Sigma delta quantization for compressed sensing

Abstract

Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size 8. The robust recovery result of Candes et ale and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, £1 recovery yields an approximation of the original sparse signal with an accuracy of 0 (8). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in }RN, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)a for any 0 < a < 1 if m~r k(log N)l/(l-a) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via rth-order Sobolev dual frames.

One‐Bit Compressed Sensing by Linear Programming

“…However, the number of bits per measurement there is not one (or constant); this number depends on the sparsity level s and the dynamic range of the signal x. Similarly, in the work of Gunturk et al [14,15] on sigma-delta quantization, the number of bits per measurement depends on the dynamic range of x. On the other hand, by considering sigmadelta quantization and multiple bits, the Gunturk et al are able to provide excellent guarantees on the speed of decay of the error δ as s/m decreases.…”

Joint Sampling Rate and Bit-Depth Optimization in Compressive Video Sampling

“…Thus, a more precise model of CS acquisition [16] is (6) where is a -bit scalar quantization function (applied element-wise in (6)) that maps real-valued CS measurements to the discrete alphabet with . Since the quantizer is scalar, we can write the bit-budget constraint as (7) Although we will focus on scalar quantization in this paper, alternative quantization techniques such as sigma-delta [17] or non-monotonic scalar quantization [18] have also been proposed for CS, as have many algorithms specialized to CS quantization problems. The main RDO theme presented here will be generally applicable to these techniques and algorithms as well.…”

An uncertainty principle for finite frames

“…the Moore-Penrose inverse, is optimal but alternate duals have also been shown to be useful [10] in shaping error. In recent work, non-canonical duals that minimize a finite version of df dx have been applied in analog to digital conversion [2,[7][8][9] to reduce quantization error in reconstruction. We combine these two notions and generate alternate duals that are optimal with respect to the finite time-frequency measure developed below.…”