Two-dimensional phase unwrapping by quad-tree decomposition

One problem to be tackled when interferometric phase-shifting techniques are used is the method in which the phase can be reconstructed. Because an inverse trigonometric function appears in the formulation, the final data are not the phase, but the phase modulo 2pi. A new phase-unwrapping algorithm based on a two-step procedure is presented. In the first step, the digital image to be analyzed is divided into continuous patches by a quad-tree-like recursive procedure; in the second step, the same level patches are joined together by an error-norm-minimizing approach to obtain larger, almost continuous ones. The basic idea of the procedure is to simplify the problem by factoring the complete image into square, variable-size, homogeneous areas (i.e., regions with no internal phase jump) so that only interfaces need to be dealt with. By hierarchically recombining the so-obtained subimages, an unwrapped phase field can be obtained. After a complete description of the algorithm, some examples of its use on synthesized digital images are illustrated. As the algorithm can be used with and without quality masks and the error-minimizing step can use different norms, a full class of unwrapping algorithms can be implemented by this approach.