Phase unwrapping by means of scaling information and global optimization algorithms

Many applications of SAR interferometry and differential interferometry lead to a set of sparse phase measurements. Sparse phase data can be retrieved e.g. when processing very low coherence interferometric datasets acquired at long time intervals, or, in more sophisticated form, in the processing of long multitemporal stacks of SAR differential interferograms through the permanent scatterers (PS) technique. Usually, such,sparse phase data have to be unwrapped, and then interpolated on a regular grid. We investigate the utility of the scaling information available on the absolute phase, in the process of unwrapping a set of sparse, wrapped phase measurements. Scaling information is an important tool for the description of natural processes exhibiting fractal-like behaviour. One notable example is the interferometric phase contribution due to tropospheric inhomogeneities. Scaling properties can be estimated experimentally on a set of points through computation of the variogram. If it can be assumed that the absolute phase field obeys a defined scaling power law, then the wrapping operator will cause the variogram to depart from the modelled behaviour. Under these hypotheses, the difference between actual and modelled variogram can be used as an optimization Hamiltonian. In this work, we investigate whether the scaling information can be used as a constraint in retrieving the absolute (i.e. unwrapped) phase field from a set of sparse measurements. In particular, we consider here the problem of constructing a cost function which embodies the scaling requirement, and we test several strategies to optimise the cost.