409 research outputs found
Weighted β1-Minimization for Sparse Recovery under Arbitrary Prior Information
Weighted β1-minimization has been studied as a technique for the reconstruction of a sparse signal from compressively sampled measurements when prior information about the signal, in the form of a support estimate, is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted β1-minimization when arbitrarily many distinct weights are permitted. For example, such a setup might be used when one has multiple estimates for the support of a signal, and these estimates have varying degrees of accuracy. Our analysis yields an extension to existing works that assume only a single constant weight is used. We include numerical experiments, with both synthetic signals and real video data, that demonstrate the benefits of allowing non-uniform weights in the reconstruction procedure
Universally Elevating the Phase Transition Performance of Compressed Sensing: Non-Isometric Matrices are Not Necessarily Bad Matrices
In compressed sensing problems, minimization or Basis Pursuit was
known to have the best provable phase transition performance of recoverable
sparsity among polynomial-time algorithms. It is of great theoretical and
practical interest to find alternative polynomial-time algorithms which perform
better than minimization. \cite{Icassp reweighted l_1}, \cite{Isit
reweighted l_1}, \cite{XuScaingLaw} and \cite{iterativereweightedjournal} have
shown that a two-stage re-weighted minimization algorithm can boost
the phase transition performance for signals whose nonzero elements follow an
amplitude probability density function (pdf) whose -th derivative
for some integer . However, for signals whose
nonzero elements are strictly suspended from zero in distribution (for example,
constant-modulus, only taking values `' or `' for some nonzero real
number ), no polynomial-time signal recovery algorithms were known to
provide better phase transition performance than plain minimization,
especially for dense sensing matrices. In this paper, we show that a
polynomial-time algorithm can universally elevate the phase-transition
performance of compressed sensing, compared with minimization, even
for signals with constant-modulus nonzero elements. Contrary to conventional
wisdoms that compressed sensing matrices are desired to be isometric, we show
that non-isometric matrices are not necessarily bad sensing matrices. In this
paper, we also provide a framework for recovering sparse signals when sensing
matrices are not isometric.Comment: 6pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1010.2236, arXiv:1004.040
Weighted Minimization for Sparse Recovery with Prior Information
In this paper we study the compressed sensing problem of recovering a sparse
signal from a system of underdetermined linear equations when we have prior
information about the probability of each entry of the unknown signal being
nonzero. In particular, we focus on a model where the entries of the unknown
vector fall into two sets, each with a different probability of being nonzero.
We propose a weighted minimization recovery algorithm and analyze its
performance using a Grassman angle approach. We compute explicitly the
relationship between the system parameters (the weights, the number of
measurements, the size of the two sets, the probabilities of being non-zero) so
that an iid random Gaussian measurement matrix along with weighted
minimization recovers almost all such sparse signals with overwhelming
probability as the problem dimension increases. This allows us to compute the
optimal weights. We also provide simulations to demonstrate the advantages of
the method over conventional optimization.Comment: 5 Pages, Submitted to ISIT 200
A sharp recovery condition for sparse signals with partial support information via orthogonal matching pursuit
This paper considers the exact recovery of -sparse signals in the
noiseless setting and support recovery in the noisy case when some prior
information on the support of the signals is available. This prior support
consists of two parts. One part is a subset of the true support and another
part is outside of the true support. For -sparse signals with
the prior support which is composed of true indices and wrong indices,
we show that if the restricted isometry constant (RIC) of the
sensing matrix satisfies \begin{eqnarray*}
\delta_{k+b+1}<\frac{1}{\sqrt{k-g+1}}, \end{eqnarray*} then orthogonal matching
pursuit (OMP) algorithm can perfectly recover the signals from
in iterations. Moreover, we show the above
sufficient condition on the RIC is sharp. In the noisy case, we achieve the
exact recovery of the remainder support (the part of the true support outside
of the prior support) for the -sparse signals from
under appropriate conditions. For the
remainder support recovery, we also obtain a necessary condition based on the
minimum magnitude of partial nonzero elements of the signals
Improved Sparse Recovery Thresholds with Two-Step Reweighted Minimization
It is well known that minimization can be used to recover
sufficiently sparse unknown signals from compressed linear measurements. In
fact, exact thresholds on the sparsity, as a function of the ratio between the
system dimensions, so that with high probability almost all sparse signals can
be recovered from iid Gaussian measurements, have been computed and are
referred to as "weak thresholds" \cite{D}. In this paper, we introduce a
reweighted recovery algorithm composed of two steps: a standard
minimization step to identify a set of entries where the signal is
likely to reside, and a weighted minimization step where entries
outside this set are penalized. For signals where the non-sparse component has
iid Gaussian entries, we prove a "strict" improvement in the weak recovery
threshold. Simulations suggest that the improvement can be quite
impressive-over 20% in the example we consider.Comment: accepted in ISIT 201
Recursive Recovery of Sparse Signal Sequences from Compressive Measurements: A Review
In this article, we review the literature on design and analysis of recursive
algorithms for reconstructing a time sequence of sparse signals from
compressive measurements. The signals are assumed to be sparse in some
transform domain or in some dictionary. Their sparsity patterns can change with
time, although, in many practical applications, the changes are gradual. An
important class of applications where this problem occurs is dynamic projection
imaging, e.g., dynamic magnetic resonance imaging (MRI) for real-time medical
applications such as interventional radiology, or dynamic computed tomography.Comment: To appear in IEEE Trans. Signal Processin
Efficient Spectrum Availability Information Recovery for Wideband DSA Networks: A Weighted Compressive Sampling Approach
Compressive sampling has great potential for making wideband spectrum sensing
possible at sub-Nyquist sampling rates. As a result, there have recently been
research efforts that leverage compressive sampling to enable efficient
wideband spectrum sensing. These efforts consider homogenous wideband spectrum,
where all bands are assumed to have similar PU traffic characteristics. In
practice, however, wideband spectrum is not homogeneous, in that different
spectrum bands could present different PU occupancy patterns. In fact, the
nature of spectrum assignment, in which applications of similar types are often
assigned bands within the same block, dictates that wideband spectrum is indeed
heterogeneous. In this paper, we consider heterogeneous wideband spectrum, and
exploit its inherent, block-like structure to design efficient compressive
spectrum sensing techniques that are well suited for heterogeneous wideband
spectrum. We propose a weighted minimization sensing information
recovery algorithm that achieves more stable recovery than that achieved by
existing approaches while accounting for the variations of spectrum occupancy
across both the time and frequency dimensions. In addition, we show that our
proposed algorithm requires a lesser number of sensing measurements when
compared to the state-of-the-art approaches
Sliced-Inverse-Regression-Aided Rotated Compressive Sensing Method for Uncertainty Quantification
Compressive-sensing-based uncertainty quantification methods have become a
pow- erful tool for problems with limited data. In this work, we use the sliced
inverse regression (SIR) method to provide an initial guess for the alternating
direction method, which is used to en- hance sparsity of the Hermite polynomial
expansion of stochastic quantity of interest. The sparsity improvement
increases both the efficiency and accuracy of the compressive-sensing- based
uncertainty quantification method. We demonstrate that the initial guess from
SIR is more suitable for cases when the available data are limited (Algorithm
4). We also propose another algorithm (Algorithm 5) that performs dimension
reduction first with SIR. Then it constructs a Hermite polynomial expansion of
the reduced model. This method affords the ability to approximate the
statistics accurately with even less available data. Both methods are
non-intrusive and require no a priori information of the sparsity of the
system. The effec- tiveness of these two methods (Algorithms 4 and 5) are
demonstrated using problems with up to 500 random dimensions.Comment: In section 4, numerical examples 3-5, replaced the mean of the error
with the quantiles and mean of the error. Added section 4.6 to compare
different method
Compressed sensing for longitudinal MRI: An adaptive-weighted approach
Purpose: Repeated brain MRI scans are performed in many clinical scenarios,
such as follow up of patients with tumors and therapy response assessment. In
this paper, the authors show an approach to utilize former scans of the patient
for the acceleration of repeated MRI scans.
Methods: The proposed approach utilizes the possible similarity of the
repeated scans in longitudinal MRI studies. Since similarity is not guaranteed,
sampling and reconstruction are adjusted during acquisition to match the actual
similarity between the scans. The baseline MR scan is utilized both in the
sampling stage, via adaptive sampling, and in the reconstruction stage, with
weighted reconstruction. In adaptive sampling, k-space sampling locations are
optimized during acquisition. Weighted reconstruction uses the locations of the
nonzero coefficients in the sparse domains as a prior in the recovery process.
The approach was tested on 2D and 3D MRI scans of patients with brain tumors.
Results: The longitudinal adaptive CS MRI (LACS-MRI) scheme provides
reconstruction quality which outperforms other CS-based approaches for rapid
MRI. Examples are shown on patients with brain tumors and demonstrate improved
spatial resolution. Compared with data sampled at Nyquist rate, LACS-MRI
exhibits Signal-to-Error Ratio (SER) of 24.8dB with undersampling factor of
16.6 in 3D MRI.
Conclusions: The authors have presented a novel method for image
reconstruction utilizing similarity of scans in longitudinal MRI studies, where
possible. The proposed approach can play a major part and significantly reduce
scanning time in many applications that consist of disease follow-up and
monitoring of longitudinal changes in brain MRI
Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations
Compressive sensing has become a powerful addition to uncertainty
quantification in recent years. This paper identifies new bases for random
variables through linear mappings such that the representation of the quantity
of interest is more sparse with new basis functions associated with the new
random variables. This sparsity increases both the efficiency and accuracy of
the compressive sensing-based uncertainty quantification method. Specifically,
we consider rotation-based linear mappings which are determined iteratively for
Hermite polynomial expansions. We demonstrate the effectiveness of the new
method with applications in solving stochastic partial differential equations
and high-dimensional () problems
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