Fast weighted least squares for solving the phase unwrapping problem

Phase unwrapping is important for synthetic aperture radar interferometry. All current methods of phase evaluation produce a distribution module 2π, so that the reconstruction of the correct phase field from the wrapped phase field, by applying a suitable unwrapping process, is needed. The problem of recovering the absolute phase field is ill-posed and can be solved only by introducing additional constraints on the final solution. Assuming a Gaussian model for noise and that the gradient of the true phase is everywhere less than π in magnitude, a solution can be found which approximates the first order wrapped gradient of the phase data. An approximation of the unwrapped phase can be retrieved by adopting a weighted least square approach, in which the phase data are weighted to avoid unwrapping across regions of corrupted phase. The corrupted phase data are caused by such SAR phenomena as layover, radar shadow and temporal decorrelation. D.C. Ghiglia et al. (1994), introduced the first practical algorithm for this weighted least squares approach to phase unwrapping. Their algorithm employes an iterative method based on Fourier or cosine transforms and a preconditioned conjugate gradient (PCG) method to solve the weighted least square equation. The PCG converges rapidly with phase unwrapping problems that do not have large phase discontinuities, but on phase data with large discontinuities it results very slow and requires many iterations to converge. M.D. Pritt (1996) proposed a multigrid technique to speed up the whole process, essentially when the phase has noise and large discontinuities. His algorithm is a full multi-grid technique which allows a theoretical decreasing of computing time up to 25 times in respect to the PCG one. In this paper a new numerical solver is presented, which can be used to approach the weighted least square problem as well as other more refined stabilisers. Their algorithm can be applied to any positive-definite quadratic cost functional used to solve the PU problem