# A Self-Supervised One-Shot Learning Approach for Seismic Noise Reduction
## 2 Method
### Noise2 Self  
The N2S approach builds on the premises established by self-supervised learning techniques such as N2N; however, this novel method enables efficient noise removal without requiring even pairs of noisy images generated from the same object or scene. Instead, N2S capitalizes on the inherent structure of single noisy images and leverages the statistical independence of noise at different locations within the same image. Thus, this method does not require different versions of the same image--either clean or noisy counterparts--because it fosters an internal learning environment that highlights the noise pattern inherent in individual images. An important part of the N2S approach is the hypothesis that noise in any specific pixel or pixel group is statistically independent of the corresponding signal or image content in adjacent pixels. This premise facilitates the prediction of the value of a pixel on the basis of its neighboring pixels, excluding the noise specific to the target pixel(s). Consequently, N2S employs a \"\(j\)-invariant\" prediction model, where the value of a pixel is estimated withina context in which the pixel itself is intentionally omitted. This ensures that the training leverages a proxy for clean data derived from the noisy image itself. The concept of \(J\) invariance is pivotal in our understanding of how to approach the statistical relationship between a pixel and its adjoining pixels. In particular, \(J\) invariance upholds the postulation that the value of a given pixel can be accurately predicted without incorporating the specific information of that pixel by instead leveraging data from adjacent pixels. This suggests that the statistical noise inherent within the target pixel does not interfere with the predictive process. To formalize, let \(\mathcal{J}\) denote a partition of dimensions \(\{1,\dots,m\}\). Let \(x\) represent the initial noisy image, and let \(x_{J}\) denote a subsample of \(x\) confined to \(J\in\mathcal{J}\). A function \(f:\mathbb{R}^{m}\to\mathbb{R}^{m}\) is considered J-invariant if the value of \(f(x)_{J}\) is independent of the value of \(x_{J}\). Thus, \(f\) is \(\mathcal{J}\)-invariant if it exhibits J-invariance for every \(J\in\mathcal{J}\). Figure 1a illustrates the concepts being discussed. Here, the feature space of an image is divided into two independent sets: \(J\), containing the features of interest and \(J^{C}\), representing the remaining features. Within this space, we have points \(x\) that are mapped by the function \(f\). The function \(f\) is \(J\)-invariant because, when applying this transformation, the features in \(J\) remain unaltered. In other words, \(f\) preserves the information in the subset \(J\) while utilizing information from \(J^{C}\) to perform tasks such as denoising. For the practical application of \(J\) invariance via N2S, the model is trained to deduce the clean values of target pixels or areas using only the information from surrounding pixels, thus omitting the target pixels themselves. This objective is achieved by applying a \(J\)-invariant mask during training, which forces the model to discern missing or modified pixel information on the basis of the statistical characteristics of nearby pixels, effectively separating the signal from the noise. Figure 1b shows an example of independent features; the gray boxes represent regions processed independently, with the mask varying depending on the specific masking process used. The training objective function under \(J\) invariance is depicted as follows:  
\[\operatorname*{argmin}_{\theta}\mathbb{E}_{x}\{\mathcal{L}(f_{\theta}(x)_{J}, x_{J})\}, \tag{4}\]  
where \(\mathcal{L}\) denotes the loss function that measures the discrepancy between the model prediction \(f_{\theta}(x)_{J}\) and the masked noisy observation \(x_{J}\). The model that reconstructs clean images from their noisy counterparts, parameterized by \(\theta\), is refined by minimizing this loss.