph{independent} of
the -simplex or -degree polynomial curve given. By using
Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we
obtain high-quality feasible solutions for any and prove
numerical global optimality for . The results obtained for show
that, for any given -degree polynomial curve, the MINVO basis is
able to obtain an enclosing simplex whose volume is and times
smaller than the ones obtained by the Bernstein and B-Spline bases,
respectively. When , these ratios increase to and
, respectively.Comment: 25 pages, 16 figure
Fast and exact continuous collision detection with Bernstein sign classification
We present fast algorithms to perform accurate CCD queries between triangulated models. Our formulation uses properties of the Bernstein basis and Bézier curves and reduces the problem to evaluating signs of polynomials. We present a geometrically exact CCD algorithm based on the exact geometric computation paradigm to perform reliable Boolean collision queries. Our algorithm is more than an order of magnitude faster than prior exact algorithms. We evaluate its performance for cloth and FEM simulations on CPUs and GPUs, and highlight the benefits
Compact Parameterized Black-Box Modeling via Fourier-Rational Approximations
We present a novel black-box modeling approach for frequency responses that depend on additional parameters with periodic behavior. The methodology is appropriate for representing with compact low-order equivalent models the behavior of electromagnetic systems observed at well-defined ports and/or locations, including dependence on geometrical parameters with rotational symmetry. Examples can be polarization or incidence angles of a plane wave, or stirrer rotation in reverberation chambers. The proposed approach is based on fitting a Fourier-rational model to sampled frequency responses, where frequency dependence is represented through rational functions and parameter dependence through a Fourier series. Several examples from different applications are used to validate and demonstrate the approach
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
New Models for High-Quality Surface Reconstruction and Rendering
The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface
New Models for High-Quality Surface Reconstruction and Rendering
The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface
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