## 4 Experimental Setup and Result Analysis
### Image Rectification Data Preparation  
Designing customized data generation methods tailored to specific application scenarios has been widely recognized as an effective strategy for addressing real-world challenges in visual tasks. Building upon this idea, given the difficulty in acquiring large-scale, high-quality wide-angle images paired with their rectified ground-truth counterparts, this study adopts a synthetic data generation approach for training the wide-angle image rectification network. Specifically, existing distortion correction techniques are leveraged to construct a synthetic dataset [6; 45; 46; 47].  
Specifically, original image samples are first selected from the MS-COCO dataset [48], and radial distortion in wide-angle images is simulated using a fourth-order polynomial model. Experimental results demonstrate that this model can effectively approximate most common projection models and offers high fitting accuracy.  
For the distortion coefficients \(\phi=[k_{1},k_{2},k_{3},k_{4}]\), the value ranges are determined based on empirical settings from prior studies [45; 49]. Specifically, we set \(k_{1}\in[-10^{-5},-10^{-8}]\), while the remaining coefficients are assigned ranges based on empirical observations: \(k_{2}\in[-10^{-9},-10^{-12}]\) or \(\in[10^{-12},10^{-9}]\). The ranges for \(k_{3}\) and \(k_{4}\) follow a similar configuration, ensuring that the synthesized distortions closely approximate those found in real-world wide-angle imagery.  
Specifically, four distortion coefficients are randomly sampled from the following value ranges to construct a diverse set of radial distortion models:  
\[\begin{array}{l}k_{1}\in[-1\times 10^{-4},-1\times 10^{-8}]\\ k_{2}\in[1\times 10^{-12},10^{-8}]\;ork_{2}\in[-1\times 10^{-8},-1\times 10^{-12}]\\ k_{3}\in[1\times 10^{-16},\;1\times 10^{-12}]\;ork_{3}\in[-1\times 10^{-12},-1 \times 10^{-16}]\\ k_{4}\in[1\times 10^{-20},\;1\times 10^{-16}]\;ork_{4}\in[-1\times 10^{-16},-1 \times 10^{-20}]\\ x_{u}=x_{d}(1+k_{1}r_{d}^{2}+k_{2}r_{d}^{4}+k_{3}r_{d}^{6}+k_{4}r_{d}^{8}+\dots) \\ y_{u}=y_{d}(1+k_{1}r_{d}^{2}+k_{2}r_{d}^{4}+k_{3}r_{d}^{6}+k_{4}r_{d}^{8}+\dots) \end{array} \tag{13}\]  
In Equation (13), \((x_{u},y_{u})\) and \((x_{d},y_{d})\) denote the coordinates of the undistorted and distorted image points, respectively, and \(r_{d}\) represents the Euclidean distance from the distorted point to the distortion center. The parameters \(k_{1},k_{2},k_{3},k_{4}\) are coefficients in the polynomial distortion model, which controls the intensity and shape of the radial distortion.  
The polynomial distortion model is one of the most commonly used approaches for correcting fisheye camera images. Based on mathematical modeling, this method fits the radial distortion present in fisheye imagery using a polynomial function, enabling pixel-level geometric correction. The correction process typically involves two steps: first, converting image coordinates to a normalized coordinate system; then, applying the polynomial function to correct the radial distortion and restore a projection closer to the real-world geometry.  
Due to its simplicity and effective fitting performance in handling fisheye distortion, this model has been widely applied in industrial vision and wide-angle image reconstruction tasks. Although more complex distortion models exist, the field has largely standardized around two primary types of polynomial distortion formulations.