# A Dual-Branch Self-Boosting Network Based on Noise2 Noise for Unsupervised Image Denoising
## 2 Related Work  
### Noise2 Noise  
[PERSON] [21] introduced Noise2 Noise, a model for unsupervised noise reduction, enabling image restoration without dependency on a clear image as the training target. This innovative approach solely relies on paired independent noisy images originating from the same ground truth. Noise2 Noise operates under the assumption of accessing a collection of noisy image pairs \(y_{1}=x+n_{1},y_{2}=x+n_{2}\), where \(n_{1},n_{2}\) denote independent noise vectors. The optimization of \(\theta\) in Noise2 Noise aims to minimize the following loss:  
\[\frac{1}{n}\sum_{i=1}^{n}\left\|f_{\theta}\left(y_{1}^{i}\right)-y_{2}^{i} \right\|_{2}^{2} \tag{2}\]  
where \(f_{\theta}\) denotes the denoising network parameterized by \(\theta\). Considering Equation (2), if the dataset is sufficiently large, we have  
\[ \begin{split}\arg\min_{\theta}&\mathbb{E}\left\|f_{ \theta}\left(x_{i}+n_{2}^{i}\right)-\left(x_{i}+n_{1}^{i}\right)\right\|_{2}^{ 2}\\ &=\arg\min_{\theta}\mathbb{E}\|f_{\theta}(y_{i})-x_{i}\|_{2}^{2}- 2\mathbb{E}\Big{[}{n_{1}^{i}}^{T}f_{\theta}(y_{i})\Big{]}.\end{split}  \tag{3}\]  
Because \(n_{1}^{i}\) and \(n_{2}^{i}\) are two independent noises, so \(\mathbb{E}\big{[}{n_{1}^{i}}\mid x_{i}+\ n_{2}^{i}\big{]}=\mathbb{E}\big{[}{n_ {1}^{i}}\mid x_{i}\big{]}=0\). Therefore, we have \[ \begin{split}\mathbb{E}\Big{[}n_{1}^{iT}f_{\theta}(y_{i})\Big{]}& =\mathbb{E}\Big{[}\mathbb{E}\Big{[}n_{1}^{i}\mid f_{\theta}(y_{i}) \Big{]}^{T}f_{\theta}(y_{i})\Big{]}\\ &=\mathbb{E}\Big{[}\mathbb{E}\Big{[}n_{1}^{i}\mid x_{i}+n_{2}^{i }\Big{]}^{T}f_{\theta}(y_{i})\Big{]}\\ &=0.\end{split}  \tag{4}\]  
In expectation of such noisy instances, and assuming zero mean noise, training a network in a supervised manner to map a noisy image to another noisy image is equivalent to mapping it to a clean image, i.e.,  
\[\underset{\theta}{\arg\min}\mathbb{E}\Big{[}\|f_{\theta}(y_{1})-x\|_{2}^{2} \Big{]}=\underset{\theta}{\arg\min}\mathbb{E}\Big{[}\|f_{\theta}(y_{1})-y_{2} \|_{2}^{2}\Big{]}. \tag{5}\]  
The N2N denoising model, trained on pairs of noisy images without access to ground truth clean images, utilizes an unsupervised learning approach that offers certain advantages. In theory, N2N training can achieve the same performance as Noise2 Clean (N2C) training given an infinitely large dataset. However, in practice, due to limited training set size, N2N falls short of N2C. While N2N [21] demonstrates great flexibility in image denoising, its results may not always be optimal. One key limitation of the Noise2 Noise (N2N) model is its restricted generalization ability when confronted with noise characteristics that differ significantly from the training data. Furthermore, in order to achieve optimal denoising performance, the N2N technique may require the selection of different loss functions tailored to specific application scenarios. Additionally, the N2N model assumes that the noise statistics follow an independent and identically distributed pattern. However, real-world noise often exhibits more complex, spatially correlated characteristics. Consequently, the denoising performance of N2N inevitably degrades when dealing with this type of spatially correlated noise. It is also worth noting that the performance of the N2N model is influenced by the quality and capacity of its underlying backbone network architecture. In the current implementation, a UNet architecture is utilized as the backbone network, as illustrated in Figure 1. However, the use of a more advanced network structure has the potential to enhance the denoising performance of the N2N model further. The limitations of the N2N denoising model mentioned above can be attributed to its overreliance on a simplistic loss function definition, which only establishes constraint between the network output and input. This limited constraint fails to ensure the stability and generalization capability of the model during training.