  Recovering complex-valued image recovery from noisy indirect data is
important in applications such as ultrasound imaging and synthetic aperture
radar. While there are many effective algorithms to recover point estimates of
the magnitude, fewer are designed to recover the phase. Quantifying uncertainty
in the estimate can also provide valuable information for real-time decision
making. This investigation therefore proposes a new Bayesian inference method
that recovers point estimates while also quantifying the uncertainty for
complex-valued signals or images given noisy and indirect observation data. Our
method is motivated by the Bayesian LASSO approach for real-valued sparse
signals, and here we demonstrate that the Bayesian LASSO can be effectively
adapted to recover complex-valued images whose magnitude is sparse in some
(e.g.~the gradient) domain. Numerical examples demonstrate our algorithm's
robustness to noise as well as its computational efficiency.
