Compressed Sensing (CS) is a sampling paradigm used for acquiring sparse or compressible signals from a seemingly incomplete set of measurements. In any practical application with our digitally driven technology, these "compressive measurements" are quantized and thus they do not have infinite precision. So far, the theory of quantization in CS has mainly focused on compressive sampling systems designed with random measurement matrices. In this note, we turn our attention to "deterministic compressed sensing". Specifically, we focus on quantization in CS with chirp sensing matrices and present quantization approaches and numerical experiments.
In this paper we derive weighted and reweighted AMP algorithms for signal reconstruction from compressed sensing measurements. Weighted AMP incorporates prior support information into the AMP algorithm and iteratively solves the weighted I<sub>1</sub> minimization which is much faster than the usual linear programming algorithms used to solve this problem. We also introduce a reweighting scheme for regular and weighted AMP algorithms which enhances the recovery performance of both regular and weighted AMP while still maintaining the low complexity nature of AMP algorithms.
In this paper, we study the support recovery conditions of weighted ℓ<sub>1</sub> minimization for signal reconstruction
from compressed sensing measurements when multiple support estimate sets with different accuracy are available.
We identify a class of signals for which the recovered vector from ℓ<sub>1</sub> minimization provides an accurate support
estimate. We then derive stability and robustness guarantees for the weighted ℓ<sub>1</sub> minimization problem with
more than one support estimate. We show that applying a smaller weight to support estimate that enjoy higher
accuracy improves the recovery conditions compared with the case of a single support estimate and the case with
standard, i.e., non-weighted, ℓ<sub>1</sub> minimization. Our theoretical results are supported by numerical simulations on
synthetic signals and real audio signals.
In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable <i>r<sup>th</sup></i> order ΣΔ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual corresponding to the well known Blackman filter.
Sigma-Delta (ΣΔ) schemes are shown to be an effective approach for quantizing finite frame expansions. Basic error estimates show that first order ΣΔ schemes can achieve quantization error of order 1/<i>N</i>, where <i>N</i> is the frame size. Under certain technical assumptions, improved quantization error estimates of order (log<i>N</i>)/<i>N</i><sup>1.25</sup> are obtained. For the second order ΣΔ scheme with linear quantization rule, error estimates of order 1/<i>N</i><sup>2</sup> can be achieved in certain circumstances. Such estimates rely critically on being able to construct sufficiently small invariant sets for the scheme. New experimental results indicate a connection between the orbits of state variables in ΣΔ schemes and the structure of constant input invariant sets.
Conference Committee Involvement (1)
26 August 2007 | San Diego, California, United States