## 1 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. However, in the case of mixed signals, their Fourier spectra may overlap, and thus 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]. Previous studies have mostly focused on the separation of one-dimensional (time domain) mixed signals. [PERSON] and [PERSON] dealt with the problem of image restoration by using the two-dimensional (2-D) FrFT [7]. The FrFT of a 2-D signal _f_(_x,y_) can be defined as [8]. \[ \begin{array}{rl}F^{a_{x}a_{y}}[f(x,y)](u,v)&=\iint\;K_{a_{x}a_{y}}(x,y;u,v)f( x,y)dxdy\\ &=\iint\;K_{a_{x}}(x,u)K_{a_{y}}(y,v)f(x,y)dxdy,\end{array}  \tag{1}\]where the kernel function \(K_{\alpha_{x}}(x,u)\) is defined as  
\[K_{\alpha_{x}}(x,u)=\left\{ \begin{array}{ll}\frac{e^{-i/\langle\frac{x\pi \pi\pi(\phi_{x})}{\sigma_{x}}-\frac{\phi_{x}}{\sigma_{x}}\rangle}}{\sqrt{|\sin x _{x}|}}e^{i\pi\left[(x^{2}+u^{2})\cot\phi_{x}-2 xu\csc\phi_{x}\right]},&\phi_{x} \
eq n\pi\\ &\delta(x-u),&\phi_{x}=2n\pi\\ &\delta(x+u),&\phi_{x}=(2n+1)\pi\end{array} \right. \tag{2}\]  
and \(\phi_{x}=\alpha_{x}\pi/2\) denotes the angle corresponding to the fractional order \(\alpha_{x}\) in the \(x\) direction. Note that the definition of \(K_{\alpha_{y}}(y,u)\) is similar to that of \(K_{\alpha_{x}}(x,u)\) and the range of \(\alpha_{x}\) and \(\alpha_{y}\) values is within [0, 2]. Recently, optical technologies based on measuring the interference fringe patterns have been dramatically developed, especially for optical metrology, because the results are usually encoded as fringe patterns. In some cases, a single fringe pattern may contain multiple fringe sets. The fringe patterns are 2-D signals and are usually detected as 2-D images by using the image sensors. [PERSON] et al. firstly use the FrFT to automatically separate the 2-D mixed Gaussian chirp signals [9]. [PERSON] used the 2-D windowed Fourier transform to determine the phase and phase derivatives in fringe pattern analysis [10]. [PERSON] et al. proposed several algorithmic solutions based on the notion of Hilbert-Huang transform for fringe pattern processing and analysis in optical interferometry [11]. [PERSON] [12] and [PERSON] [13] proposed the methods for the mixed 2-D signals composed of the overlapping fringe patterns. [PERSON]'s method utilizes the 2-D continuous wavelet transform (CWT) and is especially useful for separating the mixed Moire patterns. However, improvement is still required for the cases of symmetrical circular fringe patterns. In [PERSON]'s method [13], the original mixed fringe patterns are firstly transformed to the spatial-frequency domain by using the 2-D FrFT. Then, the maxima projection method is proposed to search for the fractional Fourier spectra under all the possible fractional orders. The circular fringes are of the same width and their center positions can be determined according to the peak values detected in the spectra. With the determined fractional order, fringe width, and the center position, each of the separated circular fringes can be reconstructed. This study deals with a more general case than the previous method for automatically separating the 2-D mixed multiple fringe patterns by using the FrFT. The previous method proposed by [PERSON] [13] does not consider the condition of phase reversal phenomena among the fringe patterns. Thus, they only consider the fringe patterns, which are all in phase, and thus cannot reconstruct the mixed multiple fringe patterns, in which some are out of phase.