Sigma delta quantization for compressed sensing

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 δ. The robust recovery result of Candès et ale and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ℓ¹ recovery yields an approximation of the original sparse signal with an accuracy of O (δ). 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 R^N, the reconstruction accuracy can be improved by a factor of (m/k)^(r-1/2)α for any 0 < α < 1 if m ≳_γ k(log N)^1/(1-α) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via γth-order Sobolev dual frames.