Under-Sampled Phase Retrieval of Single Interference Fringe Based on Hilbert Transform

Abstract

Phase retrieval from single interference fringe is important and effective method in obtaining the real phase distribution. The original phase can be retrieved by the line integral of its gradient expressed as sine and cosine components, which were gained by the Hilbert transform twice from a single interference fringe pattern. However, this method fails when the phase transformation of the interference fringe is too fast. In this paper, a novel method to recover the continuous phase of the whole field is proposed to solve the above problems. The shear interference technique is introduced into the phase retrieval method to build an exponential 2-D complex light field of natural base for the phase slope obtained by the Hilbert transform. Then, the expressions of phase slopes in x- and y-directions are constructed as a discrete Poisson equation. Therefore, the calculation of phase retrieval is equivalent to solve the discrete Poisson equation mathematically. Finally, the real phase is gotten by the weighted discrete cosine transform (WDCT) of the discrete Poisson equation. The simulation results verify the validity of this method and show that the proposed method can achieve the phase retrieval of the phase discontinuity in x- and y-directions, which leads to the under-sampled problem. It can restore the whole field phase distribution rapidly and accurately. Moreover, this method is applied to phase retrieval of interferometric synthetic aperture radar (InSAR) with the under-sampled problem in this paper. The experimental results show that this method can recover the phase of InSAR with the under-sampled problems caused by terrain abrupt change and so on. Compared with other commonly used methods, it achieved satisfactory results. This method provides a new idea for solving the under-sampled problem in the phase retrieval from a single-frame interference fringe.