Windowed Fourier filtered and quality guided phase unwrapping algorithm: on locally high-order polynomial phase

A windowed Fourier filtered and quality guided (WFF-QG) phase unwrapping algorithm was proposed recently [Appl. Opt. 47, 5420-5428 (2008)], based on the windowed Fourier transform [Appl. Opt. 47, 5408-5419 (2008)] where the phase is assumed to be locally quadric. We consider a locally higher order polynomial phase. After the phase is filtered and unwrapped by the WFF-QG, it is postprocessed by a congruence operation (CO), so that the unwrapped phase is congruent to the original wrapped phase. The unwrapped phase can now be assumed to be a locally high-order polynomial, and, consequently, least squares fitting (LSF) is proposed to suppress the noise. This postprocessing algorithm is abbreviated as CO-LSF. The CO-LSF is theoretically a reasonable choice to improve the WFF-QG results, especially when the noise is severe. This is because for severe noise, a large window is necessary for reliable phase extraction in the WFF-QG. However, this large window makes the quadric phase assumption less reasonable and leads to a large phase error. The CO-LSF thus helps to reduce the phase error by more reasonably assuming that the phase is a high-order polynomial. The polynomial order of 4 is suggested for the CO-LSF, as higher order polynomials do not give significant improvement to the WFF-QG. One disadvantage of the CO-LSF is that it is more sensitive to phase discontinuities than the WFF-QG.