On regularisation of coherent imagery with proximal methods

In complex-valued coherent inverse problems such as synthetic aperture radar (SAR), one may often have prior information only on the magnitude image which shows the features of interest such as strength of reflectivity. In contrast, there may be no more prior knowledge of the phase beyond it being a uniform random variable. However, separately regularising the magnitude, via some function \(G:=H(|\cdot|)\), would appear to lead to a potentially challenging non-linear phase fitting problem in each iteration of even a linear least-squares reconstruction problem. We show that under certain sufficient conditions the proximal map of such a function \(G\) may be calculated as a simple phase correction to that of \(H\). Further, we provide proximal map of (almost) arbitrary \(G:=H(|\cdot|)\) which does not meet these sufficient conditions. This may be calculated through a simple numerical scheme making use of the proximal map of \(H\) itself, and thus we provide a means to apply practically arbitrary regularisation functions to the magnitude when solving coherent reconstruction problems via proximal optimisation algorithms. This is demonstrated using publicly available real SAR data for generalised Tikhonov regularisation applied to multi-channel SAR, and both a simple level set formulation and total generalised variation applied to the standard single-channel case.