141 research outputs found
Compressive Inverse Scattering II. SISO Measurements with Born scatterers
Inverse scattering methods capable of compressive imaging are proposed and
analyzed. The methods employ randomly and repeatedly (multiple-shot) the
single-input-single-output (SISO) measurements in which the probe frequencies,
the incident and the sampling directions are related in a precise way and are
capable of recovering exactly scatterers of sufficiently low sparsity.
For point targets, various sampling techniques are proposed to transform the
scattering matrix into the random Fourier matrix. The results for point targets
are then extended to the case of localized extended targets by interpolating
from grid points. In particular, an explicit error bound is derived for the
piece-wise constant interpolation which is shown to be a practical way of
discretizing localized extended targets and enabling the compressed sensing
techniques.
For distributed extended targets, the Littlewood-Paley basis is used in
analysis. A specially designed sampling scheme then transforms the scattering
matrix into a block-diagonal matrix with each block being the random Fourier
matrix corresponding to one of the multiple dyadic scales of the extended
target. In other words by the Littlewood-Paley basis and the proposed sampling
scheme the different dyadic scales of the target are decoupled and therefore
can be reconstructed scale-by-scale by the proposed method. Moreover, with
probes of any single frequency \om the coefficients in the Littlewood-Paley
expansion for scales up to \om/(2\pi) can be exactly recovered.Comment: Add a new section (Section 3) on localized extended target
Phase Retrieval by Linear Algebra
The null vector method, based on a simple linear algebraic concept, is
proposed as a solution to the phase retrieval problem.
In the case with complex Gaussian random measurement matrices, a
non-asymptotic error bound is derived, yielding an asymptotic regime of
accurate approximation comparable to that for the spectral vector method
Relaxation Time of Quantized Toral Maps
We introduce the notion of the relaxation time for noisy quantum maps on the
2d-dimensional torus - a generalization of previously studied dissipation time.
We show that relaxation time is sensitive to the chaotic behavior of the
corresponding classical system if one simultaneously considers the
semiclassical limit ( -> 0) together with the limit of small noise
strength (\ep -> 0).
Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime
\hbar^{-1}\ep\hbar$ << 1,
quantum and classical relaxation times behave very differently. In the special
case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps),
we obtain the exact asymptotics of the quantum relaxation time and precise the
regime of correspondence between quantum and classical relaxations.Comment: LaTeX, 27 pages, former term dissipation time replaced by relaxation
time, new introduction and reference
Self-Averaging Scaling Limits of Two-Frequency Wigner Distribution for Random Paraxial Waves
Two-frequency Wigner distribution is introduced to capture the asymptotic
behavior of the space-frequency correlation of paraxial waves in the radiative
transfer limits. The scaling limits give rises to deterministic transport-like
equations. Depending on the ratio of the wavelength to the correlation length
the limiting equation is either a Boltzmann-like integral equation or a
Fokker-Planck-like differential equation in the phase space. The solutions to
these equations have a probabilistic representation which can be simulated by
Monte Carlo method. When the medium fluctuates more rapidly in the longitudinal
direction, the corresponding Fokker-Planck-like equation can be solved exactly.Comment: typos correcte
Coherence-Pattern Guided Compressive Sensing with Unresolved Grids
Highly coherent sensing matrices arise in discretization of continuum imaging
problems such as radar and medical imaging when the grid spacing is below the
Rayleigh threshold.
Algorithms based on techniques of band exclusion (BE) and local optimization
(LO) are proposed to deal with such coherent sensing matrices. These techniques
are embedded in the existing compressed sensing algorithms such as Orthogonal
Matching Pursuit (OMP), Subspace Pursuit (SP), Iterative Hard Thresholding
(IHT), Basis Pursuit (BP) and Lasso, and result in the modified algorithms
BLOOMP, BLOSP, BLOIHT, BP-BLOT and Lasso-BLOT, respectively.
Under appropriate conditions, it is proved that BLOOMP can reconstruct
sparse, widely separated objects up to one Rayleigh length in the Bottleneck
distance {\em independent} of the grid spacing. One of the most distinguishing
attributes of BLOOMP is its capability of dealing with large dynamic ranges.
The BLO-based algorithms are systematically tested with respect to four
performance metrics: dynamic range, noise stability, sparsity and resolution.
With respect to dynamic range and noise stability, BLOOMP is the best
performer. With respect to sparsity, BLOOMP is the best performer for high
dynamic range while for dynamic range near unity BP-BLOT and Lasso-BLOT with
the optimized regularization parameter have the best performance. In the
noiseless case, BP-BLOT has the highest resolving power up to certain dynamic
range.
The algorithms BLOSP and BLOIHT are good alternatives to
BLOOMP and BP/Lasso-BLOT: they are faster than both BLOOMP and BP/Lasso-BLOT
and shares, to a lesser degree, BLOOMP's amazing attribute with respect to
dynamic range.
Detailed comparisons with existing algorithms such as Spectral Iterative Hard
Thresholding (SIHT) and the frame-adapted BP are given
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