# Unsupervised Low-Light Image Enhancement in the Fourier Transform Domain
## 3 Proposed Method  
In this paper, our idea is to utilize the CycleGAN model to enhance luminance without paired images. Taking advantage of the particularity that the noise after the Fourier transform is mainly concentrated in the phase domain, we propose an unsupervised low-light image enhancement framework in the Fourier transform domain. On the one hand, the CycleGAN model is used to enhance luminance in the amplitude domain. On the other hand, we use the three-layer convolution neural network in the phase domain of the Fourier transform to denoise. The specific network framework is shown in Figure 1. For the input low-light image \(I_{low}=f(x,y)\), the fast Fourier transform (FFT) formula is shown as:  
\[F(u,v)=\sum\sum f(x,y)e^{-i2\pi(\frac{u}{P}+\frac{v}{Q})}, \tag{1}\]  
where \(F(u,v)\) denotes the complex value in the frequency domain, \(u\) and \(v\) represent the horizontal and vertical frequency components, respectively, \(x\) and \(y\) represent the position coordinates in the horizontal and vertical directions, respectively, in the spatial domain, \(P\) denotes the width of the image, and \(Q\) denotes the height of the image. Let \(R(u,v)\) and \(I(u,v)\) represent the real part and the imaginary part of the \(F(u,v)\), respectively; we can obtain the amplitude spectrum \(A_{low}^{real}\) and phase spectrum \(P_{low}^{real}\) as follows:  
\[A_{low}^{real}=\sqrt{R^{2}(u,v)+I^{2}(u,v)} \tag{2}\]  
\[P_{low}^{real}=\arctan\frac{I(u,v)}{R(u,v)}. \tag{3}\]  
In Figure 1a, generator \(G_{1}\) is for generating the normal-light amplitude image \(A_{nor}^{fake}\) from the low-light amplitude image \(A_{low}^{real}\), and generator \(G_{2}\) is for generating the low-light amplitude image \(A_{low}^{rec}\) again.  
\[A_{nor}^{fake}=G_{1}(A_{low}^{real}),A_{low}^{rec}=G_{2}(A_{nor}^{fake}) \tag{4}\]  
There is a discriminator \(D_{1}\) that identifies whether it is the generated normal-light image or the initial normal-light image. In the phase domain, we use the three-layer convolution denoiser to denoise the phase image \(P_{low}^{real}\). To ensure the consistency of \(I_{low}\) and \(I_{rec}\) in terms of noise distribution and color offset, the residual noise map \(N^{{}^{\prime}}\) adds to phase spectrum \(\hat{P}_{low}^{real}\), which denotes the residual noise map that is calculated as:  
\[N^{{}^{\prime}}=P_{low}^{real}-\hat{P}_{low}^{real}. \tag{5}\]The final recovered image \(I_{low}^{rec}\) is obtained by the inverse fast Fourier transform (IFFT). And the formula of IFFT is calculated as:  
\[I_{tor}^{fake}=A_{nor}^{fake}*e^{iP^{real}},I_{low}^{rec}=A_{low}^{rec}*e^{iP^{ pre}_{low}}, \tag{6}\]  
where \(A_{nor}^{fake}\) is the enhanced normal-light amplitude image generated by generator \(G_{1}\), and \(A_{low}^{rec}\) is the recovered low-light amplitude image generated by generator \(G_{2}\). \(\hat{P}_{low}^{real}\) is the denoised phase image of the low-light image, and \(P_{low}^{rec}\) is the same for the phase image of the initial low-light image \(I_{low}\).  
Figure 1b represents the reverse process; the structure of generator \(G_{1}\) and \(G_{2}\), discriminator \(D_{2}\), and denoiser are all the same to Figure 1a. The added noise \(N\) denotes the random Gaussian distribution noise map.