## 3 The Proposed Method
### Self-Training with Noise for Depth Completion
The 1D Gaussian function is expressed as follows:  
\[\mathrm{G(x)}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\mathrm{e}^{-\frac{\sigma^{2}}{2 \sigma^{2}}}. \tag{4}\]  
In 2D space, the Gaussian function is defined as the product of two independent 1D Gaussian functions, each corresponding to a distinct dimension:  
\[\mathrm{G(x,y)}=\frac{1}{2\pi\sigma^{2}}\mathrm{e}^{-\frac{\sigma^{2}+\sigma^{ 2}}{2\sigma^{2}}}, \tag{5}\]  
where r denotes the horizontal distance from the origin, d implies the vertical distance from the origin, and \(\sigma\) signifies the standard deviation of the Gaussian distribution. When extended to two dimensions, Equation (5) creates a surface characterized by concentric circular contours, indicating a Gaussian distribution that radiates symmetrically from the central point. (2) Stochastic perturbation on sparse depth: To improve the model's resilience and generalization capability, we introduce stochastic noise perturbation into the sparse depth data. This process involves randomly selecting a subset, with 1-k% of the available depth points, and modifying each chosen point by a small random noise within a range of +/-\(\varepsilon\), where \(\varepsilon\) = 0-5 units. By introducing these controlled perturbations, we generate a diverse set of slightly modified depth maps encountered during training, effectively simulating the natural variability and uncertainties in real-world depth measurements. This method fosters greater model resilience and mitigates overfitting by reducing the model's reliance on specific sparse data patterns, allowing adaptation to a broad spectrum of depth scenarios. This approach substantially improves both the performance and the precision of our framework in depth completion tasks. Our improvement strategy involves incorporating noise into the student model while retaining a level of complexity equal to or greater than that of the teacher model. This knowledge expansion technique enables the student to outperform the teacher by equipping it with adequate capacity and subjecting it to demanding conditions through controlled noise. Figure 3 evaluates noise-processing effects through visual comparisons. Figure 3a,b contrast the original RGB image with its Gaussian-blurred counterpart, where edge sharpness and fine details are attenuated to create a smoothed appearance. Subsequent columns analyze prediction outcomes across varying inputs: Figure 3c: baseline results using unmodified data; Figure 3d: predictions with Gaussian-blurred RGB inputs highlighting detail loss impacts; and Figure 3e: outputs combining the original RGB image with stochastically perturbed depth data. Incorporating noise into the student model during training establishes a more rigorous learning environment, promoting the acquisition of robust and generalized features. This regularization technique mitigates overfitting on training data and enhances the model's adaptability to diverse input scenarios. Furthermore, the student model, possessing a complexity level equal to or greater than that of the teacher model, fully leverages the knowledge transferred from the teacher. By expanding the student model's capacity, learning becomes more efficient, allowing it to capture and retain more features and patterns. Our approach surpasses the constraints of the teacher model by equipping the student model with both the capability and the challenging conditions necessary for superior learning. This approach promotes the development of a more robust and precise depth completion model and highlights the effectiveness of well-designed student-teacher training frameworks in performance improvements. Figure 3: Comparative results of noise-processing techniques. (**a**) displays the unprocessed RGB image, while (**b**) shows its Gaussian-blurred counterpart.