In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma-Delta (Σ∆) quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.
Compressed sensing is now established as an effective method for dimension reduction when the underlying signals are sparse or compressible with respect to some suitable basis or frame. One important, yet under-addressed problem regarding the compressive acquisition of analog signals is how to perform quantization. This is directly related to the important issues of how "compressed" compressed sensing is (in terms of the total number of bits one ends up using after acquiring the signal) and ultimately whether compressed sensing can be used to obtain compressed representations of suitable signals. Building on our recent work, we propose a concrete and practicable method for performing "analog-to-information conversion" . Following a compressive signal acquisition stage, the proposed method consists of a quantization stage, based on Σ∆ (sigma-delta) quantization, and a subsequent encoding (compression) stage that fits within the framework of compressed sensing seamlessly. We prove that, using this method, we can convert analog compressive samples to compressed digital bitstreams and decode using tractable algorithms based on convex optimization. We prove that the proposed AIC provides a nearly optimal encoding of sparse and compressible signals. Finally, we present numerical experiments illustrating the effectiveness of the proposed analog-to-information converter. Encoding (Compression):The quantized measurements require log 2 |C 0 | bits to be represented. Often, we can reduce this bit budget by incorporating an encoding stage. We denote by E : C 0 → C, the encoding map, where C is a finite set called the codebook, usually satisfying log 2 |C| ≪ log 2 |C 0 |. The goal of encoding is to reduce the number of bits while still permitting accurate reconstruction. We focus on simple encoding schemes implemented via, e.g., discrete Johnson-Lindenstrauss embeddings [3] (cf. [4, 5, 6, 7]).Reconstruction: The final stage of a compressive AIC is the reconstruction or decoding stage where we recover an approximation to the original signal x. To that end we use a map ∆ : C → R N .
The null space property (NSP) and the restricted isometry property (RIP) are two properties which have received considerable attention in the compressed sensing literature. As the name suggests, NSP is a property that depends solely on the null space of the measurement procedure and as such, any two matrices which have the same null space will have NSP if either one of them does. On the other hand, RIP is a property of the measurement procedure itself, and given an RIP matrix it is straightforward to construct another matrix with the same null space that is not RIP. We say a matrix is RIP-NSP if it has the same null space as an RIP matrix. We show that such matrices can provide robust recovery of compressible signals under Basis pursuit. More importantly, we constructively show that the RIP-NSP is stronger than NSP with the aid of this robust recovery result, which shows that RIP is fundamentally stronger than NSP.
Sigma Delta (Σ∆) quantization, a quantization method which first surfaced in the 1960s, has now been used widely in various digital products such as cameras, cell phones, radars, etc. The method samples an input signal at a rate higher than the Nyquist rate, thus achieves great robustness to quantization noise.Compressed Sensing (CS) is a frugal acquisition method that utilizes the possible sparsity of the signals to reduce the required number of samples for a lossless acquisition. One can deem the reduced number as an effective dimensionality of the set of sparse signals and accordingly, define an effective oversampling rate as the ratio between the actual sampling rate and the effective dimensionality. A natural conjecture is that the error of Sigma Delta quantization, previously shown to decay with the vanilla oversampling rate, should now decay with the effective oversampling rate when carried out in the regime of compressed sensing. Confirming this intuition is one of the main goals in this direction.The study of quantization in CS has so far been limited to proving error convergence results for Gaussian and sub-Gaussian sensing matrices, as the number of bits and/or the number of samples grow to infinity. In this paper, we provide a first result for the more realistic Fourier sensing matrices. The major idea is to randomly permute the Fourier samples before feeding them into the quantizer. We show that the random permutation can effectively increase the low frequency power of the measurements, thus enhance the quality of Σ∆ quantization.
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