2,858 research outputs found
Universal and efficient compressed sensing by spread spectrum and application to realistic Fourier imaging techniques
We advocate a compressed sensing strategy that consists of multiplying the
signal of interest by a wide bandwidth modulation before projection onto
randomly selected vectors of an orthonormal basis. Firstly, in a digital
setting with random modulation, considering a whole class of sensing bases
including the Fourier basis, we prove that the technique is universal in the
sense that the required number of measurements for accurate recovery is optimal
and independent of the sparsity basis. This universality stems from a drastic
decrease of coherence between the sparsity and the sensing bases, which for a
Fourier sensing basis relates to a spread of the original signal spectrum by
the modulation (hence the name "spread spectrum"). The approach is also
efficient as sensing matrices with fast matrix multiplication algorithms can be
used, in particular in the case of Fourier measurements. Secondly, these
results are confirmed by a numerical analysis of the phase transition of the
l1- minimization problem. Finally, we show that the spread spectrum technique
remains effective in an analog setting with chirp modulation for application to
realistic Fourier imaging. We illustrate these findings in the context of radio
interferometry and magnetic resonance imaging.Comment: Submitted for publication in EURASIP Journal on Advances in Signal
Processin
On sparsity averaging
Recent developments in Carrillo et al. (2012) and Carrillo et al. (2013)
introduced a novel regularization method for compressive imaging in the context
of compressed sensing with coherent redundant dictionaries. The approach relies
on the observation that natural images exhibit strong average sparsity over
multiple coherent frames. The associated reconstruction algorithm, based on an
analysis prior and a reweighted scheme, is dubbed Sparsity Averaging
Reweighted Analysis (SARA). We review these advances and extend associated
simulations establishing the superiority of SARA to regularization methods
based on sparsity in a single frame, for a generic spread spectrum acquisition
and for a Fourier acquisition of particular interest in radio astronomy.Comment: 4 pages, 3 figures, Proceedings of 10th International Conference on
Sampling Theory and Applications (SampTA), Code available at
https://github.com/basp-group/sopt, Full journal letter available at
http://arxiv.org/abs/arXiv:1208.233
Modulated Unit-Norm Tight Frames for Compressed Sensing
In this paper, we propose a compressed sensing (CS) framework that consists
of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a
column-wise orthonormal matrix. We prove that this structure satisfies the
restricted isometry property (RIP) with high probability if the number of
measurements for -sparse signals of length
and if the column-wise orthonormal matrix is bounded. Some existing structured
sensing models can be studied under this framework, which then gives tighter
bounds on the required number of measurements to satisfy the RIP. More
importantly, we propose several structured sensing models by appealing to this
unified framework, such as a general sensing model with arbitrary/determinisic
subsamplers, a fast and efficient block compressed sensing scheme, and
structured sensing matrices with deterministic phase modulations, all of which
can lead to improvements on practical applications. In particular, one of the
constructions is applied to simplify the transceiver design of CS-based channel
estimation for orthogonal frequency division multiplexing (OFDM) systems.Comment: submitted to IEEE Transactions on Signal Processin
Uniform Recovery from Subgaussian Multi-Sensor Measurements
Parallel acquisition systems are employed successfully in a variety of
different sensing applications when a single sensor cannot provide enough
measurements for a high-quality reconstruction. In this paper, we consider
compressed sensing (CS) for parallel acquisition systems when the individual
sensors use subgaussian random sampling. Our main results are a series of
uniform recovery guarantees which relate the number of measurements required to
the basis in which the solution is sparse and certain characteristics of the
multi-sensor system, known as sensor profile matrices. In particular, we derive
sufficient conditions for optimal recovery, in the sense that the number of
measurements required per sensor decreases linearly with the total number of
sensors, and demonstrate explicit examples of multi-sensor systems for which
this holds. We establish these results by proving the so-called Asymmetric
Restricted Isometry Property (ARIP) for the sensing system and use this to
derive both nonuniversal and universal recovery guarantees. Compared to
existing work, our results not only lead to better stability and robustness
estimates but also provide simpler and sharper constants in the measurement
conditions. Finally, we show how the problem of CS with block-diagonal sensing
matrices can be viewed as a particular case of our multi-sensor framework.
Specializing our results to this setting leads to a recovery guarantee that is
at least as good as existing results.Comment: 37 pages, 5 figure
Measure What Should be Measured: Progress and Challenges in Compressive Sensing
Is compressive sensing overrated? Or can it live up to our expectations? What
will come after compressive sensing and sparsity? And what has Galileo Galilei
got to do with it? Compressive sensing has taken the signal processing
community by storm. A large corpus of research devoted to the theory and
numerics of compressive sensing has been published in the last few years.
Moreover, compressive sensing has inspired and initiated intriguing new
research directions, such as matrix completion. Potential new applications
emerge at a dazzling rate. Yet some important theoretical questions remain
open, and seemingly obvious applications keep escaping the grip of compressive
sensing. In this paper I discuss some of the recent progress in compressive
sensing and point out key challenges and opportunities as the area of
compressive sensing and sparse representations keeps evolving. I also attempt
to assess the long-term impact of compressive sensing
On Variable Density Compressive Sampling
We advocate an optimization procedure for variable density sampling in the
context of compressed sensing. In this perspective, we introduce a minimization
problem for the coherence between the sparsity and sensing bases, whose
solution provides an optimized sampling profile. This minimization problem is
solved with the use of convex optimization algorithms. We also propose a
refinement of our technique when prior information is available on the signal
support in the sparsity basis. The effectiveness of the method is confirmed by
numerical experiments. Our results also provide a theoretical underpinning to
state-of-the-art variable density Fourier sampling procedures used in magnetic
resonance imaging
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
Recommended from our members
Compressed sensing: Reconstruction of non-uniformly sampled multidimensional NMR data
© 2018 Wiley Periodicals, Inc. Nuclear magnetic resonance (NMR) spectroscopy is widely used across the physical, chemical, and biological sciences. A core component of NMR studies is multidimensional experiments, which enable correlation of properties from one or more NMR-active nuclei. In high-resolution biomolecular NMR, common nuclei are1H,15N, and13C, and triple resonance experiments using these three nuclei form the backbone of NMR structural studies. In other fields, a range of other nuclei may be used. Multidimensional NMR experiments provide unparalleled information content, but this comes at the price of long experiment times required to achieve the necessary resolution and sensitivity. Non-uniform sampling (NUS) techniques to reduce the required data sampling have existed for many decades. Recently, such techniques have received heightened interest due to the development of compressed sensing (CS) methods for reconstructing spectra from such NUS datasets. When applied jointly, these methods provide a powerful approach to dramatically improve the resolution of spectra per time unit and under suitable conditions can also lead to signal-to-noise ratio improvements. In this review, we explore the basis of NUS approaches, the fundamental features of NUS reconstruction using CS and applications based on CS approaches including the benefits of expanding the repertoire of biomolecular NMR experiments into higher dimensions. We discuss some of the recent algorithms and software packages and provide practical tips for recording and processing NUS data by CS
Fast Fiber Orientation Estimation in Diffusion MRI from kq-Space Sampling and Anatomical Priors
High spatio-angular resolution diffusion MRI (dMRI) has been shown to provide
accurate identification of complex fiber configurations, albeit at the cost of
long acquisition times. We propose a method to recover intra-voxel fiber
configurations at high spatio-angular resolution relying on a kq-space
under-sampling scheme to enable accelerated acquisitions. The inverse problem
for reconstruction of the fiber orientation distribution (FOD) is regularized
by a structured sparsity prior promoting simultaneously voxelwise sparsity and
spatial smoothness of fiber orientation. Prior knowledge of the spatial
distribution of white matter, gray matter and cerebrospinal fluid is also
assumed. A minimization problem is formulated and solved via a forward-backward
convex optimization algorithmic structure. Simulations and real data analysis
suggest that accurate FOD mapping can be achieved from severe kq-space
under-sampling regimes, potentially enabling high spatio-angular dMRI in the
clinical setting.Comment: 10 pages, 5 figures, Supplementary Material
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