44 research outputs found
Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing
We demonstrate that sub-wavelength optical images borne on
partially-spatially-incoherent light can be recovered, from their far-field or
from the blurred image, given the prior knowledge that the image is sparse, and
only that. The reconstruction method relies on the recently demonstrated
sparsity-based sub-wavelength imaging. However, for
partially-spatially-incoherent light, the relation between the measurements and
the image is quadratic, yielding non-convex measurement equations that do not
conform to previously used techniques. Consequently, we demonstrate new
algorithmic methodology, referred to as quadratic compressed sensing, which can
be applied to a range of other problems involving information recovery from
partial correlation measurements, including when the correlation function has
local dependencies. Specifically for microscopy, this method can be readily
extended to white light microscopes with the additional knowledge of the light
source spectrum.Comment: 16 page
On Conditions for Uniqueness in Sparse Phase Retrieval
The phase retrieval problem has a long history and is an important problem in
many areas of optics. Theoretical understanding of phase retrieval is still
limited and fundamental questions such as uniqueness and stability of the
recovered solution are not yet fully understood. This paper provides several
additions to the theoretical understanding of sparse phase retrieval. In
particular we show that if the measurement ensemble can be chosen freely, as
few as 4k-1 phaseless measurements suffice to guarantee uniqueness of a
k-sparse M-dimensional real solution. We also prove that 2(k^2-k+1) Fourier
magnitude measurements are sufficient under rather general conditions
Imaging with a small number of photons
Low-light-level imaging techniques have application in many diverse fields,
ranging from biological sciences to security. We demonstrate a single-photon
imaging system based on a time-gated inten- sified CCD (ICCD) camera in which
the image of an object can be inferred from very few detected photons. We show
that a ghost-imaging configuration, where the image is obtained from photons
that have never interacted with the object, is a useful approach for obtaining
images with high signal-to-noise ratios. The use of heralded single-photons
ensures that the background counts can be virtually eliminated from the
recorded images. By applying techniques of compressed sensing and associated
image reconstruction, we obtain high-quality images of the object from raw data
comprised of fewer than one detected photon per image pixel.Comment: 9 pages, 4 figure
Nonlinear Basis Pursuit
In compressive sensing, the basis pursuit algorithm aims to find the sparsest
solution to an underdetermined linear equation system. In this paper, we
generalize basis pursuit to finding the sparsest solution to higher order
nonlinear systems of equations, called nonlinear basis pursuit. In contrast to
the existing nonlinear compressive sensing methods, the new algorithm that
solves the nonlinear basis pursuit problem is convex and not greedy. The novel
algorithm enables the compressive sensing approach to be used for a broader
range of applications where there are nonlinear relationships between the
measurements and the unknowns
Reconstruction of Integers from Pairwise Distances
Given a set of integers, one can easily construct the set of their pairwise
distances. We consider the inverse problem: given a set of pairwise distances,
find the integer set which realizes the pairwise distance set. This problem
arises in a lot of fields in engineering and applied physics, and has
confounded researchers for over 60 years. It is one of the few fundamental
problems that are neither known to be NP-hard nor solvable by polynomial-time
algorithms. Whether unique recovery is possible also remains an open question.
In many practical applications where this problem occurs, the integer set is
naturally sparse (i.e., the integers are sufficiently spaced), a property which
has not been explored. In this work, we exploit the sparse nature of the
integer set and develop a polynomial-time algorithm which provably recovers the
set of integers (up to linear shift and reversal) from the set of their
pairwise distances with arbitrarily high probability if the sparsity is
O(n^{1/2-\eps}). Numerical simulations verify the effectiveness of the
proposed algorithm.Comment: 14 pages, 4 figures, submitted to ICASSP 201
Phase Retrieval From Binary Measurements
We consider the problem of signal reconstruction from quadratic measurements
that are encoded as +1 or -1 depending on whether they exceed a predetermined
positive threshold or not. Binary measurements are fast to acquire and
inexpensive in terms of hardware. We formulate the problem of signal
reconstruction using a consistency criterion, wherein one seeks to find a
signal that is in agreement with the measurements. To enforce consistency, we
construct a convex cost using a one-sided quadratic penalty and minimize it
using an iterative accelerated projected gradient-descent (APGD) technique. The
PGD scheme reduces the cost function in each iteration, whereas incorporating
momentum into PGD, notwithstanding the lack of such a descent property,
exhibits faster convergence than PGD empirically. We refer to the resulting
algorithm as binary phase retrieval (BPR). Considering additive white noise
contamination prior to quantization, we also derive the Cramer-Rao Bound (CRB)
for the binary encoding model. Experimental results demonstrate that the BPR
algorithm yields a signal-to- reconstruction error ratio (SRER) of
approximately 25 dB in the absence of noise. In the presence of noise prior to
quantization, the SRER is within 2 to 3 dB of the CRB