With the continuous improvement of Chinese economic level, the sports industry is booming. The development trend of modern football has gradually shown the integration of global football. How we use corner kicks effectively is also on the agenda. The application of corner kick tactics has an important influence on football players’ corner kick goals. Among them, tactical coordination methods are widely used by football players in corner kicks [1]. This makes it difficult to accurately judge the position information of the ball during the corner kick. In this case, how to effectively extract the trajectory of the corner kick has become a fundamental problem to be solved urgently in the modern sports field at this stage. The extraction method of corner kick video image trajectory based on Kalman filter theory can combine Kalman filter extraction and linear difference theory [2]. This method is the fundamental way to solve the above problems. Therefore, relevant experts and scholars have highly valued this model in this field. We propose a Kalman filter-based method for extracting trajectories from video images of football players’ corner kicks. The experimental results show that the proposed method can better extract the movement trajectories of football players in the process of corner kicks. The extraction efficiency of our proposed method is high.

We first give the recursive form of the sliding window kernel ridge regression theory under the condition of Gaussian kernel function [3]. Then we extract the football trajectory during the football player's corner kick in the form of a sliding window. We take each frame's football trajectory in the neighboring frames as input. In this paper, the position of the next frame of the football trajectory is extracted based on the kernel-ridge regression theory. From this, we complete the extraction of the video image trajectory of the football player's corner kick. The specific process is as follows:

Dynamic structural equations describe the trajectory of a football player's corner kick shot. The current position of the football is related to the work of the football's nearest

_{t}_{t}

In the high-dimensional feature space _{t}_{−N+1},…, _{t}_{−1}, _{t}^{T}. Assuming

The football trajectory is under the constraints of
_{t}_{−N+1},…, _{t}_{−1}, _{t}^{T}. We use equation (5) to express the final cost function

Based on the estimated

In

In extracting the trajectory of the football player's corner kick video image, we first use the particle filter as the trajectory tracking framework of the football player's corner kick. This paper uses visual features to obtain ball candidates for each frame of images during a football player's corner kick. In this paper, the particle filter theory is used to extract and track the football player's corner kick trajectory to form the initial course [5]. The specific process is as follows:

_{i}^{2} → {1, 2, …, _{i}

Where _{i}

We have pixel _{x}_{y}

Then the modulus value

We can obtain the

_{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}_{−1}) during the football player's corner kick. Initialize the probability density function _{0} | _{0}) = _{0}) as known. Then the football target probability density function _{k}_{1k}

We take the initial trajectory of the football player's corner kick as the basis for extracting the video image trajectory of the football player's corner kick. We select and confirm the accurate ball trajectory of the football player's corner kick by selecting the football player's corner kick trajectory [8]. On this basis, we combine Kalman filter extraction and linear difference theory to fill in the ball's position that is missed in each frame of the football player's corner kick. From this, we complete the extraction of the video image trajectory of the football player's corner kick. The tooling process is as follows:

_{f}_{f}_{f}

_{i}_{u}_{v}

In the formula

We base on the set of corner kick soccer trajectories obtained by Equation (16). This paper combines Kalman filter and linear interpolation theory to fill the missed frames between two corner kick soccer circuits [9]. Assume that there are two soccer goal trajectories _{u}_{v}_{max,u} < _{min,v}. We first combine the Kalman filter to obtain the extracted values of the football target track _{u}_{v}_{max,u} < _{min,v}]. We use
_{max,u} < _{min,v}]. Then we find the two points in the extraction interval of the football player's corner kick when the distance between the two football trajectories is the closest. They correspond to the _{u}_{v}

Where:

We can obtain the values of _{v}_{u}_{v}

When

We can accurately fill the missed ball positions between a football player's corner kick trajectories. This constitutes a complete football player's corner kick trajectory.

The following experiments demonstrate the effectiveness of the method proposed in this paper for extracting the trajectory of a football player's corner kick video image based on Kalman filtering. In the Madab7.0 environment, we build an experimental platform to remove the video image trajectory of a football player's corner kick. The experimental data are taken from the Premier League match between Liverpool and Watford in the 2019–2020 season [10]. The resolution is 720×404. We use the Kalman filter method and the least-squares method to extract the trajectory of the video image of the football player's corner kick. Due to the influence of noise, we conducted 30 experiments on the two sets of simulated trajectories, respectively. We compare the average mean square error (%), and root means square error (%) of the two methods for extracting the video image trajectory of the football player's corner kick. The average mean square error and root mean square error are calculated as

Where

Statistical results of video image trajectory extraction error statistics by Kalman filtering method

Number of experiments/time | Mean Square Error/% | RMSE/% |
---|---|---|

5 | 3.21 | 4.21 |

10 | 3.09 | 3.79 |

15 | 2.81 | 2.44 |

20 | 2.44 | 2.21 |

25 | 1.94 | 2.1 |

30 | 1.01 | 0.94 |

Statistical results of video image trajectory extraction error statistics by least-squares method

Number of experiments/time | Mean Square Error/% | RMSE/% |
---|---|---|

5 | 4.21 | 48.21 |

10 | 6.32 | 38.12 |

15 | 4.12 | 24.44 |

20 | 4.98 | 20.34 |

25 | 4.21 | 19.44 |

30 | 3.98 | 17.64 |

The extraction errors of the Kalman filter method are all smaller than the error of the least-squares method for extracting the video image trajectory of the football player's corner kick [12]. This is mainly because the Kalman filter method confirms the accurate ball trajectory of the football player's corner kick by selecting the football player's corner kick trajectory. On this basis, the Kalman filter extraction and linear difference theory are combined to fill in the ball's position that is missed in each frame of the football player's corner kick. Thus, the video image trajectory of the football player's corner kick is extracted. This makes the extraction error of the video image trajectory of the football player's corner kick shot by the Kalman filter method smaller.

We use the Kalman filter method and the least-squares method to extract the trajectory of the football player's corner kick [13]. We compare the two strategies for the extraction efficiency (%) of the video image trajectory of the football player's corner kick. The comparison results are shown in FIG. 1.

The extraction efficiency of the video image trajectory of a football player's corner kick using the least-squares method is lower than that of the Kalman filter method. This is mainly because the Kalman filter method first uses visual features to obtain the candidate balls of each image frame during the football player's corner kick. On this basis, the Kalman filter extraction and linear difference theory are combined to fill in the ball's position that is missed in each frame of the football player's corner kick. In this way, the video image trajectory of the football player's corner kick is extracted. In this way, the Kalman filtering method has high efficiency in removing the rotation of the video image of the football player's corner kick.

When the current method extracts the trajectory of the video image of the corner kick of the football player, there is a problem of a significant error in the extraction of the video image trajectory of the corner kick. This paper proposes a Kalman filter-based method for extracting the soccer player's corner kick video image trajectory. The experimental results show that the proposed method can better extract the football player's corner kick trajectory and the extraction efficiency is high.

#### Statistical results of video image trajectory extraction error statistics by least-squares method

Number of experiments/time | Mean Square Error/% | RMSE/% |
---|---|---|

5 | 4.21 | 48.21 |

10 | 6.32 | 38.12 |

15 | 4.12 | 24.44 |

20 | 4.98 | 20.34 |

25 | 4.21 | 19.44 |

30 | 3.98 | 17.64 |

#### Statistical results of video image trajectory extraction error statistics by Kalman filtering method

Number of experiments/time | Mean Square Error/% | RMSE/% |
---|---|---|

5 | 3.21 | 4.21 |

10 | 3.09 | 3.79 |

15 | 2.81 | 2.44 |

20 | 2.44 | 2.21 |

25 | 1.94 | 2.1 |

30 | 1.01 | 0.94 |

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