Quantized compressive sensing (QCS) deals with the problem of coding
compressive measurements of low-complexity signals with quantized, finite
precision representations, i.e., a mandatory process involved in any practical
sensing model. While the resolution of this quantization clearly impacts the
quality of signal reconstruction, there actually exist incompatible
combinations of quantization functions and sensing matrices that proscribe
arbitrarily low reconstruction error when the number of measurements increases.
This work shows that a large class of random matrix constructions known to
respect the restricted isometry property (RIP) is "compatible" with a simple
scalar and uniform quantization if a uniform random vector, or a random dither,
is added to the compressive signal measurements before quantization. In the
context of estimating low-complexity signals (e.g., sparse or compressible
signals, low-rank matrices) from their quantized observations, this
compatibility is demonstrated by the existence of (at least) one signal
reconstruction method, the projected back projection (PBP), whose
reconstruction error decays when the number of measurements increases.
Interestingly, given one RIP matrix and a single realization of the dither, a
small reconstruction error can be proved to hold uniformly for all signals in
the considered low-complexity set. We confirm these observations numerically in
several scenarios involving sparse signals, low-rank matrices, and compressible
signals, with various RIP matrix constructions such as sub-Gaussian random
matrices and random partial discrete cosine transform (DCT) matrices.

研究了在估计稀疏或可压缩信号时，使用某些随机矩阵构造、添加均匀随机向量或随机噪声来量化信号，可以使信号重建误差降低。并分别在多种情境下进行数值验证。