# Novel Method for Bridge Structural Full-Field Displacement Monitoring and Damage Identification
## 2 Full-Field Displacement Monitoring of a Structure Based on Natural Texture Feature Monitoring
### Calculation Method of the Structure's FFDV  
The feature points obtained using the SIFT feature-extraction method constitute the source data of the structural FFDV. The matching of feature points only completes the connection between homonymous points on the structural surface, while the relative spatial position change at the homonymous points before and after deformation cannot be determined. We established a standard reference system under different operating conditions to determine the relative positions of the feature points before and after deformation. We used fixed points to simplify the spatial geometric transformation of the feature points.  
Establishing a fixed point provides a location basis for the feature points of the structure before and after deformation. To accurately calculate the displacement value of the feature points of the structure before and after deformation, we propose a mathematical model of the relative position change of the feature points, as shown in Figure 3.  
In Figure 3, the red squares represent fixed points. Theoretically, a fixed point \(O\) can represent any feature point on the chessboard. The center of the chessboard was selected as the fixed point. As shown in Figure 3, the displacement vector \(A_{i}A^{\prime}_{i}\) is composed of starting point \(A_{i}\) before deformation and ending point \(A^{\prime}_{i}\) after deformation. This displacement vector is the basic element of the structure's FFDV, as shown by the blue vector in Figure 3, where \(M_{i}\) represents the feature points before and after deformation, which are generated based on the following algorithm: For bridge images (\(i_{1}\) and \(i_{2}\)) before and after deformation, the feature points are first extracted from the images and expressed as \(A_{i}\) and \(A^{\prime}_{i}\), respectively; subsequently, the initial matching for \(C\) = {(\(A_{i}\), \(A^{\prime}_{i}\)):i = 1, 2,..., \(n\)} is obtained, where \(A_{i}\)\(\in\)\(i_{1}\) and \(A^{\prime}_{i}\)\(\in\)\(i_{2}\). Feature points \(A_{i}\) and \(A^{\prime}_{i}\) are projected to the fixed point \(O\) to obtain feature-point coordinates, \(A_{i}\)(\(x_{i}\),\(y_{i}\)) and \(A^{\prime}_{i}\)(\(x^{\prime}_{i}\),\(y^{\prime}_{i}\)), respectively, and coordinate values (\(x_{i}\),\(y_{i}\)) and (\(x^{\prime}_{i}\),\(y^{\prime}_{i}\)) are extracted between fixed point \(O\) and each of the feature points \(A_{i}\) and \(A^{\prime}_{i}\). Next, \(M_{i}\) = [(\(x^{\prime}_{i}-x_{i}\)),(\(y^{\prime}_{i}-y_{i}\))]\({}_{i}\) is obtained. \(M_{i}\) is a displacement vector with two parameters, i.e., length \(L\) and angle \(\theta\), which can be obtained from the known feature-point coordinates \(A_{i}\)(\(x_{i}\),\(y_{i}\)) and \(A^{\prime}_{i}\)(\(x^{\prime}_{i}\),\(y^{\prime}_{i}\)). Set \(M\) = {\(M_{i}\):\(I\) = 1, 2,..., \(n\)} is composed of \(M_{i}\) as the initial calculated full-field displacement of the structure. The FFDV of the structure surface can be extracted using the method described above by solving vector set \(M\).