4,727 research outputs found
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
Compressed Sensing aims to capture attributes of -sparse signals using
very few measurements. In the standard Compressed Sensing paradigm, the
\m\times \n measurement matrix \A is required to act as a near isometry on
the set of all -sparse signals (Restricted Isometry Property or RIP).
Although it is known that certain probabilistic processes generate \m \times
\n matrices that satisfy RIP with high probability, there is no practical
algorithm for verifying whether a given sensing matrix \A has this property,
crucial for the feasibility of the standard recovery algorithms. In contrast
this paper provides simple criteria that guarantee that a deterministic sensing
matrix satisfying these criteria acts as a near isometry on an overwhelming
majority of -sparse signals; in particular, most such signals have a unique
representation in the measurement domain. Probability still plays a critical
role, but it enters the signal model rather than the construction of the
sensing matrix. We require the columns of the sensing matrix to form a group
under pointwise multiplication. The construction allows recovery methods for
which the expected performance is sub-linear in \n, and only quadratic in
\m; the focus on expected performance is more typical of mainstream signal
processing than the worst-case analysis that prevails in standard Compressed
Sensing. Our framework encompasses many families of deterministic sensing
matrices, including those formed from discrete chirps, Delsarte-Goethals codes,
and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in
Signal Processing, the special issue on Compressed Sensin
Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions
In this paper, we study the problem of compressed sensing using binary
measurement matrices and -norm minimization (basis pursuit) as the
recovery algorithm. We derive new upper and lower bounds on the number of
measurements to achieve robust sparse recovery with binary matrices. We
establish sufficient conditions for a column-regular binary matrix to satisfy
the robust null space property (RNSP) and show that the associated sufficient
conditions % sparsity bounds for robust sparse recovery obtained using the RNSP
are better by a factor of compared to the
sufficient conditions obtained using the restricted isometry property (RIP).
Next we derive universal \textit{lower} bounds on the number of measurements
that any binary matrix needs to have in order to satisfy the weaker sufficient
condition based on the RNSP and show that bipartite graphs of girth six are
optimal. Then we display two classes of binary matrices, namely parity check
matrices of array codes and Euler squares, which have girth six and are nearly
optimal in the sense of almost satisfying the lower bound. In principle,
randomly generated Gaussian measurement matrices are "order-optimal". So we
compare the phase transition behavior of the basis pursuit formulation using
binary array codes and Gaussian matrices and show that (i) there is essentially
no difference between the phase transition boundaries in the two cases and (ii)
the CPU time of basis pursuit with binary matrices is hundreds of times faster
than with Gaussian matrices and the storage requirements are less. Therefore it
is suggested that binary matrices are a viable alternative to Gaussian matrices
for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
A Transfer Operator Methodology for Optimal Sensor Placement Accounting for Uncertainty
Sensors in buildings are used for a wide variety of applications such as
monitoring air quality, contaminants, indoor temperature, and relative
humidity. These are used for accessing and ensuring indoor air quality, and
also for ensuring safety in the event of chemical and biological attacks. It
follows that optimal placement of sensors become important to accurately
monitor contaminant levels in the indoor environment. However, contaminant
transport inside the indoor environment is governed by the indoor flow
conditions which are affected by various uncertainties associated with the
building systems including occupancy and boundary fluxes. Therefore, it is
important to account for all associated uncertainties while designing the
sensor layout. The transfer operator based framework provides an effective way
to identify optimal placement of sensors. Previous work has been limited to
sensor placements under deterministic scenarios. In this work we extend the
transfer operator based approach for optimal sensor placement while accounting
for building systems uncertainties. The methodology provides a probabilistic
metric to gauge coverage under uncertain conditions. We illustrate the
capabilities of the framework with examples exhibiting boundary flux
uncertainty
Sparse Recovery Analysis of Preconditioned Frames via Convex Optimization
Orthogonal Matching Pursuit and Basis Pursuit are popular reconstruction
algorithms for recovery of sparse signals. The exact recovery property of both
the methods has a relation with the coherence of the underlying redundant
dictionary, i.e. a frame. A frame with low coherence provides better guarantees
for exact recovery. An equivalent formulation of the associated linear system
is obtained via premultiplication by a non-singular matrix. In view of bounds
that guarantee sparse recovery, it is very useful to generate the
preconditioner in such way that the preconditioned frame has low coherence as
compared to the original. In this paper, we discuss the impact of
preconditioning on sparse recovery. Further, we formulate a convex optimization
problem for designing the preconditioner that yields a frame with improved
coherence. In addition to reducing coherence, we focus on designing well
conditioned frames and numerically study the relationship between the condition
number of the preconditioner and the coherence of the new frame. Alongside
theoretical justifications, we demonstrate through simulations the efficacy of
the preconditioner in reducing coherence as well as recovering sparse signals.Comment: 9 pages, 5 Figure
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