## 3 Results and Discussion  
In this section, the proposed algorithm was used to test the six types of pattern images (\(F_{i}\), \(F_{r}\), \(F_{c}\), \(F_{s1}\), \(F_{s2}\) and \(F_{s3}\)) shown in Figure 11a-f. The center point maps reflecting the periodic characteristics of the pattern are shown in Figure 11g-l. Since \(\theta_{c}\) in Figure 10a is an external angle and does not affect the overall accuracy of the angle to be measured, this paper measured and evaluated the \(\theta_{h}\) and \(\theta_{v}\) shown in Figure 10a. To evaluate the proposed algorithm, we compare the proposed algorithm with manual measurement results. Manual measurement of the period length was achieved by clicking the center points of the patterns on the computer screen. As illustrated in Figure 12b, for example, we obtained the center points \(H_{i}\), \(V_{i}\), (\(i=1\), 2,... ) and then calculated the weft period length as \(T_{h}=d_{HHj}\), where \(d_{HHj}\) is the distance between the \(H_{i}\) and \(H_{j}\) (\(j=i+1\), \(i=1\), 2,... ).And the warp period length as \(T_{v}=d_{VIVj}\), where \(d_{VIVj}\) is the distance between the \(V_{i}\) and \(V_{j}\) (\(j=i+1\), \(i=1\), 2,... ). \(T_{h}\) and \(T_{v}\) were measured twenty times, and the final result was the average of the measurements. Similarly, manual measurement of the \(\theta_{h}\) and \(\theta_{v}\) also was accomplished by clicking the center points of the patterns on the computer screen. The center points in the same period direction were fitted as a straight line and the angle between the line and the x-axis was calculated (see Figure 11a). We measured each angle twenty times and then calculated the average values to obtain the result. The standard deviation \(\delta\) was used to analyze the accuracy of the manual measurements of the period length and skew angle. Figure 11: Six types of pattern images and their center point maps. Figure 12: Manual measurement of parameters. (**a**) Skew angle, (**b**) period length. The expression of the standard deviation is shown as follows:  
\[\delta=\sqrt{\frac{1}{N}{\sum\limits_{i=1}^{N}{{{({X_{i}}-\overline{X})}^{2}}}}} \tag{4}\]  
where \(\overline{X}\) is the average value of \(X\). Table 2 shows the manual measurement standard deviation of period length of images (\(F_{r}\), \(F_{c}\), \(F_{s1}\), \(F_{s2}\) and \(F_{s3}\)), where \(\delta_{Tha}\), \(\delta_{Thy}\), and \(\delta_{Thy}\) were the minimum, maximum and average standard deviation, respectively, of the werf period length measurement. \(\delta_{Tva}\), \(\delta_{Ty\beta}\), and \(\delta_{Ty\gamma}\) were defined similarly for the warp period length measurement. As shown in Table 2, the standard deviation obtained by manual measurement was very small. Therefore, it was reasonable to use the manual measurement results as the evaluation standard. Table 3 shows the period length measurement results of images (\(F_{i}\), \(F_{r}\), \(F_{c}\), \(F_{s1}\), \(F_{s2}\) and \(F_{s3}\) in Figure 11). The method based on image autocorrelation, the method based on distance matching function and the proposed method were compared with the manual measurement results, where \(T\) is the period length, the subscript \(h\) represents the werf direction, the subscript \(v\) represents the warp direction, \(m\) stands for manual measurement method, \(p\) stands for the proposed method, \(z\) stands for autocorrelation, \(d\) stands for distance matching function. Table 4 shows the relative error value between different period length measurement methods and manual measurements of images (\(F_{i}\), \(F_{r}\), \(F_{c}\), \(F_{s1}\), \(F_{s2}\) and \(F_{s3}\) in Figure 11), where the \(e_{therm}\) represents the relative error of \(T_{hp}\) with \(T_{hm}\). \(e_{h2m}\) represents the relative error of \(T_{h2}\) with \(T_{hm}\). \(e_{hdm}\) represents the relative error of \(T_{hd}\) with \(T_{hm}\). Similarly, \(e_{therm}\) is the relative error of \(T_{vp}\) with \(T_{vm}\). \(e_{h2m}\) is the relative error of \(T_{v2}\) with \(T_{vm}\).