Goldstein algorithm was a classical path tracking algorithm proposed by Goldstein et al in 1988. It identifies the positive and negative residuals connecting adjacent residue pair or multiple residues and achieving the residual charge balance, and then it generates the optimal branch. Residual points represent the discontinuity at the beginning or end. If they are connected appropriately as the integral roadblock, as long as the unwrapping bypasses these roadblocks, we can get the independent phase unwrapping path, preventing error spread along the path.??

The basic steps are:?

(1) Identifying the residual points;?

(2) Generating the branches cut;?

(3) Integrating around the branches cut;?

The detail steps are as follows: First, to look for residual points according to certain order, define a 2x2 pixels wrapped phase as nodes and to connect the four elements in series, which is the smallest closed path in the image. Second, to accumulate the wrapped phase gradient along the minimum closed path. If the sum is zero, the four points are the same, otherwise the upper left corner of the pixel is called residual point.?

When the first residual point is found, we continue to search for other residual points; if another residual point is found, connect them as a branch (if the point’s polarity is opposite to the original one, the branch line is marked as non-polarity and the searching continues; if the point’s polarity is same to the original one, the search center moves to the new found point and continues searching). Repeat the process until all of the residual points are balanced and connected with branches. Finally, unwrap bypass these roadblocks, and then we can get the original phase.?

In Goldstein's branch-cut method, two steps are very important: (1) When the search window finds a new residual point, connect it with the residual point in the middle of the window, whether the residual points are connected with other residues. (2) When the search window reaches the edge of the picture, connect the residual point with the edge of the picture to stop the integral path.

[1]Hu J, Zhang Y, Wu J, et al. Improved branch-cut method algorithm applied in phase unwrapping[C]//MIPPR 2015: Pattern Recognition and Computer Vision. International Society for Optics and Photonics, 2015, 9813: 98131A.

[2]Goldstein R M, Zebker H A, Werner C L. Satellite radar interferometry: Two-dimensional phase unwrapping[J]. Radio science, 1988, 23(4): 713-720.