Separation of Two-Dimensional Mixed Circular Fringe Patterns Based on Spectral Projection Property in Fractional Fourier Transform Domain

In this paper, a method for automatically separating the mixed circular fringe patterns based on the fractional Fourier transform (FrFT) analysis is proposed. Considering the mixed two-dimensional (2-D) Gaussian-based circular fringe patterns, detected by using an image sensor, we propose a method that can efficiently determine the number and parameters of each separated fringe patterns by using the FrFT due to the observed higher sparsity in the frequency domain than that in the spatial domain. First, we review the theory of FrFT and the properties of the 2-D circular fringe patterns. By searching the spectral intensities of the various fractional orders in the FrFT projected along both the frequency axes, the proposed method can automatically determine the total fringe number, the central position, binary phase, and the maximum fringe width of each 2-D circular fringe pattern. In the experimental results, both the computer-simulated and optically mixed fringe patterns are used to verify the proposed method. In addition, the additive Gaussian noise effects on the proposed method are investigated. The proposed method can still successfully separate the mixed fringe pattern when the signal-to-noise ratio is higher than 7 dB.