14,370 research outputs found
Convex Optimization Approaches for Blind Sensor Calibration using Sparsity
We investigate a compressive sensing framework in which the sensors introduce
a distortion to the measurements in the form of unknown gains. We focus on
blind calibration, using measures performed on multiple unknown (but sparse)
signals and formulate the joint recovery of the gains and the sparse signals as
a convex optimization problem. We divide this problem in 3 subproblems with
different conditions on the gains, specifially (i) gains with different
amplitude and the same phase, (ii) gains with the same amplitude and different
phase and (iii) gains with different amplitude and phase. In order to solve the
first case, we propose an extension to the basis pursuit optimization which can
estimate the unknown gains along with the unknown sparse signals. For the
second case, we formulate a quadratic approach that eliminates the unknown
phase shifts and retrieves the unknown sparse signals. An alternative form of
this approach is also formulated to reduce complexity and memory requirements
and provide scalability with respect to the number of input signals. Finally
for the third case, we propose a formulation that combines the earlier two
approaches to solve the problem. The performance of the proposed algorithms is
investigated extensively through numerical simulations, which demonstrates that
simultaneous signal recovery and calibration is possible with convex methods
when sufficiently many (unknown, but sparse) calibrating signals are provided
Power-Constrained Sparse Gaussian Linear Dimensionality Reduction over Noisy Channels
In this paper, we investigate power-constrained sensing matrix design in a
sparse Gaussian linear dimensionality reduction framework. Our study is carried
out in a single--terminal setup as well as in a multi--terminal setup
consisting of orthogonal or coherent multiple access channels (MAC). We adopt
the mean square error (MSE) performance criterion for sparse source
reconstruction in a system where source-to-sensor channel(s) and
sensor-to-decoder communication channel(s) are noisy. Our proposed sensing
matrix design procedure relies upon minimizing a lower-bound on the MSE in
single-- and multiple--terminal setups. We propose a three-stage sensing matrix
optimization scheme that combines semi-definite relaxation (SDR) programming, a
low-rank approximation problem and power-rescaling. Under certain conditions,
we derive closed-form solutions to the proposed optimization procedure. Through
numerical experiments, by applying practical sparse reconstruction algorithms,
we show the superiority of the proposed scheme by comparing it with other
relevant methods. This performance improvement is achieved at the price of
higher computational complexity. Hence, in order to address the complexity
burden, we present an equivalent stochastic optimization method to the problem
of interest that can be solved approximately, while still providing a superior
performance over the popular methods.Comment: Accepted for publication in IEEE Transactions on Signal Processing
(16 pages
Optimized Compressed Sensing Matrix Design for Noisy Communication Channels
We investigate a power-constrained sensing matrix design problem for a
compressed sensing framework. We adopt a mean square error (MSE) performance
criterion for sparse source reconstruction in a system where the
source-to-sensor channel and the sensor-to-decoder communication channel are
noisy. Our proposed sensing matrix design procedure relies upon minimizing a
lower-bound on the MSE. Under certain conditions, we derive closed-form
solutions to the optimization problem. Through numerical experiments, by
applying practical sparse reconstruction algorithms, we show the strength of
the proposed scheme by comparing it with other relevant methods. We discuss the
computational complexity of our design method, and develop an equivalent
stochastic optimization method to the problem of interest that can be solved
approximately with a significantly less computational burden. We illustrate
that the low-complexity method still outperforms the popular competing methods.Comment: Submitted to IEEE ICC 2015 (EXTENDED VERSION
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
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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