9,781 research outputs found
ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm
Multivariate problems are typically governed by anisotropic features such as
edges in images. A common bracket of most of the various directional
representation systems which have been proposed to deliver sparse
approximations of such features is the utilization of parabolic scaling. One
prominent example is the shearlet system. Our objective in this paper is
three-fold: We firstly develop a digital shearlet theory which is rationally
designed in the sense that it is the digitization of the existing shearlet
theory for continuous data. This implicates that shearlet theory provides a
unified treatment of both the continuum and digital realm. Secondly, we analyze
the utilization of pseudo-polar grids and the pseudo-polar Fourier transform
for digital implementations of parabolic scaling algorithms. We derive an
isometric pseudo-polar Fourier transform by careful weighting of the
pseudo-polar grid, allowing exploitation of its adjoint for the inverse
transform. This leads to a digital implementation of the shearlet transform; an
accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we
introduce various quantitative measures for digital parabolic scaling
algorithms in general, allowing one to tune parameters and objectively improve
the implementation as well as compare different directional transform
implementations. The usefulness of such measures is exemplarily demonstrated
for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu
Fractional Fourier detection of L\'evy Flights: application to Hamiltonian chaotic trajectories
A signal processing method designed for the detection of linear (coherent)
behaviors among random fluctuations is presented. It is dedicated to the study
of data recorded from nonlinear physical systems. More precisely the method is
suited for signals having chaotic variations and sporadically appearing regular
linear patterns, possibly impaired by noise. We use time-frequency techniques
and the Fractional Fourier transform in order to make it robust and easily
implementable. The method is illustrated with an example of application: the
analysis of chaotic trajectories of advected passive particles. The signal has
a chaotic behavior and encounter L\'evy flights (straight lines). The method is
able to detect and quantify these ballistic transport regions, even in noisy
situations
Fractional dynamics of coupled oscillators with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power-wise interaction. The corresponding term in dynamical
equations is proportional to . It is shown that the
equation of motion in the infrared limit can be transformed into the medium
equation with the Riesz fractional derivative of order , when
. We consider few models of coupled oscillators and show how their
synchronization can appear as a result of bifurcation, and how the
corresponding solutions depend on . The presence of fractional
derivative leads also to the occurrence of localized structures. Particular
solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear
Schrodinger) equation are derived. These solutions are interpreted as
synchronized states and localized structures of the oscillatory medium.Comment: 34 pages, 18 figure
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