# Study on Bridge Displacement Monitoring Algorithms Based on Multi-Targets Tracking
## 2 Digital Image Correlation  
The basic principle of DIC is to mesh the ROI in the reference image, treat the motion of each sub-region as a rigid motion, and calculate the correlation according to the pre-defined correlation function through a certain search method for each sub-region. In the deformed image, the location of the sub-region after deformation is the region whose cross-correlation coefficient with the sub-region in the reference image is the maximum, and then the displacement of the sub-region is obtained. By calculating all the sub-regions, the deformation information of the whole field can be obtained.  
In the DIC method, we need to determine a correlation coefficient function, which is a criterion to measure the matching degree between reference subset and target subset and then find out the extremum of correlation coefficient by corresponding integral pixel displacement search method to extract displacement information. About the extremum, we take the maximum value in positive correlation coefficient function and the minimum value in negative correlation coefficient function. Different expressions of correlation coefficients have different performance and its performance will directly affect the search accuracy and computation efficiency of the DIC method. Two commonly used correlation coefficient functions, Zero Normalized Cross-Correlation (ZNCC) and Zero Normalized Sum of Squared Differences (ZNSSD), are presented in Equation (1) [27; 28; 29]:  
\[\begin{array}{l}C_{ZNCC}=\sum\limits_{i=-M}^{M}\sum\limits_{j=-M}^{M}[\frac{ [\frac{f(x_{i}y_{j})-f_{m}}{\Delta f}\{g(x_{i}^{\prime}y_{j}^{\prime})-g_{m} \}}{\Delta f\Delta g}],\\ C_{ZNSSD}=\sum\limits_{i=-M}^{M}\sum\limits_{j=-M}^{M}[\frac{ [\frac{f(x_{i}y_{j})-f_{m}}{\Delta f}-\frac{g(x_{i}^{\prime}y_{j}^{\prime})-g_{ m}}{\Delta g}]^{2}},\end{array} \tag{1}\]  
where \(f()\) and \(g()\) mean the gray level intensity of a pixel for the reference and deformed images,  
\[\begin{array}{l}f_{m}=\frac{1}{(2M+1)^{2}}\sum\limits_{i=-M}^{M}\sum \limits_{j=-M}^{M}f(x_{i},y_{j}),\\ g_{m}=\frac{1}{(2M+1)^{2}}\sum\limits_{i=-M}^{M}\sum\limits_{j=-M}^{M}g(x_{i} ^{\prime},y_{j}^{\prime}),\\ \Delta f=\sqrt{\sum\limits_{i=-M}^{M}\sum\limits_{j=-M}^{M}[f(x_{i},y_{j})-f_ {m}]^{2}},\\ \Delta g=\sqrt{\sum\limits_{i=-M}^{M}\sum\limits_{j=-M}^{M}[g(x_{i}^{\prime},y_{j}^{\prime})-g_{m}]^{2}}.\end{array}\]  
The advantages of this method are that the algorithm is simple and easy to realize, it is the whole field of view and the non-contact deformation monitoring, and the monitoring precision is high.  
The disadvantages of this method are that it is a deformation monitoring method according to the speckle image of the structure. If the texture of the structure surface is good, the speckle image with a good effect can be obtained, and the results of deformation monitoring will also be good. However, when the texture of the structure surface is general or no obvious texture, it is necessary to make artificial spots, otherwise the displacement can not be monitored by this method. In addition, it can be seen from the principle that the essence of this method is to carry out a large number of repeated calculations, and the overall calculation is very large. If the image pixels are large, the correlation computation of the two images would be extremely large so it can not meet the requirements of monitoring speed at all.