## 2 Related Work  
The image denoising problem can often be represented as an energy minimization problem of  
\[x^{*}=\min_{x}E(x;y)+R(x) \tag{2}\]  
where \(y\) is a given noisy image, \(E(x;y)=||x-y||^{2}\) is used to compare the generated image to \(y\), and \(R(x)\) is a regularizer. In [23], [PERSON] et al. proposed a randomly initialized DNNcalled DIP to solve image-denoising problems. In the DIP framework, the optimization problem is expressed as follows:  
\[\theta^{*}=\underset{\theta}{\text{argmin}}E(f_{\theta}(z);y),x^{*}=f_{\theta^{*}} (z) \tag{3}\]  
where \(f\) denotes the DNN model, \(\theta\) is the random initialized network parameter, \(z\) denotes the random noise input of the network, \(\theta^{*}\) is the learned network parameter, and \(x^{*}\) is the result of the recovery process. Given a noisy image \(y\), the DIP learning process attempts to find the proper value of \(\theta^{*}\) that can reproduce the noisy image. It has been demonstrated that a DIP network contains prior knowledge, such as the low-level structure of natural images. It can remove noise from corrupted images with a certain number of iterations. However, the denoising performance of DIP is not as satisfactory as that of state-of-the-art methods; that is, a single run leads to a far lower PSNR value than state-of-the-art methods, such as BM3D [1], especially for synthetic noisy image denoising. This is because the DIP optimization process can be considered a process of finding the best-quality instance in the image space. The distance between the low-quality target image and the clean image was larger than that between the high-quality target image and the clean image. Figure 1 shows the variation in the PSNR values of the two denoised images generated from the DIP optimization processes versus the number of training iterations. The two DIP optimization processes used two different images as learning targets. One is a preliminary image, and the other is a noisy image. The preliminary image was a denoised image generated using a pre-trained supervised image denoising method. As shown in Figure 1, DIP achieves a PSNR of 28.93 dB after 3083 iteration steps when using the preliminary image or the noisy image as the learning target. The PSNR is increased to 30.16 dB in iteration 8999 when using the preliminary image as the learning target; however, the PSNR decreases to 23.59 dB in iteration 8999 when using the noisy image as the learning target. This implies that the denoised image generated from the DIP gradually converges to the target image as the number of iterations increases. Therefore, using images with better quality as the DIP learning target can avoid overfitting and help achieve a better denoising result. DIP shows that the unsupervised DNN is capable of capturing low-level image statistics. However, given the same network input and learning target, DIP cannot generate identical denoised images owing to the randomly initialized network weights. To illustrate its uncertainty, we compared two denoised images generated from the DIP optimization processes. Figure 2 shows the variations in mean square error (MSE) between the network outputs of the two DIP optimization processes as the number of DIP iterations increases. In the two DIP learning procedures, we used the same noisy image as the learning target  
Figure 1: Learning curves for the reconstruction task using natural and noise images. PSNR of a natural looking image is higher compared to that of the noise image. and the same random noisy image as the input. We can see from Figure 2 that MSEs show a downward trend before 4000 iterations, whereas MSEs remain \(0.5\times 10^{-3}\) to \(1.5\times 10^{-3}\) after 4000 iterations. It is apparent that different DIP optimization processes generate different denoising results. It is impossible to obtain the same network output at the same iteration steps, even when given the same network input \(z\) and the same learning target.