| Two-dimensional phase unwrapping is a key step, and often the most significant error source, in the analysis of synthetic-aperture-radar interferograms. In the interferometric technique, very accurate measurements of the Earth's topography or its surface deformation are derived from radar-image phase data. Phase, however, is defined only modulo 2π rad, so a resulting 2-D array of measurements is wrapped with respect to some modulus or ambiguity. These data must be unwrapped to provide meaningful information. For this purpose, we introduce a new, nonlinear constrained-optimization approach in which (i) defined cost functions map particular unwrapped solutions to scalar costs and (ii) a solver routine computes minimum-cost solutions. Previous efforts have focused mainly on simple cost functions that have yielded efficient—but not necessarily accurate—algorithms. These inaccuracies seriously degrade the effectiveness of the interferometric technique and can preclude useful geophysical interpretation of the data. We propose a new set of nonconvex, statistically based cost functions through which we treat phase unwrapping as a maximum a posteriori probability estimation problem. That is, we derive approximate, application-specific statistical models for the problem variables. Based on these models, we cast phase unwrapping as an optimization problem whose objective is to find the most physically probable unwrapped solution given the observable quantities: wrapped phase, image intensity, and interferogram coherence. We prove that the resulting problem is NP-hard, and we develop nonlinear network-flow solver techniques for approximating solutions to this problem. Extending our statistical framework and network methods, we also present a tiling heuristic for applying our algorithm to large data sets. Performance tests on topographic and deformation data acquired by the ERS-1 and ERS-2 satellites suggest that our algorithm yields superior accuracy and competitive efficiency as compared to other existing algorithms. |
