## 3 The Proposed Method
### Self-Training with Noise for Depth Completion  
Algorithm 1 presents the self-training process incorporating noise for depth completion, inspired by the noisy-student approach originally developed to improve ImageNet classification [39]. ```
Input: Labeled data, \(\big{\{}\left(\text{rsd}_{i};\hat{g}_{i}\right)\big{\}}_{1}^{n}\), and unlabeled data, \(\big{\{}\overset{\sim}{\text{rsd}_{j}}\big{\}}_{1}^{m}\), where rsd denotes the combination of RGB images and sparse depth, and \(g\) represents the corresponding ground truth. Output: Student model \(\theta_{*}^{s}\)
1. For k = 1 to K do
2. Train the teacher model, \(\theta_{*}^{t}\), to minimize the loss function that is applied to the annotated dataset: \[\frac{1}{n}\sum\limits_{i=1}^{n}l(g_{i}f(\text{rsd}_{i},\theta^{t}))\] (1)
3. Generate pseudo-labels for the unannotated dataset using the trained teacher model: \[\overset{\sim}{\hat{g}}_{j}=f(\overset{\sim}{\text{rsd}_{j}},\theta_{*}^{t}), \ \forall j=1,\cdots,\ m\] (2)
4. Train the student model, \(\theta_{*}^{s}\), by leveraging both labeled and pseudo-labeled data to minimize the loss function while introducing noise into the student model's training process: \[L_{task}=\frac{1}{n}\sum\limits_{i=1}^{n}l(g_{i}f(rsd_{i},\theta^{t}))+\frac{1 }{m}\sum\limits_{j=1}^{m}l(\overset{\sim}{\widetilde{g}}_{j},f^{nosed}( \overset{\sim}{rsd}_{j},\theta^{s}))\] (3)
5. k = k + 1;
6. End for
```  
**Algorithm 1**Self-Training with Noise for Depth Completion  
The algorithm introduces an advanced self-training framework in which the teacher model generates high-quality pseudo-labels using clean data. The student model is trained to replicate labels from both annotated and pseudo-labeled datasets, even when exposed to noisy inputs. This approach significantly boosts the student model's ability to generalize effectively, surpassing the teacher model's capabilities. Two types of noise augmentation are introduced to the student: Gaussian blur is applied to RGB images [40] and stochastic perturbation is used for sparse depth [41]. Applying noise augmentation to the two input data types enhances the model's ability to utilize the limited labeled data more effectively, promoting the extraction of more generalized features. Expanding the training dataset for the student model is crucial. Therefore, random noise is injected into the input variables of the student model during training, ensuring variations in the data at every time step. Incorporating noise into the student model generates new samples in proximity to existing samples, effectively smoothing the structure of the input space. This smoothing effect simplifies the learning process by making the mapping function easier to model. Furthermore, the injected noise mimics potential errors encountered in real-world depth information collection, enhancing the model's robustness. The two types of noise used in this process are as follows:  
(1) Gaussian blur on RGB images: Gaussian blur is a commonly utilized image processing technique that applies a Gaussian function to modify pixel values within the image. This technique reduces sharp edges and fine details, producing a visually softened effect. The Gaussian function, known for its bell-shaped distribution, computes the weight of each pixel based on its distance from the kernel's center. Pixels closer to the center have a greater influence on the resulting blurred value. Incorporating Gaussian blur into RGB images during self-training introduces controlled noise, which strengthens the model's capacity to generalize and extract robust features for depth completion. This technique enhances the model's resilience variations in input data, boosting performance and precision in depth estimation tasks.