  This work focuses on the reconstruction of sparse signals from their 1-bit
measurements. The context is the one of 1-bit compressive sensing where the
measurements amount to quantizing (dithered) random projections. Our main
contribution shows that, in addition to the measurement process, we can
additionally reconstruct the signal with a binarization of the sensing matrix.
This binary representation of both the measurements and sensing matrix can
dramatically simplify the hardware architecture on embedded systems, enabling
cheaper and more power efficient alternatives. Within this framework, given a
sensing matrix respecting the restricted isometry property (RIP), we prove that
for any sparse signal the quantized projected back-projection (QPBP) algorithm
achieves a reconstruction error decaying like O(m-1/2)when the number of
measurements m increases. Simulations highlight the practicality of the
developed scheme for different sensing scenarios, including random partial
Fourier sensing.
