2-D PHASE UNWRAPPING AND PHASE ALIASING

The phase of complex signals is measured modulo-2pi (wrapped phase); c ontinuous-phase information is obtained by adding properly chosen mult iples of 2pi shift to the wrapped phase. Unwrapping searches for the 2 pi combinations that minimize the discontinuity of the unwrapped phase as only the unwrapped phase can be analyzed and interpreted by furthe r processing. The key problem of phase unwrapping is phase aliasing, a condition mainly caused by rapid phase variations. The extension of t he one-dimensional (1-D) phase unwrapping algorithms to a two-dimensio nal (2-D) domain by 1-D slicing gives unsatisfactory results even in t he presence of low-phase aliasing, whereas 2-D phase unwrapping deals with the complete problem, overcoming the limitations of 1-D unwrappin g. The 2-D unwrapped phase is obtained as the solution of a variationa l problem that minimizes the differences between the gradients of the wrapped and unwrapped phase. The Euler equation is then integrated usi ng the boundary conditions obtained from the wrapped phase. In additio n to determining a unique unwrapped phase, this approach has the advan tage that it limits the influence of phase aliasing. It is also more a ttractive than iterative 1-D unwrapping since it limits the propagatio n of unwrapping errors. Error propagation in phase unwrapping can stro ngly influence the result of any phase processing. Examples in this pa per apply 2-D phase unwrapping to problems of refraction statics and i nterferometrical imaging using a remote system (SAR) and demonstrate h ow limited error propagation allows phase processing.