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.

翻译：从含噪间接数据中恢复复值图像在超声成像和合成孔径雷达等应用中具有重要意义。虽然已有许多有效算法可用于恢复幅度的点估计，但专门设计用于恢复相位的算法相对较少。估计结果的不确定性量化可为实时决策提供宝贵信息。因此，本研究提出一种新的贝叶斯推断方法，该方法能在从含噪间接观测数据中恢复复值信号或图像点估计的同时，有效量化其不确定性。我们的方法受适用于实值稀疏信号的贝叶斯LASSO方法启发，本文论证了贝叶斯LASSO可被有效改造，用于恢复幅度在某个域（如梯度域）具有稀疏性的复值图像。数值实验表明，我们的算法对噪声具有鲁棒性，且计算效率高。