1,854 research outputs found
Channel-Optimized Vector Quantizer Design for Compressed Sensing Measurements
We consider vector-quantized (VQ) transmission of compressed sensing (CS)
measurements over noisy channels. Adopting mean-square error (MSE) criterion to
measure the distortion between a sparse vector and its reconstruction, we
derive channel-optimized quantization principles for encoding CS measurement
vector and reconstructing sparse source vector. The resulting necessary optimal
conditions are used to develop an algorithm for training channel-optimized
vector quantization (COVQ) of CS measurements by taking the end-to-end
distortion measure into account.Comment: Published in ICASSP 201
Quantization and Compressive Sensing
Quantization is an essential step in digitizing signals, and, therefore, an
indispensable component of any modern acquisition system. This book chapter
explores the interaction of quantization and compressive sensing and examines
practical quantization strategies for compressive acquisition systems.
Specifically, we first provide a brief overview of quantization and examine
fundamental performance bounds applicable to any quantization approach. Next,
we consider several forms of scalar quantizers, namely uniform, non-uniform,
and 1-bit. We provide performance bounds and fundamental analysis, as well as
practical quantizer designs and reconstruction algorithms that account for
quantization. Furthermore, we provide an overview of Sigma-Delta
() quantization in the compressed sensing context, and also
discuss implementation issues, recovery algorithms and performance bounds. As
we demonstrate, proper accounting for quantization and careful quantizer design
has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing
and Its Applications", 201
Compression-Based Compressed Sensing
Modern compression algorithms exploit complex structures that are present in
signals to describe them very efficiently. On the other hand, the field of
compressed sensing is built upon the observation that "structured" signals can
be recovered from their under-determined set of linear projections. Currently,
there is a large gap between the complexity of the structures studied in the
area of compressed sensing and those employed by the state-of-the-art
compression codes. Recent results in the literature on deterministic signals
aim at bridging this gap through devising compressed sensing decoders that
employ compression codes. This paper focuses on structured stochastic processes
and studies the application of rate-distortion codes to compressed sensing of
such signals. The performance of the formerly-proposed compressible signal
pursuit (CSP) algorithm is studied in this stochastic setting. It is proved
that in the very low distortion regime, as the blocklength grows to infinity,
the CSP algorithm reliably and robustly recovers instances of a stationary
process from random linear projections as long as their count is slightly more
than times the rate-distortion dimension (RDD) of the source. It is also
shown that under some regularity conditions, the RDD of a stationary process is
equal to its information dimension (ID). This connection establishes the
optimality of the CSP algorithm at least for memoryless stationary sources, for
which the fundamental limits are known. Finally, it is shown that the CSP
algorithm combined by a family of universal variable-length fixed-distortion
compression codes yields a family of universal compressed sensing recovery
algorithms
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
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