Three-dimensional noise-immune phase unwrapping algorithm.

The classical problem of phase unwrapping in two dimensions, that of how to create a path-independent unwrapped map, is extended to the case of a three-dimensional phase distribution. Whereas in two dimensions the path dependence problem arises from isolated phase singularity points, in three dimensions the phase singularities are shown to form closed loops in space. A closed path that links one such loop will cross a nonzero number of phase discontinuities. In two dimensions, path independence is achieved when branch-cut lines are placed between singular points of opposite sign; an equivalent path-independent algorithm for three dimensions is developed that places branch-cut surfaces so as to prevent unwrapping through the phase singularity loops. The placing of the cuts is determined uniquely by the phase data, which contrasts with the two-dimensional case for which there are many possible ways in which to pair up the singular points. The performance of the new algorithm is demonstrated on three-dimensional phase data from a high-speed phase-shifting speckle pattern interferometer.