9,054 research outputs found
Super-Resolution of Positive Sources: the Discrete Setup
In single-molecule microscopy it is necessary to locate with high precision
point sources from noisy observations of the spectrum of the signal at
frequencies capped by , which is just about the frequency of natural
light. This paper rigorously establishes that this super-resolution problem can
be solved via linear programming in a stable manner. We prove that the quality
of the reconstruction crucially depends on the Rayleigh regularity of the
support of the signal; that is, on the maximum number of sources that can occur
within a square of side length about . The theoretical performance
guarantee is complemented with a converse result showing that our simple convex
program convex is nearly optimal. Finally, numerical experiments illustrate our
methods.Comment: 31 page, 7 figure
Multiple Illumination Phaseless Super-Resolution (MIPS) with Applications To Phaseless DOA Estimation and Diffraction Imaging
Phaseless super-resolution is the problem of recovering an unknown signal
from measurements of the magnitudes of the low frequency Fourier transform of
the signal. This problem arises in applications where measuring the phase, and
making high-frequency measurements, are either too costly or altogether
infeasible. The problem is especially challenging because it combines the
difficult problems of phase retrieval and classical super-resolutionComment: To appear in ICASSP 201
Super-Resolution of Mutually Interfering Signals
We consider simultaneously identifying the membership and locations of point
sources that are convolved with different low-pass point spread functions, from
the observation of their superpositions. This problem arises in
three-dimensional super-resolution single-molecule imaging, neural spike
sorting, multi-user channel identification, among others. We propose a novel
algorithm, based on convex programming, and establish its near-optimal
performance guarantee for exact recovery by exploiting the sparsity of the
point source model as well as incoherence between the point spread functions.
Numerical examples are provided to demonstrate the effectiveness of the
proposed approach.Comment: ISIT 201
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
- β¦