Digital camera positioning is widely used in traffic supervision (electronic police) and other aspects. The so-called digital camera positioning refers to taking pictures of objects with digital cameras and determining the positions of some feature points on the surface of objects. The most commonly used positioning method is binocular positioning, that is, positioning with two cameras. For the feature point on the object, two cameras fixed in different positions are used to shoot the image of the object, and the coordinates of the point on the image planes of the two cameras are obtained respectively. As long as the exact relative positions of the two cameras are known, the coordinates of the feature points in the fixed camera coordinate system can be obtained by geometric method, that is, the position of the feature points can be determined. Therefore, it is the key to locate the binocular and accurately determine the relative position of the two cameras. This process is called system calibration.

One method of calibration is to draw several points on a flat plate, and take pictures with these two cameras at the same time to get the image points of these points on its image plane respectively. Using the geometric relationship between the two groups of image points, the relative position of the two cameras can be obtained. However, no matter on the object surface or the image surface, we can't get the "point" directly without the geometric dimension. The practical method is to draw several circles (called targets) on the object surface, and their centers are geometric points. However, their images are generally distorted, as shown in figure 1, so the image of the center of the circle must be accurately found from the images of these circles on the target in order to achieve calibration.

Figure 1 Image of the circle on the target

Someone designed the target like this: take 1 square with a side length of 100mm, make a circle with four vertices (corresponding to A, C, D and E) as the center and 12mm as the radius. Make a circle with B, which is 30mm away from point A on the AC side, as the center and 12mm as the radius, as shown in Figure 2.

Figure 2 Schematic diagram of the target

Take pictures with a fixed digital camera, as shown in Figure 3.

Figure 3 Target image

Please:

(1) Establish a mathematical model and algorithm, and determine that the image coordinates centered on the target are on the image plane of the camera, where the origin of the coordinate system is on the focal point of the camera and the x-y plane is parallel to the image plane;

(2) For the target and its images given in Figure 2 and Figure 3 respectively, calculate the image coordinates with the center of the target on the image plane. The image distance of the camera (that is, the distance from the focus to the image plane) is 1577 pixel unit (1mm is about 3.78 pixel unit), and the camera resolution is1024× 786.

(3) Design a method to test your model, and discuss the accuracy and stability of this method;

(4) The mathematical model and method are established, and the relative positions of two fixed cameras and the target are given.