In order to obtain higher recognition quality, an image preprocessing is used to suppress and eliminate noise from image. A Gaussian smoothing is a widely used to reduce image noise, which is represented by Gaussian filter [9]. Gaussian filter is a windowed filter of linear class with weighted nature, which is calculated according to Gaussian distribution. Gaussian filter in the most general function form of

G σ ( x ) = 1 2 π σ e − x 2 2 σ 2 , (8)

in one-dimensional is able to fulfill this criterion optimally, where σ is the standard deviation of the Gaussian curve. In two-dimensions the Gaussian filter is defined by

G σ ( x , y ) = 1 2 σ 2 π e − x 2 + y 2 2 σ 2 (9)

This filter can be separable by

G σ ( x , y ) = G σ ( x ) G σ ( y ) . (10)

In order to remove small details from the image and fill small gaps in lines or curves, the Gaussian filter should be used before other operations. The above mentioned Gaussian filter is used in continuous domain. In one discrete digital image, a discrete Gaussian low-pass filter should be used. The value of every pixel in the image will be replaced by the average of the intensity levels in the neighborhood defined by the filter mask. Because the smoothing process leads to “un-sharp” transition in intensities, smoothing filters have the undesirable side effect, which blur edges of object. In order to keep the balance between eliminating random noise and preserving sharpness of edges, the Gaussian filter with mask size 3 × 3

1 9 × ( 1 1 1 1 1 1 1 1 1 ) (11)

should be used in our experiment. It can be best seen by substituting the coefficient of the mask into the characteristic response R [10] with the sum of products as

R = ∑ k = 1 9 w k z k = W T Z , (12)

where W and Z are nine-dimensional vectors formed from the coefficients of a 3 × 3 filter mark and the image intensities are encompassed by the mask respectively. Then we obtain

R = 1 9 ∑ i = 1 9 z i . (13)

Figure 1 shows the results of applying a Gaussian smoothing in digital image. As expected from the Gaussian filter with mask size 3 × 3 , the detail of noise is blurred.

The purpose of edge detection is to identify points in digital image at which the image brightness changes sharply. In the past 30 years, there are many methods for edge detection, but most of them can be classified into two categories, the Template Matching (TM) and the Differential Gradient (DG) [11]. In either

case, the aim is to find where the intensity gradient magnitude g is large enough to be taken as a reliable indicator of the edge. In the DG approach, the gradient information is estimated vectorially. There are some well-known edge operators, such as Sobel edge operator, Canny edge operator, Robbers edge operator, Prewitt edge operator and so on. In practice, Sobel gradient operator is most frequently used according to its simplicity and effectiveness. The local edge magnitude may be calculated vectorially using the nonlinear transformation [11]

g = g x 2 + g y 2 , (14)

In order to save computational effort, the approximate formula

g = | g x | + | g y | , (15)

could be used in practice. Figure 2 illustrates the edge segmentation, which are obtained from input original image Figure 1(a) and smoothed image Figure 1(b). It is clear that Figure 2(b) shows fewer noisy in the segmentation.

In order to reduce the computation running time and enhance the accuracy, the contours are obtained by using software OpenCV. RCD will be executed in the connected contour to find possible circles. The computational complexity is reduced from all pixels to contours.

As illustrated in Figure 3(a), an image with all contours from Figure 2(b) is generated. Figure 3(b) shows another more complex image with a different fiducial mark. All contours extracted from Figure 3(b) are shown in Figure 3(c). The circle is divided into some broken edges. Every connected contour is drawn with its own particular color in images.

As above mentioned, we describe how to determine possible circles according to

the extracted contours. Let V denote the set of all edge contours in the image. The procedure randomly picks four pixels from each contour to make a decision whether the contour belongs to a possible circle based on the RCD algorithm. If the four pixels meet Equation (6) and Equation (7), all pixels from all contours will be collected. We continue the above process until the contour is determined as a possible circle or the global threshold T f has been reached. The procedure proceeds to next contour and the global Threshold T f is reset at the beginning.

Because the proposed algorithm samples only on the connected contours, it can overcome the interference between different objects and improve the accuracy of detection. The computational complexities are reduced and the execution-time is saved.

The above technique can detect all possible circles around the true fiducial mark. There exists a bias problem, because of noise and nonstandard circular circle. The circularity will be calculated to determine the best matching circle.

Usually the used form features can be grouped into two categories, the contour characteristic and the regional characteristic. The contour characteristic of image is mainly aimed at the boundary of image, and the regional characteristic of image is related to the whole shape region. The circular degree could be used to describe the circle. There are four circular degree measures [12], such as circularity, boundary energy, density and ratio of area to average distance quadratic sum. The characteristic of circularity is easy to be realized and more suitable for engineering application. It is needed to explain that the extraction of characteristic of circularity is based on image pre-processing and edge detection.

The characteristic of circularity is defined by all boundary pixels of the region D. An image may be defined as one two-dimensional function f ( x , y ) , and the amplitude of f at any pair of coordinates ( x , y ) is called the gray level of the image at the pixel. Suppose that the image is of size N × M elements. Set a pair of coordinates ( x k , y k ) of any pixel in the image. The barycentric coordinates are defined as

x ¯ : = ∑ i = 1 N ∑ j = 1 M i ⋅ f ( i , j ) ∑ i = 1 N ∑ j = 1 M f ( i , j ) ,                             y ¯ : = ∑ i = 1 N ∑ j = 1 M j ⋅ f ( i , j ) ∑ i = 1 N ∑ j = 1 M f ( i , j ) . (16)

The average distance from the regional barycentric to the boundary pixel is in the form of

μ D : = 1 K ∑ k = 0 K − 1 ‖ ( x k , y k ) − ( x ¯ , y ¯ ) ‖   , (17)

where ‖ ( x 1 , y 1 ) − ( x 2 , y 2 ) ‖ : = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 , and the mean square deviation of distance from the regional barycentric to the boundary pixel is in the form of

δ D : = 1 K ∑ k = 0 K − 1 [ ‖ ( x k , y k ) − ( x ¯ , y ¯ ) ‖ − μ D ] 2 (18)

can be simplified to

δ D : = 1 K ∑ k = 0 K − 1 [ ( x k − x ¯ ) 2 + ( y k − y ¯ ) 2 ] 2 − μ D 2 . (19)

The calculation formula for the characteristic of circularity is defined by

C : = δ D μ D . (20)

When the regional D tends to be a circle, the characteristic of circularity C is monotonically decreasing and tends to be infinitesimal. It’s not affected by regional translation, rotation and variation of scales. It means that we can determine the position of the circle in the image by the minimum point of characteristics of circularity in local region.

There are still some shortcomings in the standard RCD. Because the standard RCD randomly picks four edge pixels each time. The search process ended when it found the true circle or reached the number of failures T f . It means that the searching result may be quite different according to the different four starting pixels. If the staring pixels are close to the target circle, the circle may be detected rapidly. Sometimes, the start four pixels are too far from the target circle and the input image contains too many useful and useless pixels. The RCD algorithm may end without finding the true circle. In order to find possible circle rapidly, these thresholds T d , T r and T a should not be too small. Then many incorrect circles would be found.

By suppressing noise and finding of the connected contours, which contains at least a certain number of points, it will reduces the calculation time and find fewer possible circles. At last, after verifying the circularity, the real circle will be selected. So the improved RCD provides the better stability and accuracy than standard RCD. As the image grows larger and more complex, the time will become more longer. But the improved RCD will significantly reduce the time. The following experiment will be used to verify these conclusions.

In this part the marks will be detected by using standard RCD algorithm. As shown in Figure 5 there are many correct and incorrect circles in images. In Figure 5(c) the true mark circle is not included in the detected possible circles. There is no way to discard the incorrect circles and find the true mark. According to the threshold T c , there are twenty possible detected circles at most, which are drawn in Figure 5. In Tables 2-5 only ten possible circles are described.

In the following experiments, the fiducial marks will be detected by our improved RCD algorithm and corresponding circularity. As illustrated in Figure 6, the left four images (a)-(d) denote the results, which are obtained without calculating minimal circularity. Because of unsharpness of edge, the correct and bias circles are detected, which have similar radius and position. It is difficult to determine the best true circle without additional decision condition. After calculating the minimal circularity, the true marks are located in the right four images (e)-(h). The detected possible circles are shown in Tables 6-9. Compared to the former standard RCD algorithm, the number of possible circles is fewer and the results are close to the true mark. By calculating minimal circularity, in Table 6 the true circle with center pixel (71.500, 80.500) and radius 26.163 pixels is detected. In Table 7 the true circle with center pixel (71.369, 81.622) and radius 27.132 pixels is detected. In Table 8 the true circle with center pixel (75.601, 81.013) and

radius 26.951 pixels is detected. In Table 9 the true circle with center pixel (114.992, 108.471) and radius 34.629 pixels is detected.

In order to compare the execution-time, all concerned experiments are performed on the Intel i5-5200U CPU with 2.20 GHz and 8 GB RAM. The adopted operating system is MS-Windows 7 and the programming environment is VS2010. In order to accurately evaluate the execution-time, we run each image 100 times and calculate the average time in Table 10. For locating mark in small images, the execution-time is almost the same for both algorithms. The time will be significantly reduced, when the mark is detected in a larger and complex image.