## 4 Research on Image Acquisition and Theoretical Methods
### Scene 3D Reconstruction Study  
After acquiring the post-blast depth images of the tunnel, the 3D reconstruction of the tunnel's palm face and contour surface is conducted using the motion-based Structure from Motion (SfM) method. This method involves extracting and matching feature points from a series of 2D depth images containing motion information, ultimately reconstructing a highly accurate 3D point cloud model of the target object. The SfM method is particularly effective because it leverages the motion between images to identify and align corresponding feature points, resulting in a detailed and precise 3D representation [40]. This process ensures that the complex structures within the tunnel are accurately captured, providing valuable data for further analysis and evaluation of the tunnel's condition post-blast. By utilizing this method, we can achieve a comprehensive and reliable 3D model that supports various engineering and research applications. (1) Feature Point Detection and Scale Space Extreme Detection: Identifying feature points that exist in the same image at different scales involves generating a series of images at varying scales by continuously down-sampling and smoothing the original depth map [41]. This process ultimately identifies the feature points present in each image. The Gaussian kernel is the unique kernel capable of producing multiple scales. The image is convolved with Gaussian kernels of different scale space factors (standard deviations) to obtain depth maps with varying levels of blur [42]. These depth maps are then converted to the Gaussian scale space, as shown in the formula. The Gaussian scale space allows for the detection of feature points across different scales, ensuring that the most significant points are accurately identified regardless of their size or orientation. This method is crucial for robust feature matching in subsequent 3D reconstruction processes, as it provides a reliable set of feature points that can be used to align and integrate multiple images into a cohesive 3D model. \[\mathrm{L(x,y,\sigma)=G(x,y,\sigma)*I(x,y)} \tag{1}\]  
\[\mathrm{G(x,y,\sigma)=\frac{1}{2\pi\sigma}e^{-(x^{2}+y^{2})/2\sigma^{2}}} \tag{2}\]  
\(\mathrm{G(x,y,\sigma)}\) is the Gaussian kernel function and represents the Gaussian scale space of the image, and is the point corresponding to the point in image I, and is the scale coordinate, whose value is 0 in the original depth map, when the value becomes smaller the resolution will be higher accordingly, and the details in the image will be more obvious. Remove unstable points. The extreme points in the scale space may be feature points, so it is necessary to search the 26 neighbors of each point in the Gaussian difference scale space, and if the point is a local extreme point, it is saved as a candidate key point. However, since the image is a discrete sampling of a continuous signal, the pixel-level extreme points  
Figure 5: Image Acquisition Schematic. obtained in the previous step may be inaccurate, and further computation is needed to determine the exact location of the key point. For each extreme point, a second-order Taylor expansion of the DoG function is performed:  
\[\text{D(x)}=\text{D}+\frac{\partial\text{D}^{\text{T}}}{\partial\text{X}}\text{ X}+\frac{1}{2}\text{X}^{\text{T}}\frac{\partial^{2}\text{D}}{\partial\text{X}^{2}} \tag{3}\]  
By taking the derivative of the equation and making the derivative zero, the solution obtained is the offset of the extreme point:  
\[\hat{\text{X}}=\frac{\partial^{2}\text{D}^{-1}}{\partial\text{X}^{2}}\frac{ \partial\text{D}}{\partial\text{X}} \tag{4}\]  
\(\text{X}=(\text{x},\text{y},\sigma)\), if the offset is greater than 0.5 in either dimension, then the extreme point has been offset to another pixel and needs to be culled.