Probabilistic Cost Functions for Network Flow Phase Unwrapping

The well-studied interferometric synthetic aperture radar (InSAR) problem for digital elevation map generation involves the derivation of topography from radar phase. The topography is a function of the full phase, whereas the measured phase is known module 2π, necessitating the process of recovering full phase values via phase unwrapping. This mathematical process becomes difficult through the presence of noise and phase discontinuities. The authors' research is motivated by recent research that models phase unwrapping as a network flow minimization problem. The cost function to be optimized is a weighted L1-norm of the phase discontinuities. Determining these cost weights is critical, yet past work in the literature does not reflect the statistics of the unwrapping problem. The purpose of this paper is to propose a new method to compute the flow weights from a theoretical foundation. Specifically, they formulate phase unwrapping as a maximum likelihood (ML) estimation problem, which they mathematically rewrite as a network flow problem with a specific choice of weights. The approach is based on estimating the probability of phase discontinuities, which can be derived as a function of coherence and topographic slope from the known statistical properties of SAR phase.