  The problem of recovering a structured signal from its linear measurements in
the presence of speckle noise is studied. This problem appears in many imaging
systems such as synthetic aperture radar and optical coherence tomography. The
current acquisition technology oversamples signals and converts the problem
into a denoising problem with multiplicative noise. However, this paper
explores the possibility of reducing the number of measurements below the
ambient dimension of the signal. The sophistications that appear in the study
of multiplicative noises have so far impeded theoretical analysis of such
problems. This paper aims to present the first theoretical result regarding the
recovery of signals from their undersampled measurements under the speckle
noise. It is shown that if the signal class is structured, in the sense that
the signals can be compressed efficiently, then one can obtain accurate
estimates of the signal from fewer measurements than the ambient dimension. We
demonstrate the effectiveness of the methods we propose through simulation
results.
