Noise-shaping Quantization Methods for Frame-based and Compressive Sampling Systems

Chou, Evan; Güntürk, C. Sınan; Krahmer, Felix; Saab, Rayan; Yılmaz, Özgür

doi:10.48550/arxiv.1502.05807

Cited by 2 publications

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“…Recently, noise-shaping quantizers [15] have been successfully employed in the compressive sensing framework. Traditionally such quantizers, for example Σ∆ quantizers, are used for analog-to-digital conversion of bandlimited signals and they are designed to push (most of) the quantization error outside of the signal spectrum (see, e.g., [43]).…”

Section: Noise-shaping Quantization Methods For Compressed Sensingmentioning

confidence: 99%

“…Over the last decade it has been established that noise shaping quantizers, including Σ∆ quantizers as well as a new family of quantizers based on beta encoders [14] cf. [15] (not our focus in this paper) can be used in the more general setting of linear sampling systems associated with both frames and compressed sensing.…”

Section: Noise-shaping Quantization Methods For Compressed Sensingmentioning

confidence: 99%

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Quantization of compressive samples with stable and robust recovery

Saab

Wang

Yılmaz

2018

Applied and Computational Harmonic Analysis

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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.

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Section: Noise-shaping Quantization Methods For Compressed Sensingmentioning

confidence: 99%

Section: Noise-shaping Quantization Methods For Compressed Sensingmentioning

confidence: 99%

Quantization of compressive samples with stable and robust recovery

Saab

Wang

Yılmaz

2018

Applied and Computational Harmonic Analysis

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“…Accordingly, coarse quantizers are typically cheap to implement robustly on analog hardware. However, obtaining nearly optimal rate distortion characteristics, i.e., exponential decay of approximation error as a function of the bit budget, is highly non-trivial (see, e.g., [25]). We now provide a brief (non-exhaustive) overview of the literature that is most related to our work.…”

Section: Relevant Prior Workmentioning

confidence: 99%

“…We focus primarily on memoryless scalar quantization and on Σ∆ quantization but we also give some attention to the special case of one-bit quantization due to the attention it has recently received. More detailed reviews can be found in [26,25].…”

Section: Relevant Prior Workmentioning

confidence: 99%

“…We begin by observing that many of the quantizers studied in the literature fall under the umbrella of memoryless scalar quantization, while only some use a noise-shaping (see, e.g., [25]) approach, of which Σ∆ quantization is an example. All these quantizers can be used within the fine quantization paradigm as one can make the quantizer step size as small as the desired reconstruction accuracy demands require.…”

Section: Relevant Prior Workmentioning

confidence: 99%

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From Compressed Sensing to Compressed Bit-Streams: Practical Encoders, Tractable Decoders

Saab

Wang

Yılmaz

2018

IEEE Trans. Inform. Theory

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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 .

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Noise-shaping Quantization Methods for Frame-based and Compressive Sampling Systems

Cited by 2 publications

References 0 publications

Quantization of compressive samples with stable and robust recovery

Quantization of compressive samples with stable and robust recovery

From Compressed Sensing to Compressed Bit-Streams: Practical Encoders, Tractable Decoders

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