11,833 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
Ab initio compressive phase retrieval
Any object on earth has two fundamental properties: it is finite, and it is
made of atoms. Structural information about an object can be obtained from
diffraction amplitude measurements that account for either one of these traits.
Nyquist-sampling of the Fourier amplitudes is sufficient to image single
particles of finite size at any resolution. Atomic resolution data is routinely
used to image molecules replicated in a crystal structure. Here we report an
algorithm that requires neither information, but uses the fact that an image of
a natural object is compressible. Intended applications include tomographic
diffractive imaging, crystallography, powder diffraction, small angle x-ray
scattering and random Fourier amplitude measurements.Comment: 7 pages, 4 figures, presented at the XXI IUCr Congress, Aug. 2008,
Osaka Japa
Video Compressive Sensing for Dynamic MRI
We present a video compressive sensing framework, termed kt-CSLDS, to
accelerate the image acquisition process of dynamic magnetic resonance imaging
(MRI). We are inspired by a state-of-the-art model for video compressive
sensing that utilizes a linear dynamical system (LDS) to model the motion
manifold. Given compressive measurements, the state sequence of an LDS can be
first estimated using system identification techniques. We then reconstruct the
observation matrix using a joint structured sparsity assumption. In particular,
we minimize an objective function with a mixture of wavelet sparsity and joint
sparsity within the observation matrix. We derive an efficient convex
optimization algorithm through alternating direction method of multipliers
(ADMM), and provide a theoretical guarantee for global convergence. We
demonstrate the performance of our approach for video compressive sensing, in
terms of reconstruction accuracy. We also investigate the impact of various
sampling strategies. We apply this framework to accelerate the acquisition
process of dynamic MRI and show it achieves the best reconstruction accuracy
with the least computational time compared with existing algorithms in the
literature.Comment: 30 pages, 9 figure
Imaging With Nature: Compressive Imaging Using a Multiply Scattering Medium
The recent theory of compressive sensing leverages upon the structure of
signals to acquire them with much fewer measurements than was previously
thought necessary, and certainly well below the traditional Nyquist-Shannon
sampling rate. However, most implementations developed to take advantage of
this framework revolve around controlling the measurements with carefully
engineered material or acquisition sequences. Instead, we use the natural
randomness of wave propagation through multiply scattering media as an optimal
and instantaneous compressive imaging mechanism. Waves reflected from an object
are detected after propagation through a well-characterized complex medium.
Each local measurement thus contains global information about the object,
yielding a purely analog compressive sensing method. We experimentally
demonstrate the effectiveness of the proposed approach for optical imaging by
using a 300-micrometer thick layer of white paint as the compressive imaging
device. Scattering media are thus promising candidates for designing efficient
and compact compressive imagers.Comment: 17 pages, 8 figure
- …