## 2 Orthogonal Projection and Global Deformation Acquisition Method of Structures
### Edge Detection  
The edge of structural image is an important carrier of overall deformation information. Edge detection is a method to analyze the main features of images [21]; it can greatly reduce the amount of data, eliminate information not related to deformation monitoring, and retain the basic attributes of structure. The basic task of edge detection is to recognize the step change of the gray value of the structure edge in the image, which can be further used to obtain the feature edge of the structure. According to [22], the step edge is related to the peak value of the first-order derivative of the gray level of the image, and the degree of change of the gray value can be expressed by gradient. The gradient of image function is a vector with direction and size, as shown below:  
\[G(x,y)=\left[\begin{array}{c}G_{x}\\ G_{y}\end{array}\right]=\left[\begin{array}{c}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{array}\right]. \tag{15}\]  
It can be seen that the direction of the vector \(G(x,y)\) is the change rate of the gray value of function \(f(x,y)\).  
The amplitude of the gradient can be expressed as:  
\[\left|G(x,y)\right|=\sqrt{G_{x}^{2}+G_{y}^{2}}. \tag{16}\]  
In this paper, the absolute value is used to approximate the gradient amplitude:  
\[\left|G(x,y)\right|\approx\max\left(\left|G_{x}\right|,\left|G_{y}\right| \right). \tag{17}\]  
The direction of the gradient can be derived as:  
\[\alpha(x,y)=arctan(G_{y}/G_{x}) \tag{18}\]  
where \(\alpha\) is the angle between the direction vector and the \(x\)-axis.  
From the above formulas, it is suggested that the degree of the change in gray levels can be detected by the discrete approximation function of gradient. At the edge of the structure, the gray value will change [23], resulting in the maximum value of the gradient function. On this end, the edge can be extracted through the above features.  
From the above algorithm, we can see that the essence of the image edge is the point of discontinuous gray level, or where the gray level changes dramatically. The drastic change of gray level of edge means that near the edge point, the signal has high frequency components in the spatial domain. Therefore, the edge detection method is essentially to detect the high-frequency component of the signal, but it is difficult to distinguish the high-frequency component of the gray signal from the environmental noise of the actual structure photogrammetry, which makes it difficult to accurately extract the edge information of the structure. Taking one-dimensional signal of the structure image as an example, as shown in Figure 2, if point A is regarded as the edge point of the signal and there is a jump in the signal, then whether there is an edge at point B and point C needs to be treated with caution. In fact, point B and point C are probably the combination of the signal and some noise.  
Edges with continuous gradients, such as point A in Figure 2, are very rare in actual structure images. Most of the structural edge points will be accompanied by environmental noise, forming a large number of complex edge points such as points B and C. Therefore, it is necessary to study the false edges caused by noise in order to ensure the accuracy of structural deformation monitoring.