# Recognizing and Recovering Ball Motion Based on Low-Frame-Rate Monocular Camera
## 3 Method
### Matching Coupled Point Pairs  
A coupled point pair consists of two points on the lateral contours and can be mapped to a sphere in space. [PERSON] et al. [12] developed a contour point matching (CPM for short) method based on their cross section reconstruction geometry. However, their method focused on the slope of the lateral contours but tolerated the error of semi-ellipse fitting on the profile. To make up for the error, they added a coupled point pair searching process, which increased the complexity of the method but still failed to eliminate the error. This paper proposes a novel method to match coupled point pairs based on the following observations. The perspective projection of a central curve (as introduced in Section 3.1) is a streak axis. When the sphere position is far enough away from the camera, the output shapes of the orthogonal projection and the perspective projection are highly similar. Thus, the streak axis is close to the central curve of the streak, which is formed by the midpoints of the contour point pairs output by the profile fitting model (as introduced in Section 3.2.1). Since the offset between these two curves can be ignored, our idea is to consider the central curve as the streak axis. A coupled point pair is the image of two points on the same cross section. In projection, they are approximately symmetric with respect to the streak axis as the axis of symmetry [19]. Meanwhile, the points in the contour point pair output by the profile fitting model are also symmetric about the streak axis. Thus, a feasible idea is to consider the contour point pairs as coupled point pairs. With the contours of the concatenated block detected in Section 3.2.2, the central curve could be initialized by fitting with a B spline curve. However, the initial central curve is fluctuating in slope while the real streak axis is a smooth curve with a slowly varying slope. The fluctuation in the initial central curve implies a large amount of error. Fortunately, we observed that such error could be eliminated using the profile fitting model proposed in Section 3.2.1: given a center point and its slope, the new center point output is closer to the actual center curve. Based on the above observations, we propose an iterative coupled point pair matching method (ICPPM for short). The idea of the method is to obtain a more accurate smooth curve by iteration and then directly output a sequence of point pairs according to the profile fitting model in Section 3.2. In terms of specific operations, our iterative method consists of three parts: (1) initializing a central curve, (2) iterating the central curve, and (3) extracting coupled point pairs. The first step is to initialize a central curve. The contour points at the ends of the block are not related to its orientation. As a result, the ends of the fitted curve are usually overfitted. Taking Figure 8 for example, the gray points are the origin contour obtained by Canny, and the red curve is the B spline fitting result of the contour. The curves in the blue boxes are overfitted and fluctuate sharply in their slope. To deal with the overfitting problem, the method sets a weight \(\beta\) for slope \(S\) to obtain the corrected slope \(V\) as follows:  
\[V_{n}=\beta S_{n}+(1-\beta)Q \tag{15}\]  
where \(Q\) is the overall slope obtained in Section 3.2.2, and weight \(\beta\) is used as a smooth factor in considering the end of the block. Then, the center curve is iterated to eliminate the error. In the iteration process, each center point is updated to a new location. The new center point's location is determined by the parameter \(c\) in parameter set \(b\) in the profile fitting model. These new center points form a new center curve, which is then fitted with the B spline. Take Figure 9b-e as examples, the green curves are the center curve, and they become smoother in iterations. After iterations, the center curve is finally two contour points can be considered as a coupled point pair.