102,718 research outputs found
Towards a Mathematical Theory of Super-Resolution
This paper develops a mathematical theory of super-resolution. Broadly
speaking, super-resolution is the problem of recovering the fine details of an
object---the high end of its spectrum---from coarse scale information
only---from samples at the low end of the spectrum. Suppose we have many point
sources at unknown locations in and with unknown complex-valued
amplitudes. We only observe Fourier samples of this object up until a frequency
cut-off . We show that one can super-resolve these point sources with
infinite precision---i.e. recover the exact locations and amplitudes---by
solving a simple convex optimization problem, which can essentially be
reformulated as a semidefinite program. This holds provided that the distance
between sources is at least . This result extends to higher dimensions
and other models. In one dimension for instance, it is possible to recover a
piecewise smooth function by resolving the discontinuity points with infinite
precision as well. We also show that the theory and methods are robust to
noise. In particular, in the discrete setting we develop some theoretical
results explaining how the accuracy of the super-resolved signal is expected to
degrade when both the noise level and the {\em super-resolution factor} vary.Comment: 48 pages, 12 figure
Super-resolution MRI Using Finite Rate of Innovation Curves
We propose a two-stage algorithm for the super-resolution of MR images from
their low-frequency k-space samples. In the first stage we estimate a
resolution-independent mask whose zeros represent the edges of the image. This
builds off recent work extending the theory of sampling signals of finite rate
of innovation (FRI) to two-dimensional curves. We enable its application to MRI
by proposing extensions of the signal models allowed by FRI theory, and by
developing a more robust and efficient means to determine the edge mask. In the
second stage of the scheme, we recover the super-resolved MR image using the
discretized edge mask as an image prior. We evaluate our scheme on simulated
single-coil MR data obtained from analytical phantoms, and compare against
total variation reconstructions. Our experiments show improved performance in
both noiseless and noisy settings.Comment: Conference paper accepted to ISBI 2015. 4 pages, 2 figure
Lensfree on-chip microscopy over a wide field-of-view using pixel super-resolution.
We demonstrate lensfree holographic microscopy on a chip to achieve approximately 0.6 microm spatial resolution corresponding to a numerical aperture of approximately 0.5 over a large field-of-view of approximately 24 mm2. By using partially coherent illumination from a large aperture (approximately 50 microm), we acquire lower resolution lensfree in-line holograms of the objects with unit fringe magnification. For each lensfree hologram, the pixel size at the sensor chip limits the spatial resolution of the reconstructed image. To circumvent this limitation, we implement a sub-pixel shifting based super-resolution algorithm to effectively recover much higher resolution digital holograms of the objects, permitting sub-micron spatial resolution to be achieved across the entire sensor chip active area, which is also equivalent to the imaging field-of-view (24 mm2) due to unit magnification. We demonstrate the success of this pixel super-resolution approach by imaging patterned transparent substrates, blood smear samples, as well as Caenoharbditis Elegans
Comparing Numerical Methods for Isothermal Magnetized Supersonic Turbulence
We employ simulations of supersonic super-Alfvenic turbulence decay as a
benchmark test problem to assess and compare the performance of nine
astrophysical MHD methods actively used to model star formation. The set of
nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER,
and ZEUS. We present a comprehensive set of statistical measures designed to
quantify the effects of numerical dissipation in these MHD solvers. We compare
power spectra for basic fields to determine the effective spectral bandwidth of
the methods and rank them based on their relative effective Reynolds numbers.
We also compare numerical dissipation for solenoidal and dilatational velocity
components to check for possible impacts of the numerics on small-scale density
statistics. Finally, we discuss convergence of various characteristics for the
turbulence decay test and impacts of various components of numerical schemes on
the accuracy of solutions. We show that the best performing codes employ a
consistently high order of accuracy for spatial reconstruction of the evolved
fields, transverse gradient interpolation, conservation law update step, and
Lorentz force computation. The best results are achieved with divergence-free
evolution of the magnetic field using the constrained transport method, and
using little to no explicit artificial viscosity. Codes which fall short in one
or more of these areas are still useful, but they must compensate higher
numerical dissipation with higher numerical resolution. This paper is the
largest, most comprehensive MHD code comparison on an application-like test
problem to date. We hope this work will help developers improve their numerical
algorithms while helping users to make informed choices in picking optimal
applications for their specific astrophysical problems.Comment: 17 pages, 5 color figures, revised version to appear in ApJ, 735,
July 201
Bounds for Euler from vorticity moments and line divergence
The inviscid growth of a range of vorticity moments is compared using Euler
calculations of anti-parallel vortices with a new initial condition. The primary goal
is to understand the role of nonlinearity in the generation of a new hierarchy of
rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments
obey Dm ≥ Dm+1, the reverse of the usual
Ωm+1 ≥ Ωm Hölder ordering of the original
moments. Two temporal phases have been identified for the Euler calculations. In the
first phase the 1 < m < ∞ vorticity moments are ordered in a manner consistent with
the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of
possible singular growth with D2
m → � sup ӏωӏ ~ Am(Tc-t)-1 where the Am are nearly
independent of m. In the second phase, the new Dm ordering breaks down as the Ωm
converge towards the same super-exponential growth for all m. The transition is
identified using new inequalities for the upper bounds for the -dD-2m/dt that are based
solely upon the ratios Dm+1/Dm, and the convergent super-exponential growth is shown
by plotting log(d log Ωm/dt). Three-dimensional graphics show significant divergence
of the vortex lines during the second phase, which could be what inhibits the initial
power-law growth
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