35 research outputs found
Superresolution without Separation
This paper provides a theoretical analysis of diffraction-limited
superresolution, demonstrating that arbitrarily close point sources can be
resolved in ideal situations. Precisely, we assume that the incoming signal is
a linear combination of M shifted copies of a known waveform with unknown
shifts and amplitudes, and one only observes a finite collection of evaluations
of this signal. We characterize properties of the base waveform such that the
exact translations and amplitudes can be recovered from 2M + 1 observations.
This recovery is achieved by solving a a weighted version of basis pursuit over
a continuous dictionary. Our methods combine classical polynomial interpolation
techniques with contemporary tools from compressed sensing.Comment: 23 pages, 8 figure
Conservative classical and quantum resolution limits for incoherent imaging
I propose classical and quantum limits to the statistical resolution of two
incoherent optical point sources from the perspective of minimax parameter
estimation. Unlike earlier results based on the Cram\'er-Rao bound, the limits
proposed here, based on the worst-case error criterion and a Bayesian version
of the Cram\'er-Rao bound, are valid for any biased or unbiased estimator and
obey photon-number scalings that are consistent with the behaviors of actual
estimators. These results prove that, from the minimax perspective, the
spatial-mode demultiplexing (SPADE) measurement scheme recently proposed by
Tsang, Nair, and Lu [Phys. Rev. X 6, 031033 (2016)] remains superior to direct
imaging for sufficiently high photon numbers.Comment: 12 pages, 2 figures. v2: focused on imaging, cleaned up the math,
added new analytic and numerical results. v3: restructured and submitte
Compressed sensing of data with a known distribution
Compressed sensing is a technique for recovering an unknown sparse signal
from a small number of linear measurements. When the measurement matrix is
random, the number of measurements required for perfect recovery exhibits a
phase transition: there is a threshold on the number of measurements after
which the probability of exact recovery quickly goes from very small to very
large. In this work we are able to reduce this threshold by incorporating
statistical information about the data we wish to recover. Our algorithm works
by minimizing a suitably weighted -norm, where the weights are chosen
so that the expected statistical dimension of the corresponding descent cone is
minimized. We also provide new discrete-geometry-based Monte Carlo algorithms
for computing intrinsic volumes of such descent cones, allowing us to bound the
failure probability of our methods.Comment: 22 pages, 7 figures. New colorblind safe figures. Sections 3 and 4
completely rewritten. Minor typos fixe
A note on spike localization for line spectrum estimation
This note considers the problem of approximating the locations of dominant
spikes for a probability measure from noisy spectrum measurements under the
condition of residue signal, significant noise level, and no minimum spectrum
separation. We show that the simple procedure of thresholding the smoothed
inverse Fourier transform allows for approximating the spike locations rather
accurately