## 3 Methodology  
### Multiple Target Images  
First, we considered a new approach to enhance the guidance of the loss function described in Equation (5) by adding two sub-items. In other words, we added two images with higher guiding ability (higher image quality) to participate in the calculation of the loss function. Specifically, we applied two mainstream denoising methods (FFDNet and BM3D) to denoise each noisy image, thereby obtaining two preliminary denoised images \(x_{1}\) and \(x_{2}\). Next, we added the MSE values of the two preliminary denoised images (\(x_{1}\) and \(x_{2}\)) to the loss function. The new loss function can be computed with:  
\[Loss_{2}=MSE\Big{(}\hat{x}^{i},x_{0}\Big{)}+MSE\Big{(}\hat{x}^{i},x_{1}\Big{)} +MSE\Big{(}\hat{x}^{i},x_{2}\Big{)} \tag{6}\]  
where \(\hat{x}^{i}\) is the output image of the network, \(x_{0}\) represents the noisy image, and \(x_{1}\) and \(x_{2}\) are the preliminary denoised images produced by FFDNet and BM3D, respectively. As shown in Figure 4, the proposed approach starts with random weights \(\theta^{0}\), and we iteratively update them to minimize the objective function described in Equation (6). For each iteration \(i\), the weights \(\theta\) are used to generate the image \(\hat{x}^{i}=f_{\theta}(z)\), where the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Images** & **Barbara** & **Boat** & **Cameraman** & **Couple** & **Hill** & **House** & **Lena** & **Man** & **Monarch** & **Peppers** \\ \hline Noise Level & & & & & \(\sigma=30\) & & & & \\ \hline DIP & **25.76** & **28.14** & **27.31** & **27.73** & **28.36** & **31.04** & **30.80** & **28.03** & **27.91** & **28.50** \\ \hline FFDNet & **28.95** & **29.66** & **29.07** & **29.46** & **29.57** & **32.54** & **32.05** & **29.35** & **28.95** & **29.63** \\ \hline Noise Level & & & & & \(\sigma=40\) & & & & & \\ \hline DIP & **24.26** & **26.91** & **25.69** & **26.28** & **27.35** & **29.43** & **29.43** & **26.95** & **26.48** & **27.13** \\ \hline FFDNet & **27.54** & **28.41** & **27.82** & **28.15** & **28.50** & **31.40** & **30.80** & **28.19** & **27.70** & **28.34** \\ \hline \hline \end{tabular}
\end{table}
Table 1: PNSR(dB) results of DIP and FFDNet on 10 commonly used images with noise level \(\sigma=30\) and 40.  
Figure 3: PSNRs (dB) of DIP evaluated on the Barbara, Cameraman, and Peppers images with the noise level \(\sigma=50\).  
mapping \(f\) is a neural network with parameters \(\theta^{i}\) and \(z\) is a fixed tensor. The image \(\hat{x}^{i}\) is used to calculate the non-zero cost \(E\left(\hat{x}^{i};x_{0},x_{1},x_{2}\right)\). The weight \(\theta^{i}\) is then updated using the stochastic gradient descent (SGD) training method. The advantage of this method is that it utilizes preliminary denoising images to construct the loss function and can thereby adjust the evolution direction of the generative network model. This ensures that the network output image \(\hat{x}^{i}\) evolves in a reasonable direction within the solution space (close to the ground truth \(x_{gt}\)). The schematic diagram in Figure 5 shows the center of gravity of \(x_{0}\), \(x_{1}\), and \(x_{2}\), and shows the point \(\bar{x}_{0}\), where the network output image \(\hat{x}^{i}\) finally converges in the solution space after adopting the new hybrid loss function. This results in an output image \(\hat{x}^{i}\) that is closer to the undistorted image \(x_{gt}\) after a specific iteration. It should be noted that the preliminary denoised image \(x_{1}\) is obtained from FFDNet, which utilizes external information captured through a training image set, while \(x_{2}\) is obtained from BM3D, which utilizes the internal self-similarity information of the image. Consequently, the proposed method essentially utilizes both the internal and external prior constraints of the image to remove noise.  
Figure 4: Image denoising uses our generative network.  
Figure 5: Image space visualization of image denoising using the proposed method and deep image prior.