## 4 Research on Image Acquisition and Theoretical Methods
### Scene 3D Reconstruction Study
If the DoG value of the point is not large enough, it also needs to be culled:  
\[\text{D(\hat{x})}=\text{D}+\frac{1}{2}\frac{\partial\text{D}^{\text{T}}}{ \partial\text{X}}\text{X} \tag{5}\]  
As a rule of thumb, take up to this point, the part of feature point detection is completed. (2) Feature point matching: after the step of feature point extraction is completed, the matching of feature points will be carried out, using Euclidean distance for neighborhood search and building a KD vocabulary tree to narrow down the search scope and improve the search efficiency. Denote the feature points around depth image I by denote the feature points around depth image I. For each set of depth images I and J, consider each feature, find the nearest neighbor feature vector \(\text{f}_{\text{nn}}\in\text{F(J)}\) :  
\[\text{f}_{\text{nn}}= \begin{array}{c}\text{argmin}\|\text{f}_{\text{d}}- \text{f}_{\text{d}}^{\prime}\|_{2}\\ \text{f}\in\text{F(J)}\end{array}  \tag{6}\]  
Using the data structure of the established KD vocabulary tree for the nearest neighbor matching calculation, the distance of the nearest neighbor is first made to be \(\text{d}_{1}\), and then the distance to the second nearest match is found to be \(\text{d}_{2}\), and if the ratio of the distances between the two points, \(\text{d}_{1}\), and \(\text{d}_{2}\), is less than the set threshold, then it can be determined to be an acceptable matching pair [43]. (3) Scene sparse reconstruction, each 3D pointXj, obtained after feature matching, is projected onto the 2D image plane of the camera through the projection equation, which produces a certain projection error, which is the actual distance between the real point on the image and the projected point [44] (Figure 6). Figure 6: Projection Error Schematic. The sparse reconstruction of the target object based on the SFM algorithm starts with obtaining the relative rotations and translations by decomposing the essential matrix. The essential matrix satisfies the following relationship:  
\[\left(\mathrm{K}^{-1}\mathrm{x}\right)\mathrm{E}\left(\mathrm{K}^{\prime-1} \mathrm{x}^{\prime}\right)=0 \tag{7}\]  
K is the intrinsic matrix of the camera, and with the matching point pairs in the form of chi-square coordinates, the essential matrix can be obtained stably by incorporating the RANSAC algorithm. By decomposing the essential matrix, we can obtain the two-view relative R and t.  
The global rotation matrix is then calculated from the relative rotation matrix, Equation:  
\[\mathrm{R_{j}}=\mathrm{R_{ji}R_{j},i,j\in(1,2,\ldots n)} \tag{8}\]  
Splitting the Equation (8) and solving them sequentially and then combining them to get the results can reduce the reliability difference between the equations and the splitting results are as follows:  
\[\mathrm{r_{k}^{j}-R_{ij}r_{k}^{i}=0,i,j\in(1,2,3),k\in(1,2,3)} \tag{9}\]  
\(\mathrm{r_{k}}\) is the kth column of R [45; 46; 47]. Since the SFM algorithm is an iterative algorithm, it is necessary to continuously adjust the parameters and optimize the results in each step to obtain more accurate and stable results. For a viewpoint a and b trajectory on the same image, the objective of the projection error is optimized with the equation:  
\[\mathrm{g(C,X)=\sum_{i=1}^{a}\sum_{j=1}^{b}\omega_{ij}\left\|q_{ij}-P\left(C_{i },X_{j}\right)\right\|^{2}} \tag{10}\]  
When camera i observes trajectory j is taken as 1, and 0 at other times, and is the cumulative sum of the projection errors of trajectory j in camera i. The purpose of the SFM algorithm is to optimize the objective function by finding suitable camera and scene parameters for the optimal reconstruction of the 3D scene [48; 49]. After completing the 3D reconstruction of each image, they need to be fused to construct the geometric information of the whole scene, that is, the sparse 3D point cloud [42]. (4) Scene density reconstruction [50], after completing the step of sparse reconstruction, Poisson dense reconstruction of the 3D scene model is also needed to get a more detailed and complete 3D model.