2014/05/30 by Laurent Jacques, Jacques, Laurent
Computer Science · Engineering · Mathematics · Medicine · #FOS: Computer and information sciences #Image and Signal Denoising Methods #Information Theory (cs.IT) #Medical Imaging Techniques and Applications #Sparse and Compressive Sensing Techniques #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1406.0022
24 pages, 1 figure
openalex publication_date 2014/05/30 · arxiv created 2016/04/20 · arxiv updated 2016/04/21 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
This paper provides new error bounds on "consistent" reconstruction methods for signals observed from quantized random projections. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a new observation of the estimated signal under the same sensing model. Focusing on dithered uniform scalar quantization of resolution δ>0, we prove first that, given a Gaussian random frame of \mathbb RN with M vectors, the worst-case ℓ2-error of consistent signal reconstruction decays with high probability as O((N)/(M)log(M)/(√ N)) uniformly for all signals of the unit ball \mathbb BN ⊂ \mathbb RN. Up to a log factor, this matches a known lower bound in Ω(N/M) and former empirical validations in O(N/M). Equivalently, if M exceeds a minimal number of frame coefficients growing like O((N)/(ε0)log (√ N)/(ε0)), any vectors in \mathbb BN with M identical quantized projections are at most ε0 apart with high probability. Second, in the context of Quantized Compressed Sensing with M Gaussian random measurements and under the same scalar quantization scheme, consistent reconstructions of K-sparse signals of \mathbb RN have a worst-case error that decreases with high probability as O(\tfracKMlog\tfracMN√ K3) uniformly for all such signals. Finally, we show that the proximity of vectors whose quantized random projections are only approximately consistent can still be bounded with high probability. A certain level of corruption is thus allowed in the quantization process, up to the appearance of a systematic bias in the reconstruction error of (almost) consistent signal estimates.