  In this paper we study the problem of recovering sparse or compressible
signals from uniformly quantized measurements. We present a new class of convex
optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment
$p$ (BPDQ$_p$), that model the quantization distortion more faithfully than the
commonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by
minimizing the sparsity of the signal to be reconstructed subject to a
data-fidelity constraint expressed in the $\ell_p$-norm of the residual error
for $2\leq p\leq \infty$.
  We show theoretically that, (i) the reconstruction error of these new
decoders is bounded if the sensing matrix satisfies an extended Restricted
Isometry Property involving the $\ell_p$ norm, and (ii), for Gaussian random
matrices and uniformly quantized measurements, BPDQ$_p$ performance exceeds
that of BPDN by dividing the reconstruction error due to quantization by
$\sqrt{p+1}$. This last effect happens with high probability when the number of
measurements exceeds a value growing with $p$, i.e. in an oversampled situation
compared to what is commonly required by BPDN = BPDQ$_2$. To demonstrate the
theoretical power of BPDQ$_p$, we report numerical simulations on signal and
image reconstruction problems.
