## 3 Image-Servo Control and Pattern Recognition of the Positioning Mark
### Image Processing and Pattern Recognition
Therefore, some other problems should be solved; for example, the first problem of the halo effect results from the upper and lower masks being located in different depths of field as shown in Figure 10a. The second problem is that the lower mark may be covered by the upper mark in some cases as shown in Figure 10b. Therefore, solutions for these two problems are proposed as follows. To achieve the precision alignment task, it is important to determine the position of each fiducial marker with high accuracy. Detection of objects can be performed by using pattern recognition techniques such as neural networks [18; 19], linear filters [20], support vector machines [21], and the Hough Transform [22]. As the positions of the objects need  
Figure 8: Correlation between \(w(x,y)\) and \(f(x,y)\). Figure 9: Pattern recognition steps: (**a**) Original image; (**b**) Pattern image; (**c**) Binarization of original; (**d**) searched pattern. to be determined with sub-pixel precision, an accurate estimate can be obtained by computing its center of gravity [17]. However, for the case of Figure 11b, there are two recognized marks and one of the patterns is an incomplete circle; the COG method could cause a large positioning error for the incomplete circular mark. In that case, the generalized Hough Transform (GHT) is studied to find the dual positioning marks in this paper. Hough Transform (HT) is proposed by [PERSON] in 1972 and this method can be used to detect the line, circle and arbitrary shape [22; 23]. To find the circle using circle HT, the first step is to make all edge points connected together to form a closed border; the second step is transferring the coordinates \((x,y)\) of all points son the border to the space of parameter \((a,b,r)\); the final step is to obtain the intersection of all cone which is transferred according to all points on the border. For example, consider a circle equation as follows. \[(x-a)^{2}+(y-b)^{2}=r^{2} \tag{7}\]  
where \((a,b,r)\) is the vector of the center of circle with its radius. The circle HT is to represent Equation (7) to the parameter space as follows. \[H(x,y,a,b,r)=(x-a)^{2}+(y-b)^{2}-r^{2}=0 \tag{8}\]  
To find the intersection point in this parameter space \(H(x,y,a,b,r)\), an accumulator matrix is needed and the parameter space is divided into \"buckets\" using a grid. Initially, all elements in the matrix are zeros. Then, each edge point in the original space \((x,y)\) is substituted into the parameter space to obtain the corresponding parameter \((a,b,r)\). The accumulator matrix is used to count the number of \"circles\" in that passing through the corresponding grid cell in the parameter space and the number is called \"voting number\". After voting, the position of the local maxima in the accumulator matrix represents the circle centers in the original space. To illustrate how to use the circle HT, assume that three points \((x_{1},y_{1})\), \((x_{2},y_{2})\), \((x_{3},y_{3})\) are located at a circle with its radius of \(r^{\prime}\). The solution of \((a,b)\) for these three points can be described by three cones. Therefore, the intersection of these three cones is the center of the circle \((a^{\prime},b^{\prime})\) as shown in Figure 11.  
Figure 11: Cones of the three edge points in the parameter space. Figure 10: **(a)** The halo effect of dual marks; **(b)** the partial lower mark in the upper mark.