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

Chuan-Yu Chang,Chian C Ho,Chih-Hao Chuang,Hsuan-Ting Chang,Tzu-Yao Lin,Chien-Yu Chen

doi:10.3390/app11020859

Abstract

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.

Highlights

In addition to the FT, the methods based on the fractional Fourier transform (FrFT) [1], in which the fractional orders can be sought for the best separation between the noise and signal spectra, have been found to be more advantageous than the FT in solving the spectral overlapping problems [2,3,4,5,6]
When we perform the FrFTs in the x-direction, the order in the y-direction is fixed as 1; The orders with the maximum projection in the x-direction of the signal are calculated and recorded; we perform the FrFTs in the y-direction and, the order in the xdirection is fixed as 1; The orders with the maximum projection in the y-direction of the signal are calculated and recorded as well; We calculate the orders of maximum projection, αx and αy, from the local maximum values of fractional Fourier spectra through a differential operation
We propose an automatic signal separation method for separating the 2-D mixed circular fringe patterns and considering the possible phase reversal condition

Summary

Introduction

In communication systems, when signals are disturbed by noise, we can use the Fourier transform (FT) method to transfer the signals to the frequency domain for noise filtering. In the case of mixed signals, their Fourier spectra may overlap, and we cannot directly separate the noise from the signals by using direct filtering in the frequency domain. In addition to the FT, the methods based on the fractional Fourier transform (FrFT) [1], in which the fractional orders can be sought for the best separation between the noise and signal spectra, have been found to be more advantageous than the FT in solving the spectral overlapping problems [2,3,4,5,6]. Kutay and Ozaktas dealt with the problem of image restoration by using the two-dimensional (2-D)

Methods

Results

Conclusion