2-D phase unwrapping and phase aliasing

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Abstract
The phase of complex signals is measured modulo-2π (wrapped phase); continuous‐phase information is obtained by adding properly chosen multiples of 2π shift to the wrapped phase. Unwrapping searches for the 2π combinations that minimize the discontinuity of the unwrapped phase as only the unwrapped phase can be analyzed and interpreted by further processing. The key problem of phase unwrapping is phase aliasing, a condition mainly caused by rapid phase variations. The extension of the one‐dimensional (1-D) phase unwrapping algorithms to a two‐dimensional (2-D) domain by 1-D slicing gives unsatisfactory results even in the presence of low‐phase aliasing, whereas 2-D phase unwrapping deals with the complete problem, overcoming the limitations of 1-D unwrapping. The 2-D unwrapped phase is obtained as the solution of a variational problem that minimizes the differences between the gradients of the wrapped and unwrapped phase. The Euler equation is then integrated using the boundary conditions obtained from the wrapped phase. In addition to determining a unique unwrapped phase, this approach has the advantage that it limits the influence of phase aliasing. It is also more attractive than iterative 1-D unwrapping since it limits the propagation of unwrapping errors. Error propagation in phase unwrapping can strongly influence the result of any phase processing. Examples in this paper apply 2-D phase unwrapping to problems of refraction statics and interferometrical imaging using a remote system (SAR) and demonstrate how limited error propagation allows phase processing.

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