Phase Unwrapping In Two Contexts

The ambiguity of the complex logarithm, whereby its imaginary part is only specified modulo 2r, poses a problem that often arises in diverse contexts. This article reports on the inadequacy of phase unwrapping in two separate contexts. One example occurs with image restoration that is intended to counteract the deleterious effects of at-mospheric turbulence. By tracking the positions (phases) of all the interference fringes that compose the image, the mean positions might be estimated. When only a few photons are available however, the resulting uncertainties sometimes lead to 27r slips that drastically upset the mean positions. A second example occurs in rotational synthesis, where the only available information lies on a single circle in the 2-D space-frequency complex domain. If it were possible to reliably unwrap the imaginary part of the logarithms around that circle, we could easily find the brightest point of the picture domain. The procedure attempts to trace a path as the source traverses interference fringes, without ever accidently slipping a fringe. In both examples, subsidiary contributions occasionally conspire to give the 27 slips. Smooth continuity is not quite sufficient as a basis for phase unwrapping. The problem is especially important because the character of images is more dependent on the phases of their Fourier components than on the amplitudes. Perhaps it will become possible to invoke other criteria than continuity for more reliable phase unwrapping.© (1984) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.