Branch-cut algorithm with second-order residues in phase unwrapping

This Letter presents a novel, to the best of our knowledge, branch-cut algorithm for unwrapping phase maps acquired through fringe projection systems. The algorithm exposes second-order residues with vortex phase structures, computed from the original wrapped phase distribution using a second-order derivative that considers the wrap cycle. Remarkably, it reveals a consistent vorticity direction associated with these vortex structures, significantly facilitating branch-cut connections between residues, regardless of whether they carry the same or opposite charges. This innovative approach challenges a longstanding misconception in conventional branch-cut methods, which traditionally connected matched residues only when they had opposite charges. As a result, the algorithm provides a highly accurate methodology for explaining how phase behavior relates to surface characteristics. Its effectiveness is demonstrated through both computer simulations and practical experiments.