489 research outputs found
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
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
Time for dithering: fast and quantized random embeddings via the restricted isometry property
Recently, many works have focused on the characterization of non-linear
dimensionality reduction methods obtained by quantizing linear embeddings,
e.g., to reach fast processing time, efficient data compression procedures,
novel geometry-preserving embeddings or to estimate the information/bits stored
in this reduced data representation. In this work, we prove that many linear
maps known to respect the restricted isometry property (RIP) can induce a
quantized random embedding with controllable multiplicative and additive
distortions with respect to the pairwise distances of the data points beings
considered. In other words, linear matrices having fast matrix-vector
multiplication algorithms (e.g., based on partial Fourier ensembles or on the
adjacency matrix of unbalanced expanders) can be readily used in the definition
of fast quantized embeddings with small distortions. This implication is made
possible by applying right after the linear map an additive and random "dither"
that stabilizes the impact of the uniform scalar quantization operator applied
afterwards. For different categories of RIP matrices, i.e., for different
linear embeddings of a metric space
in with , we derive upper bounds on the
additive distortion induced by quantization, showing that it decays either when
the embedding dimension increases or when the distance of a pair of
embedded vectors in decreases. Finally, we develop a novel
"bi-dithered" quantization scheme, which allows for a reduced distortion that
decreases when the embedding dimension grows and independently of the
considered pair of vectors.Comment: Keywords: random projections, non-linear embeddings, quantization,
dither, restricted isometry property, dimensionality reduction, compressive
sensing, low-complexity signal models, fast and structured sensing matrices,
quantized rank-one projections (31 pages
Compressive Imaging Using RIP-Compliant CMOS Imager Architecture and Landweber Reconstruction
In this paper, we present a new image sensor architecture for fast and accurate compressive sensing (CS) of natural images. Measurement matrices usually employed in CS CMOS image sensors are recursive pseudo-random binary matrices. We have proved that the restricted isometry property of these matrices is limited by a low sparsity constant. The quality of these matrices is also affected by the non-idealities of pseudo-random number generators (PRNG). To overcome these limitations, we propose a hardware-friendly pseudo-random ternary measurement matrix generated on-chip by means of class III elementary cellular automata (ECA). These ECA present a chaotic behavior that emulates random CS measurement matrices better than other PRNG. We have combined this new architecture with a block-based CS smoothed-projected Landweber reconstruction algorithm. By means of single value decomposition, we have adapted this algorithm to perform fast and precise reconstruction while operating with binary and ternary matrices. Simulations are provided to qualify the approach.Ministerio de Economía y Competitividad TEC2015-66878-C3-1-RJunta de Andalucía TIC 2338-2013Office of Naval Research (USA) N000141410355European Union H2020 76586
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
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