30 research outputs found
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
Quantized Compressed Sensing for Partial Random Circulant Matrices
We provide the first analysis of a non-trivial quantization scheme for
compressed sensing measurements arising from structured measurements.
Specifically, our analysis studies compressed sensing matrices consisting of
rows selected at random, without replacement, from a circulant matrix generated
by a random subgaussian vector. We quantize the measurements using stable,
possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on
convex optimization. We show that the part of the reconstruction error due to
quantization decays polynomially in the number of measurements. This is in line
with analogous results on Sigma-Delta quantization associated with random
Gaussian or subgaussian matrices, and significantly better than results
associated with the widely assumed memoryless scalar quantization. Moreover, we
prove that our approach is stable and robust; i.e., the reconstruction error
degrades gracefully in the presence of non-quantization noise and when the
underlying signal is not strictly sparse. The analysis relies on results
concerning subgaussian chaos processes as well as a variation of McDiarmid's
inequality.Comment: 15 page