2,056 research outputs found
A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve
We derive a variational model to fit a composite B\'ezier curve to a set of
data points on a Riemannian manifold. The resulting curve is obtained in such a
way that its mean squared acceleration is minimal in addition to remaining
close the data points. We approximate the acceleration by discretizing the
squared second order derivative along the curve. We derive a closed-form,
numerically stable and efficient algorithm to compute the gradient of a
B\'ezier curve on manifolds with respect to its control points, expressed as a
concatenation of so-called adjoint Jacobi fields. Several examples illustrate
the capabilites and validity of this approach both for interpolation and
approximation. The examples also illustrate that the approach outperforms
previous works tackling this problem
Interpolation in waveform space: enhancing the accuracy of gravitational waveform families using numerical relativity
Matched-filtering for the identification of compact object mergers in
gravitational-wave antenna data involves the comparison of the data stream to a
bank of template gravitational waveforms. Typically the template bank is
constructed from phenomenological waveform models since these can be evaluated
for an arbitrary choice of physical parameters. Recently it has been proposed
that singular value decomposition (SVD) can be used to reduce the number of
templates required for detection. As we show here, another benefit of SVD is
its removal of biases from the phenomenological templates along with a
corresponding improvement in their ability to represent waveform signals
obtained from numerical relativity (NR) simulations. Using these ideas, we
present a method that calibrates a reduced SVD basis of phenomenological
waveforms against NR waveforms in order to construct a new waveform approximant
with improved accuracy and faithfulness compared to the original
phenomenological model. The new waveform family is given numerically through
the interpolation of the projection coefficients of NR waveforms expanded onto
the reduced basis and provides a generalized scheme for enhancing
phenomenological models.Comment: 10 pages, 7 figure
Polynomial-based non-uniform interpolatory subdivision with features control
Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present
an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge
parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm
that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation
method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique
in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special
features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired
undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that
the most convenient parameter values may be chosen as well as the intervals for insertion.
Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control
Low level direct interpolation for parametric curves
We present an algorithm for the direct interpolation of parametric curves with a CNC machine -- The algorithm expresses parametric planar curves as sequences of discrete axes movements of BLU size of the machining tool -- Therefore, the curve C(u) is directly approximated by the pulse trains, hence eliminating one source of the machining error
Low level direct interpolation for parametric curves
This article presents an algorithm for the direct interpolation of parametric planar curves C(u) with a CNC machine. It expresses parametric planar curves as sequences of machine tool axes discrete movements of BLU size. Therefore, the curve C(u) is directly approximated by the pulse trains, hence eliminating one source of the machining errors
Light-Element Abundance Variations at Low Metallicity: the Globular Cluster NGC 5466
We present low-resolution (R~850) spectra for 67 asymptotic giant branch
(AGB), horizontal branch and red giant branch (RGB) stars in the
low-metallicity globular cluster NGC 5466, taken with the VIRUS-P
integral-field spectrograph at the 2.7-m Harlan J. Smith telescope at McDonald
Observatory. Sixty-six stars are confirmed, and one rejected, as cluster
members based on radial velocity, which we measure to an accuracy of 16 km s-1
via template-matching techniques. CN and CH band strengths have been measured
for 29 RGB and AGB stars in NGC 5466, and the band strength indices measured
from VIRUS-P data show close agreement with those measured from Keck/LRIS
spectra previously taken of five of our target stars. We also determine carbon
abundances from comparisons with synthetic spectra. The RGB stars in our data
set cover a range in absolute V magnitude from +2 to -3, which permits us to
study the rate of carbon depletion on the giant branch as well as the point of
its onset. The data show a clear decline in carbon abundance with rising
luminosity above the luminosity function "bump" on the giant branch, and also a
subdued range in CN band strength, suggesting ongoing internal mixing in
individual stars but minor or no primordial star-to-star variation in
light-element abundances.Comment: 10 pages, emulateapj format, AJ accepte
Tangent-ball techniques for shape processing
Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes. Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing. Many applications of shape processing can be found in the entertainment and medical industries.
In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes.
We propose a set of ball-based operators and discuss their properties, implementations, and applications. We divide the group of ball-based operations into unary and binary as follows:
Unary operators include:
* Identifying details (sharp, salient features, constrictions)
* Smoothing shapes by removing such details, replacing them by fillets and roundings
* Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures
Binary operators include:
* Measuring the local discrepancy between two shapes
* Computing the average of two shapes
* Computing point-to-point correspondence between two shapes
* Computing circular trajectories between corresponding points that meet both shapes at right angles
* Using these trajectories to support smooth morphing (inbetweening)
* Using a curve morph to construct surfaces that interpolate between contours on consecutive slices
The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis. These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon
Analysis and new constructions of generalized barycentric coordinates in 2D
Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications
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