## 2 Dense Feature Extraction and Matching Theory and Mismatch Cause Analysis
### Secondary Removal of Feature Mismatches Based on SDCCs  
With the matched feature set obtained after coarse removal with the RANSAC method, the displacements of these feature points can be calculated, and wrong displacements reveal that mismatched feature points pairs still exist, for example, point G in Figure 3b, which can be effectively removed with prior information of structural displacement. Prior knowledge of structural displacement is the information of the structural geometric size, mechanical properties, and external load distribution, for example, the displacement direction, distribution, and angle of measured points. The prior information can be utilized in the analysis and evaluation of the feature point displacement calculated and can be utilized to effectively remove the remaining mismatches. The continuity assumption is a fundamental deformation hypothesis in structural mechanics assuming that the object maintains continuum during deformation without cracking or overlapping, or the displacements of adjacent measurement points within an infinitesimal range are continuous and uninterrupted. Taking the example of a cantilever beam in the initial and deformed states shown in Figure 4a, where the initial point \(P\) deforms to \(P^{\prime}\), and any point \(P_{1}\) within an infinitesimal range \(\Delta x\) around this point deforms to \(P_{1}\)' as shown in Figure 4b. The difference \(\Delta dy\) between the vertical deformations \(dy_{1}\) and \(dy\) of these two points should be approximately zero, according to the continuity deformation assumption; that is:  
\[\Delta dy=dy-dy_{1}\approx 0 \tag{6}\]  
Figure 4: Schematic diagram of continuity deformation of structural displacements. (**a**) Cantilever beam in the initial and deformed states. (**b**) Displacement continuity within an infinitesimal range. For continuous medium deformation, the continuity assumption within an infinitesimal region can also be expressed in strict terms of displacement \(d\), angle \(\theta\), and curvature \(\kappa\):  
\[ \begin{cases}\Delta d\approx 0\\ \Delta\theta\approx 0\\ \Delta\kappa\approx 0\end{cases}  \tag{7}\]  
The above conditions of Equation (7) are called the Structural Displacement Continuity Constraints (SDCCs), and multiple constraint conditions need to be introduced to completely remove mismatches. When the dimensions, material properties, and external loads are known, the maximum deformation \(D\) of the beam structure can be calculated by the relevant theorems and formulas of material mechanics, and the feature points whose displacements are larger than D should be considered mismatches. For simplicity, the vertical displacements \(dy_{1}\), \(dy_{2}\), and \(\Delta dy\) are calculated from two adjacent feature points and establish structural vertical displacement consistency constraints in the spatial field, referring to Equation (5) as follows:  
\[ \begin{cases}dy_{1}\leq D\text{ and }dy_{2}\leq D\\ \Delta dy=|dy_{1}-dy_{2}|\leq\varepsilon\end{cases}  \tag{8}\]  
where \(\varepsilon\) is a value lower than \(D\) and can be empirically selected between \(0\) and \(D\). In structural vertical displacement consistency constraints, the value of \(\varepsilon\) should be taken according to the specific experimental situation or actual project situation, and it indicates the displacement difference of two adjacent feature points or the continuity of the displacement field. The smaller the value of \(\varepsilon\), the stricter the requirements. Theoretically, a value of \(\varepsilon\) less than \(0.5D\) can obtain good results. However, excessively strict requirements of \(\varepsilon\) does not lead to more precise results, and a smaller value of \(\varepsilon\) will lead to the removal of some the feature matching pairs with good accuracy as mismatches, which will result in the sparsity of final retained feature pairs and is unfavorable for high-precision full-field displacement measurement.