# Experimental Validation of Non-Marker Simple Image Displacement Measurements for Railway Bridges
## 4 Verification Results by Model Bridge
### Measurement Accuracy  
Figure 10 shows examples of the accuracy verification of the non-marker image measurement when the midspan was vibrated at 35 Hz. The horizontal axis is the displacement of the contact measurement, whereas the vertical axis is the displacement of the non-marker image. Hence, the straight line with a slope of 1 indicated by the black line in Figure 10 has an error of 0. Moreover, the regression line of the measured value and the reliability interval of \(\pm\)2\(\sigma\) is shown by the red bold and dotted lines, respectively.  
Figure 9: Comparison of displacement responses during 35 Hz excitation at the midspan.  
The regression line at points other than Ch6 is similar to the straight line with a slope of 1. In this study, the pixel displacement was converted from the height of the H steel and the number of pixels at the relevant location in the image. Because the effect of this simple method on the measurement accuracy is insignificant, the absence of any practical problem is confirmed. When the effect of overexposure is significant, as in Ch6, the error of the displacement amplitude obtained by the non-marker image measurement is large; therefore, an apparent error occurs in the pixel-displacement conversion.  
The reliability interval of \(\pm 2\sigma\) represents the random error included in the non-marker image measurement. The reliability intervals of \(\pm 2\sigma\) were about \(\pm 20\)\(\upmu\)m (0.014 pixels), with the exception of Ch6 in the overexposed area. Furthermore, the amplitude dependence of the random error was not confirmed; thus, the effect of pixel locking is small. Moreover, it can be confirmed that the error is characterized by white noise. However, as shown in the previous section, Ch6 has a larger random error than the other measurement points because of overexposure.  
Figure 11 shows the standard deviation \(\sigma\) (average value of the absolute error) obtained from the regression analysis of the non-marker measurement error for the contact measurement at each measurement point. In Ch4, there was a problem with the contact displacement meter during measurement; therefore, it was excluded from the evaluation. The standard deviation, \(\sigma\), is about 0.004-0.007 pixels, with the exception of Ch6, which is the overexposed region. About 95% of the data is in the range of \(\pm 2\sigma\) for a normal error distribution. Therefore, if \(\pm 2\sigma\) or more is considered as the measurement range in which the noise and signal can be separated, the limit of non-marker image measurement is considered to be 0.015 pixels for the single-sided amplitude and 0.03 pixels for the double-sided amplitude. However, it is necessary to set the measurement points to avoid overexposed regions such as Ch6.  
Figure 11 shows the results of high-pass filtering with a cutoff frequency of 2 Hz on the measured waveform to eliminate the long-period fluctuations observed in the displacement responses of the non-marker measurement. Although the measurement  
Figure 11: Measurement error and the effect of filtering (35 Hz excitation at the midspan).  
Figure 10: Accuracy of non-marker image measurement (35 Hz excitation at the midspan).  
error of the filter response is reduced compared to the original responses, the amount of reduction is \(\sim\)0.002 pixels, and no significant reduction effect can be observed. Hence, as discussed in the previous section, most errors in the non-marker image measurement are white noise and are difficult to eliminate by simple filtering. However, the above consideration is for a minute error of about \(\pm\)10 \(\upmu\)m (0.007 pixels), and no problem exists in its application to bridge displacement measurement.