## 2 Literature Survey
### The Neighbor2 Neighbor Model  
The core idea of Noise2 Noise is that for an unobserved clean scene \(x\) and two independent observed noisy images \(y\), \(z\), a noise-reduction network that can be trained with \((y,z)\) pairs is tantamount to a network by training with \((y,z)\) pairs if the noise is 0-mean. Noise2 Noise minimizes the loss function \(\theta\):  
\[\arg\min_{\theta}\,\mathrm{E}_{x,y,z}\|f_{\theta}(y)-z\|_{2}^{2} \tag{2}\]  
where the neural network function \(f_{\theta}(y)\) is parameterized by \(\theta\).  
Neighbor2 Neighbor is an extension of Noise2 Noise, which constructs similar noisy images from a single noisy image by designing a sampler. Moreover, to solve the stubbornness of image smoothing caused by different sampling positions of similar noisy images in the sampling process, a regularization term is introduced.  
Neighbor2 Neighbor mainly considers the following two aspects: two independent noisy images of similar scenes, and a single noisy image for a single scene.  
**Assumption 1:**_There is a pure image \(x\), and its corresponding noisy image is \(y\); i.e.,_  
\[\mathrm{E}_{y|x}(y)=x \tag{3}\]  
_When a quite tiny image difference \(\varepsilon\
eq 0\) is introduced, \(x+\varepsilon\) is the pure image corresponding to another noisy image \(z\); i.e.,_  
\[\mathrm{E}_{z|x}(z)=x+\varepsilon \tag{4}\]  
_Suppose the variance of \(z\) is \(\delta^{2}\); then_  
\[\mathrm{E}_{x,y}\|f_{\theta}(y)-x\|_{2}^{2} =\mathrm{E}_{x,y,z}\|f_{\theta}(y)-z\|_{2}^{2}-\delta_{z}^{2} \tag{5}\] \[+2\epsilon\mathrm{E}_{x,y}(f_{\theta}(y)-x)\]  
_When \(\varepsilon{\rightarrow}0\) the \((y,z)\) pairing can be viewed as a neighborhood of Noise2 Noise, finding \(y\), \(z\) satisfying the \"similar but not identical\" condition allows an algorithm to train the denoising network._  
**Assumption 2:**_For a single noisy image, Neighbor2 Neighbor uses a sampler to create two \"similar but not identical\" images. The sub-images obtained by sampling adjacent spots of the raw image satisfy the constraint that the differences between each other are tiny, but the matching pure images are not identical (when \(\varepsilon{\rightarrow}0\)). For a noisy image \(y\), Neighbor2 Neighbor constructs a pair of nearest neighbor samplers \(g_{1}(^{*})\) and \(g_{2}(^{*})\), and samples two sub-images \(g_{1}(y)\), \(g_{2}(y)\). We directly construct pairs with these two sub-images. Moreover, the noise reduction network is trained in a Noise2 Noise manner; then_  
\[\arg\min_{\theta}\,\mathrm{E}_{x,y}\|f_{\theta}(g_{1}(y))-g_{2}(y)\|^{2} \tag{6}\]  
_Neighbor2 Neighbor calls this approach pseudo Noise2 Noise. Since \(g_{1}(y)\), \(g_{2}(y)\) are sampled at different locations, the resulting denoising model is not optimal and leads to excessive smoothing. Therefore, Neighbor2 Neighbor considers correcting this situation by adding a regularization term on the loss, adding a constraint based on pseudo Noise2 Noise, transforming the optimization problem with constraints into an optimization problem with regularization, and finally optimizing the loss function:_  
\[\begin{split} L&=L_{rec}+\gamma\cdot L_{reg}\\ &=\|f_{\theta}(g_{1}(y))-g_{2}(y)\|_{2}^{2}\\ &+\gamma\cdot\|f_{\theta}(g_{1}(y))-g_{2}(y)-(g_{1}(f_{\theta}(y ))-g_{2}(f_{\theta}(y)))\|_{2}^{2}\end{split} \tag{7}\]  
_where \(L_{rec}\) is the reconstruction term based on the network output and the noisy target. \(L_{reg}\) is the regularization term and \(\gamma\) is the hyperparameter that controls the strength of the regularization term._