792 research outputs found
Stable soft extrapolation of entire functions
Soft extrapolation refers to the problem of recovering a function from its
samples, multiplied by a fast-decaying window and perturbed by an additive
noise, over an interval which is potentially larger than the essential support
of the window. A core theoretical question is to provide bounds on the possible
amount of extrapolation, depending on the sample perturbation level and the
function prior. In this paper we consider soft extrapolation of entire
functions of finite order and type (containing the class of bandlimited
functions as a special case), multiplied by a super-exponentially decaying
window (such as a Gaussian). We consider a weighted least-squares polynomial
approximation with judiciously chosen number of terms and a number of samples
which scales linearly with the degree of approximation. It is shown that this
simple procedure provides stable recovery with an extrapolation factor which
scales logarithmically with the perturbation level and is inversely
proportional to the characteristic lengthscale of the function. The pointwise
extrapolation error exhibits a H\"{o}lder-type continuity with an exponent
derived from weighted potential theory, which changes from 1 near the available
samples, to 0 when the extrapolation distance reaches the characteristic
smoothness length scale of the function. The algorithm is asymptotically
minimax, in the sense that there is essentially no better algorithm yielding
meaningfully lower error over the same smoothness class. When viewed in the
dual domain, the above problem corresponds to (stable) simultaneous
de-convolution and super-resolution for objects of small space/time extent. Our
results then show that the amount of achievable super-resolution is inversely
proportional to the object size, and therefore can be significant for small
objects
The Role of Probe Attenuation in the Time-Domain Reflectometry Characterization of Dielectrics
The influence of the measurement setup on the estimation of dielectric permittivity spectra from time-domain reflectometry (TDR) responses is investigated. The analysis is based on a simplified model of the TDR measurement setup, where an ideal voltage step is applied to an ideal transmission line that models the probe. The main result of this analysis is that the propagation in the probe has an inherent band limiting effect, and the estimation of the high-frequency permittivity parameters is well conditioned only if the wave attenuation for a round trip propagation in the dielectric sample is small. This is a general result, holding for most permittivity model and estimation scheme. It has been verified on real estimation problems by estimating the permittivity of liquid dielectrics and soil samples via an high-order model of the TDR setup and a parametric inversion approac
Concepts for on-board satellite image registration, volume 1
The NASA-NEEDS program goals present a requirement for on-board signal processing to achieve user-compatible, information-adaptive data acquisition. One very specific area of interest is the preprocessing required to register imaging sensor data which have been distorted by anomalies in subsatellite-point position and/or attitude control. The concepts and considerations involved in using state-of-the-art positioning systems such as the Global Positioning System (GPS) in concert with state-of-the-art attitude stabilization and/or determination systems to provide the required registration accuracy are discussed with emphasis on assessing the accuracy to which a given image picture element can be located and identified, determining those algorithms required to augment the registration procedure and evaluating the technology impact on performing these procedures on-board the satellite
Building and Validating a Model for Investigating the Dynamics of Isolated Water Molecules
Understanding how water molecules behave in isolation is vital to understand many fundamental processes in nature. To that end, scientists have begun studying crystals in which single water molecules become trapped in regularly occurring cavities in the crystal structure. As part of that investigation, numerical models used to investigate the dynamics of isolated water molecules are sought to help bolster our fundamental understanding of how these systems behave. To that end, the efficacy of three computational methods—the Euler Method, the Euler-Aspel Method and the Beeman Method—is compared using a newly defined parameter, called the predictive stability coefficient ρ. This new parameter quantifies each algorithm\u27s stability such that the Euler-Aspel Method is determined to be relatively the most stable. Finally, preliminary results from investigating interactions between two dipole neighbors show that the computational tools that will be used for future investigations have been programmed correctly
Sampling and Reconstruction of Shapes with Algebraic Boundaries
We present a sampling theory for a class of binary images with finite rate of
innovation (FRI). Every image in our model is the restriction of
\mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the
indicator function and is some real bivariate polynomial. This particularly
means that the boundaries in the image form a subset of an algebraic curve with
the implicit polynomial . We show that the image parameters --i.e., the
polynomial coefficients-- satisfy a set of linear annihilation equations with
the coefficients being the image moments. The inherent sensitivity of the
moments to noise makes the reconstruction process numerically unstable and
narrows the choice of the sampling kernels to polynomial reproducing kernels.
As a remedy to these problems, we replace conventional moments with more stable
\emph{generalized moments} that are adjusted to the given sampling kernel. The
benefits are threefold: (1) it relaxes the requirements on the sampling
kernels, (2) produces annihilation equations that are robust at numerical
precision, and (3) extends the results to images with unbounded boundaries. We
further reduce the sensitivity of the reconstruction process to noise by taking
into account the sign of the polynomial at certain points, and sequentially
enforcing measurement consistency. We consider various numerical experiments to
demonstrate the performance of our algorithm in reconstructing binary images,
including low to moderate noise levels and a range of realistic sampling
kernels.Comment: 12 pages, 14 figure
Characterization of surface profiles using discrete measurement systems
Form error estimation techniques based on discrete point measurements can lead to significant errors in form tolerance evaluation. By modeling surface profiles as random variables, we are able to show how sample size and fitting techniques affect form error estimation. Depending on the surface characteristics, typical sampling techniques can result in estimation errors of as much as 50%;We investigate current available interpolation procedures. Kriging is an optimal interpolation for spatial data when the model of variogram is known a priori. Due to the difficulty in identifying the correct variogram model from the limited sampled data and lack of complete computer software, there is no significant advantage to apply kriging to estimate form error in the inspection process;We apply the Shannon sampling theorem and represent the surface profiles as band-limited signals. We show that the Shannon sampling function is in fact an infinite degree B-spline interpolation function and thus a best approximation for band-limited signals. Both Shannon sampling series and universal kriging (using a priori correlation function) are applied to flatness error estimation for uniform sample points measured from five common machined surfaces. The results show both methods perform similarly. The probability of over-estimating form error increases and the probability of accepting bad parts decreases using interpolation methods versus using the points directly
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