Opticist.org

Phase unwrapping is the process by which the absolute value of the phase angle of a continuous function that extends over a range of more than $2\pi$ (relative to a predefined starting point) is recovered. This absolute value is lost when the phase term is wrapped upon itself with a repeat distance of $2\pi$ due to the fundamental sinusoidal nature of the wave functions used in the measurement of physical properties. 

Numerous algorithms have been proposed for phase unwrapping of various scientific images such as optical shape reconstruction, medical image analysis, geometrical survey, etc. The number of new phase unwrapping algorithms continues to grow.  Here, the following algorithms were selected for the fringe analysis.

Sequential filling

The simplest phase unwrapping method is a sequential scan through the phase data, line by line. If any of the pixels are masked, or a phase jump is incorrectly detected, however, the unwrapping process will be interrupted or an error will propagate through the rest of the data. To improve the unwrapping results, multiple scan directions can be adopted. This method is very simple and fast. However, the unwrapping produces large errors at the regions where the phase has an incorrect jump or the pixels are masked.

A simple solution to cope with the above problem is sequential filling. With this method, after the phase is unwrapped at a pixel, the following pixel to be unwrapped will be one of the neighboring pixels that have not been unwrapped. The same procedure is repeated until the whole wrapped phases have been unwrapped. For the pixels with an incorrect jump, masks are used to exclude them before the unwrapping process, and the masked pixels are skipped during unwrapping. At the end of the unwrapping procedure, the phase at masked pixels can be restored through phase value interpolations.

The sequential filling is the fastest method among those presented here. However, the masking of the incorrect jump pixels usually requires a human-computer interactive operation.

Minimum spanning tree

Minimum spanning tree method is similar to the sequential filling method except that the next pixel to be unwrapped is not chosen arbitrarily among the neighboring pixels of the current pixel. Instead, the pixels are divided into three groups: the unwrapped pixels fall into the first group; the wrapped pixels that have at least one neighboring unwrapped pixel are put into the second group; all other wrapped pixels belong to the third group. The next pixel to be unwrapped is from the second group and this pixel should have the minimum phase difference with its unwrapped neighbor among the first group.  Because sorting is required, the minimum spanning tree algorithm takes a longer time to process. Effective sorting algorithms developed for computer data structure, such as red-black tree sorting, can be used to decrease the unwrapping time significantly.

Preconditioned-conjugate-gradient (PCG) least-sqaures iteration

Least-squares iteration is used in this method; preconditioned conjugate gradient algorithm is helpful for accelerating the iteration. Following is a brief description of this method.

The relation between the wrapped phase  and the unwrapped phase  can be expressed as                                                                                                                                                                           (Eqn.)