  Most compressed sensing algorithms do not account for the effect of
saturation in noisy compressed measurements, though saturation is an important
consequence of the limited dynamic range of existing sensors. The few
algorithms that handle saturation effects either simply discard saturated
measurements, or impose additional constraints to ensure consistency of the
estimated signal with the saturated measurements (based on a known saturation
threshold) given uniform-bounded noise. In this paper, we instead propose a new
data fidelity function which is directly based on ensuring a certain form of
consistency between the signal and the saturated measurements, and can be
expressed as the negative logarithm of a certain carefully designed likelihood
function. Our estimator works even in the case of Gaussian noise (which is
unbounded) in the measurements. We prove that our data fidelity function is
convex. We moreover, show that it satisfies the condition of Restricted Strong
Convexity and thereby derive an upper bound on the performance of the
estimator. We also show that our technique experimentally yields results
superior to the state of the art under a wide variety of experimental settings,
for compressive signal recovery from noisy and saturated measurements.
