20,973 research outputs found
Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain
Recovering a sparse signal from its low-pass projections in the Fourier
domain is a problem of broad interest in science and engineering and is
commonly referred to as super-resolution. In many cases, however, Fourier
domain may not be the natural choice. For example, in holography, low-pass
projections of sparse signals are obtained in the Fresnel domain. Similarly,
time-varying system identification relies on low-pass projections on the space
of linear frequency modulated signals. In this paper, we study the recovery of
sparse signals from low-pass projections in the Special Affine Fourier
Transform domain (SAFT). The SAFT parametrically generalizes a number of well
known unitary transformations that are used in signal processing and optics. In
analogy to the Shannon's sampling framework, we specify sampling theorems for
recovery of sparse signals considering three specific cases: (1) sampling with
arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels
and, (3) recovery from Gabor transform measurements linked with the SAFT
domain. Our work offers a unifying perspective on the sparse sampling problem
which is compatible with the Fourier, Fresnel and Fractional Fourier domain
based results. In deriving our results, we introduce the SAFT series (analogous
to the Fourier series) and the short time SAFT, and study convolution theorems
that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie
Sparse recovery and Fourier sampling
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 155-160).In the last decade a broad literature has arisen studying sparse recovery, the estimation of sparse vectors from low dimensional linear projections. Sparse recovery has a wide variety of applications such as streaming algorithms, image acquisition, and disease testing. A particularly important subclass of sparse recovery is the sparse Fourier transform, which considers the computation of a discrete Fourier transform when the output is sparse. Applications of the sparse Fourier transform include medical imaging, spectrum sensing, and purely computation tasks involving convolution. This thesis describes a coherent set of techniques that achieve optimal or near-optimal upper and lower bounds for a variety of sparse recovery problems. We give the following state-of-the-art algorithms for recovery of an approximately k-sparse vector in n dimensions: -- Two sparse Fourier transform algorithms, respectively taking ... time and ... samples. The latter is within log e log n of the optimal sample complexity when ... -- An algorithm for adaptive sparse recovery using ... measurements, showing that adaptivity can give substantial improvements when k is small. -- An algorithm for C-approximate sparse recovery with ... measurements, which matches our lower bound up to the log* k factor and gives the first improvement for ... In the second part of this thesis, we give lower bounds for the above problems and more.by Eric Price.Ph. D
Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks
We consider the approximate sparse recovery problem in Wireless Sensor Networks (WSNs) using Compressed Sensing/Compressive Sampling (CS). The goal is to recover the n \mbox{-}dimensional data values by querying only sensors based on some linear projection of sensor readings. To solve this problem, a two-tiered sampling model is considered and a novel distributed compressive sparse sampling (DCSS) algorithm is proposed based on sparse binary CS measurement matrix. In the two-tiered sampling model, each sensor first samples the environment independently. Then the fusion center (FC), acting as a pseudo-sensor, samples the sensor network to select a subset of sensors ( out of ) that directly respond to the FC for data recovery purpose. The sparse binary matrix is designed using unbalanced expander graph which achieves the state-of-the-art performance for CS schemes. This binary matrix can be interpreted as a sensor selection matrix-whose fairness is analyzed. Extensive experiments on both synthetic and real data set show that by querying only the minimum amount of sensors using the DCSS algorithm, the CS recovery accuracy can be as good as dense measurement matrices (e.g., Gaussian, Fourier Scrambles). We also show that the sparse binary measurement matrix works well on compressible data which has the closest recovery result to the known best k\mbox{-}term approximation. The recovery is robust against noisy measurements. The sparsity and binary properties of the measurement matrix contribute, to a great extent, the reduction of the in-network communication cost as well as the computational burden
Learning Sub-Sampling and Signal Recovery with Applications in Ultrasound Imaging
Limitations on bandwidth and power consumption impose strict bounds on data
rates of diagnostic imaging systems. Consequently, the design of suitable (i.e.
task- and data-aware) compression and reconstruction techniques has attracted
considerable attention in recent years. Compressed sensing emerged as a popular
framework for sparse signal reconstruction from a small set of compressed
measurements. However, typical compressed sensing designs measure a
(non)linearly weighted combination of all input signal elements, which poses
practical challenges. These designs are also not necessarily task-optimal. In
addition, real-time recovery is hampered by the iterative and time-consuming
nature of sparse recovery algorithms. Recently, deep learning methods have
shown promise for fast recovery from compressed measurements, but the design of
adequate and practical sensing strategies remains a challenge. Here, we propose
a deep learning solution termed Deep Probabilistic Sub-sampling (DPS), that
learns a task-driven sub-sampling pattern, while jointly training a subsequent
task model. Once learned, the task-based sub-sampling patterns are fixed and
straightforwardly implementable, e.g. by non-uniform analog-to-digital
conversion, sparse array design, or slow-time ultrasound pulsing schemes. The
effectiveness of our framework is demonstrated in-silico for sparse signal
recovery from partial Fourier measurements, and in-vivo for both anatomical
image and tissue-motion (Doppler) reconstruction from sub-sampled medical
ultrasound imaging data
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