260 research outputs found
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
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
Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs
We study sparse recovery with structured random measurement matrices having
independent, identically distributed, and uniformly bounded rows and with a
nontrivial covariance structure. This class of matrices arises from random
sampling of bounded Riesz systems and generalizes random partial Fourier
matrices. Our main result improves the currently available results for the null
space and restricted isometry properties of such random matrices. The main
novelty of our analysis is a new upper bound for the expectation of the
supremum of a Bernoulli process associated with a restricted isometry constant.
We apply our result to prove new performance guarantees for the CORSING method,
a recently introduced numerical approximation technique for partial
differential equations (PDEs) based on compressive sensing
Robust phase retrieval with the swept approximate message passing (prSAMP) algorithm
In phase retrieval, the goal is to recover a complex signal from the
magnitude of its linear measurements. While many well-known algorithms
guarantee deterministic recovery of the unknown signal using i.i.d. random
measurement matrices, they suffer serious convergence issues some
ill-conditioned matrices. As an example, this happens in optical imagers using
binary intensity-only spatial light modulators to shape the input wavefront.
The problem of ill-conditioned measurement matrices has also been a topic of
interest for compressed sensing researchers during the past decade. In this
paper, using recent advances in generic compressed sensing, we propose a new
phase retrieval algorithm that well-adopts for both Gaussian i.i.d. and binary
matrices using both sparse and dense input signals. This algorithm is also
robust to the strong noise levels found in some imaging applications
Compressively Sensed Image Recognition
Compressive Sensing (CS) theory asserts that sparse signal reconstruction is
possible from a small number of linear measurements. Although CS enables
low-cost linear sampling, it requires non-linear and costly reconstruction.
Recent literature works show that compressive image classification is possible
in CS domain without reconstruction of the signal. In this work, we introduce a
DCT base method that extracts binary discriminative features directly from CS
measurements. These CS measurements can be obtained by using (i) a random or a
pseudo-random measurement matrix, or (ii) a measurement matrix whose elements
are learned from the training data to optimize the given classification task.
We further introduce feature fusion by concatenating Bag of Words (BoW)
representation of our binary features with one of the two state-of-the-art
CNN-based feature vectors. We show that our fused feature outperforms the
state-of-the-art in both cases.Comment: 6 pages, submitted/accepted, EUVIP 201
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