# Experimental Validation of Non-Marker Simple Image Displacement Measurements for Railway Bridges
## 3 Processing and Test Methods  
### Image Processing Method  
In this study, the displacement is calculated from the image obtained via the DIC method [12]. A random pattern or marker is often applied to the object to be measured [12; 21]. Here, using the high-resolution image obtained by a video camera, the displacement is calculated via DIC [22] without applying a random pattern or setting markers. Figure 1 shows an example of DIC processing in the model bridge in this study. The subset A of the red frame in the figure is set at the time point \(t_{0}\). The length and width of the subset A are equal, and the size of subset A is \(L_{A}\times L_{A}\). Further, the local coordinate system whose origin is the control point, which is the center of subset A, is set as \((i^{\prime},j^{\prime})\). Moreover, subset B is set as the search area for the amount of movement of subset A at different time points. Here, the center of subset B is the origin \((i^{\prime},j^{\prime})=(0,0)\) of the local coordinate system. Furthermore, for the size \(L_{B}\times L_{B}\) of subset B, \(L_{B}>L_{A}\). In this study, \(L_{B}=2L_{A}\) was set primarily to verify the measurement accuracy of minute displacements of 0.1 pixels or less. Figure 1: Digital image correlation method processing concept. The amount of image movement \((\delta i_{t},\delta j_{t})\) at time point \(t\) is estimated by searching for the position in subset B that has the highest correlation with the lightness vector \(D_{A}\), which is the vectorization of the lightness series of each pixel of subset A. Indices for the Sum of Squared Difference (SSD), Sum of Absolute Difference (SAD), and Normalized Cross-Correlation (NCC) are used for calculating the amount of image movement [22; 23; 24]. The NCC shown in Equation (1) was used to eliminate the effect of changes in brightness of the entire subset. Let \(d_{t}(i^{\prime},j^{\prime})\) be the lightness of the reference at local coordinates \((i^{\prime},j^{\prime})\), and \(d_{t}(i^{\prime},j^{\prime})\) be the light level at time, \(t\), at the same local coordinates. The NCC in the number of movements, \((u^{\prime},v^{\prime})\), is determined by Equation (1). \[NCC\big{(}u^{\prime},v^{\prime}\big{)}=\frac{\sum_{i^{\prime},j^{\prime}}\Big{\{} \Big{(}d_{t}(i^{\prime},j^{\prime})-\overline{d}_{t}\Big{)}\Big{(}d_{t}(i^{ \prime}-u^{\prime},j^{\prime}-v^{\prime})-\overline{d}_{t}\Big{)}\Big{\}}}{ \sqrt{\sum_{j^{\prime},j^{\prime}}\Big{(}d_{t}(i^{\prime},j^{\prime})- \overline{d}_{t}\Big{)}^{2}\sum_{i^{\prime},j^{\prime}}\Big{(}d_{t}(i^{ \prime}-u^{\prime},j^{\prime}-v^{\prime})-\overline{d}_{t}\Big{)}^{2}}}, \tag{1}\]  
where \(\overline{d}_{t}\) represents the average value of the light level in subset A, and \(\overline{d}_{t}\) represents the average value of the light level within the range of size \(L_{A}\times L_{A}\) at a time point \(t\). Therefore, as shown in Equation (2), the amount of movement \((\delta i^{\prime}_{t},\delta j^{\prime}_{t})\) that maximizes the NCC is the estimated value of the displacement (in pixels) at the time point \(t\). \[\big{(}\delta i^{\prime}_{t},\delta j^{\prime}_{t}\big{)}=\text{argmax}_{u^{ \prime},v^{\prime}}\big{\{} NCC\big{(}u^{\prime},v^{\prime}\big{)}\big{\}}. \tag{2}\]  
Figure 2 shows the parabola fitting used to estimate the displacement below 1 pixel (sub-pixel estimation) [23; 25]. In parabola fitting, the 2D parabola function is fitted to the correlation value of the pixel with the highest correlation value \((\delta i^{\prime}_{t},\delta j^{\prime}_{t})\) and the surrounding 8 pixels. The positions \(\delta i_{t}\), \(\delta j_{t}\) that represent the maximum value of the fitted parabolic function are calculated as sub-pixel displacements in 1/1000-pixel units. In sub-pixel estimation using the fitting function, a phenomenon called pixel locking may occur, in which the number of movements is biased to an integer value [25]. The estimation error cancelation (EEC) method [25] was introduced in the DIC of this study to minimize the effect of pixel locking.