Weighted fidelity in non-uniformly quantized compressed sensing

Following the Compressed Sensing (CS) paradigm, this paper studies the problem of recovering sparse or compressible signals from (scalar) non-uniformly quantized measurements. We show that a simple adaptation of the Basis Pursuit De-Quantizer introduced earlier, that is, a sign sensitive weighting of their ℓ_p-norm fidelity constraint, yields good SNR improvements in the signal reconstruction. As a good indication of this improvement origin, we prove theoretically that a similar decoder, using a particular side-position-to-level oracle, displays a reduction of the reconstruction error when both the number of measurements and the moment p of the constraint increase. This follows the oversampling principle underlined in our previous study for uniformly quantized CS, with an additional gain provided by the non-uniform quantization. We conclude this paper by showing the efficiency of the approach on 1-D and 2-D signal examples.