54,588 research outputs found
Numerical Analysis of the Non-uniform Sampling Problem
We give an overview of recent developments in the problem of reconstructing a
band-limited signal from non-uniform sampling from a numerical analysis view
point. It is shown that the appropriate design of the finite-dimensional model
plays a key role in the numerical solution of the non-uniform sampling problem.
In the one approach (often proposed in the literature) the finite-dimensional
model leads to an ill-posed problem even in very simple situations. The other
approach that we consider leads to a well-posed problem that preserves
important structural properties of the original infinite-dimensional problem
and gives rise to efficient numerical algorithms. Furthermore a fast multilevel
algorithm is presented that can reconstruct signals of unknown bandwidth from
noisy non-uniformly spaced samples. We also discuss the design of efficient
regularization methods for ill-conditioned reconstruction problems. Numerical
examples from spectroscopy and exploration geophysics demonstrate the
performance of the proposed methods
Sampling in the Analysis Transform Domain
Many signal and image processing applications have benefited remarkably from
the fact that the underlying signals reside in a low dimensional subspace. One
of the main models for such a low dimensionality is the sparsity one. Within
this framework there are two main options for the sparse modeling: the
synthesis and the analysis ones, where the first is considered the standard
paradigm for which much more research has been dedicated. In it the signals are
assumed to have a sparse representation under a given dictionary. On the other
hand, in the analysis approach the sparsity is measured in the coefficients of
the signal after applying a certain transformation, the analysis dictionary, on
it. Though several algorithms with some theory have been developed for this
framework, they are outnumbered by the ones proposed for the synthesis
methodology.
Given that the analysis dictionary is either a frame or the two dimensional
finite difference operator, we propose a new sampling scheme for signals from
the analysis model that allows to recover them from their samples using any
existing algorithm from the synthesis model. The advantage of this new sampling
strategy is that it makes the existing synthesis methods with their theory also
available for signals from the analysis framework.Comment: 13 Pages, 2 figure
Four-dimensional tomographic reconstruction by time domain decomposition
Since the beginnings of tomography, the requirement that the sample does not
change during the acquisition of one tomographic rotation is unchanged. We
derived and successfully implemented a tomographic reconstruction method which
relaxes this decades-old requirement of static samples. In the presented
method, dynamic tomographic data sets are decomposed in the temporal domain
using basis functions and deploying an L1 regularization technique where the
penalty factor is taken for spatial and temporal derivatives. We implemented
the iterative algorithm for solving the regularization problem on modern GPU
systems to demonstrate its practical use
Heat-kernel approach for scattering
An approach for solving scattering problems, based on two quantum field
theory methods, the heat kernel method and the scattering spectral method, is
constructed. This approach converts a method of calculating heat kernels into a
method of solving scattering problems. This allows us to establish a method of
scattering problems from a method of heat kernels. As an application, we
construct an approach for solving scattering problems based on the covariant
perturbation theory of heat-kernel expansions. In order to apply the
heat-kernel method to scattering problems, we first calculate the off-diagonal
heat-kernel expansion in the frame of the covariant perturbation theory.
Moreover, as an alternative application of the relation between heat kernels
and partial-wave phase shifts presented in this paper, we give an example of
how to calculate a global heat kernel from a known scattering phase shift
Electromagnetic Form Factors and Charge Densities From Hadrons to Nuclei
A simple exact covariant model in which a scalar particle is modeled as a
bound state of two different particles is used to elucidate relativistic
aspects of electromagnetic form factors. The model form factor is computed
using an exact covariant calculation of the lowest-order triangle diagram and
shown to be the same as that obtained using light-front techniques. The meaning
of transverse density is explained using coordinate space variables, allowing
us to identify a true mean-square transverse size directly related to the form
factor. We show that the rest-frame charge distribution is generally not
observable because of the failure to uphold current conservation. Neutral
systems of two charged constituents are shown to obey the lore that the heavier
one is generally closer to the transverse origin than the lighter one. It is
argued that the negative central charge density of the neutron arises, in
pion-cloud models, from pions of high longitudinal momentum. The
non-relativistic limit is defined precisely and the ratio of the binding energy
to that of the mass of the lightest constituent is shown to govern the
influence of relativistic effects. The exact relativistic formula for the form
factor reduces to the familiar one of the three-dimensional Fourier transform
of a square of a wave function for a very limited range of parameters. For
masses that mimic the quark-di-quark model of the nucleon we find substantial
relativistic corrections for any value of . A schematic model of the
lowest s-states of nuclei is used to find that relativistic effects decrease
the form factor for light nuclei but increase the form factor for heavy nuclei.
Furthermore, these states are strongly influenced by relativity.Comment: 18 pages, 11 figure
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