Empirical Bayesian inference for complex-valued signals using support-informed priors

Recovering complex-valued images from multiple measurements of Fourier data is of interest in many application domains, such as synthetic aperture radar (SAR) and ultrasound imaging. While there are many algorithms designed to accurately reconstruct the magnitude, the phase information can be important for downstream processing such as coherent change detection. There is also increased interest in quantifying the recovery uncertainty. This investigation proposes a new empirical Bayesian inference method for recovering point estimates and samples of a complex-valued image posterior distribution from its corresponding multiple Fourier measurements. It also quantifies the uncertainty for both the magnitude and the phase. Our method uses a support-informed prior which we estimate directly from the multiple data acquisitions. In this way we allow for data compression, since only the sparse support information is required for the prior construction. Numerical examples demonstrate its ability to reduce the uncertainty in the magnitude of the recovery while also preserving phase information.