1. bookAHEAD OF PRINT
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Accès libre

Lagrange’s Mathematical Equations in the Sports Training of College Students

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 19 Jan 2022
Accepté: 14 Mar 2022
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

In the past research, the restoration, repair, and storage of video files were mainly used to protect related material cultural heritage. As the country grows stronger, the frequency of various sports events is increasing year by year. The development of network and computer technology makes the storage of various sports video files extremely convenient and easy. In this case, the stored video files can not only be used for cultural asset preservation but also can be used for repeated viewing and learning after the game [1]. At present, related researchers in the field of image science and computer have provided various methods and tools to effectively save various sports video files.

Despite the exponential growth of sports video files, high-speed access and access to video files are still facing many problems, so relevant technical development and high-level innovation are required. This mainly includes the following aspects: 1) Support for encoding video files and creating catalogs. Some important sports materials require special attention. 2) Support the recovery of sports video content through digital and highly reliable storage systems. 3) Support user interactive operation. Track the usage of sports video files by querying related logs. 4) Support the application of image processing technology on digital video for video recognition operations. In this way, relevant features and content structure can be extracted for classification. 5) Support the design and implementation of the man-machine interface to annotate video content and integrate relevant information.

Among these goals, the identification and classification of sports video files are particularly important. To achieve this goal, most of the existing work is done manually. This is not only costly but also the efficiency of classification is not high. For example, people’s access to the sports video files of the 2016 Olympic Games is generally intuitively searched and browsed through web pages. In this process, it is assumed that these video files are not classified or indexed, and there are tens of thousands of video file information [2]. Then it will be a huge and time-consuming task to find suitable videos to watch from these video files. Therefore, it is necessary to find new methods to identify these video files to perform features classification, metadata distribution and extraction, indexing, and other operations on them. This article attempts to combine these two technologies to solve the problem of categorizing Olympic sports videos. First, the image processing algorithm is used to calculate the feature vector that can represent the keyframe in the sports video. We use Lagrange-Gaussian (LG) function to decompose the video image and then analyze the keyframes. Secondly, machine learning technology is used to classify sports videos according to their keyframe feature matrix. It is relatively easy to use support vector machine (SVM) machine learning technology to classify video files. This paper uses the LG function and SVM technology to solve the video classification problem for the first time. This method is highly innovative. Compared with some existing classification methods, the algorithm proposed in this paper has higher classification efficiency and accuracy.

Feature vector extraction of Olympic video files

Our first task is to analyze the video file and process the keyframes in the video image, calculate and generate feature vectors to represent the content in the video. We organize these feature vectors and can be used to identify keyframes and the category to which the video belongs [3]. This article uses the LG function to decompose the image in the video and analyze the decomposed keyframe structure. According to the mathematical specification, the LG function is defined as follows: l(n)k(r,φ)=(1)k2| n |+12π| n |2[ k!(| n |+k) ]12r| n |Lk| n |(2πr2)eπr2ejnφ

Lk(n) is a generalized Lagrangian polynomial of order k and n defined by Rodriguez’s formula. Its definition is as follows: Lk(n)=i=Qk(1)i(kin+k)xii!

If n=1, then according to formulas (1) and (2), an operator can be obtained to extract the edge features of the image. These edge features include image intensity and direction. To detect the edges of complex images at different resolutions, it is suitable to use the first-order circular harmonic wavelet (CHW) function to represent the coefficients required for detection at different resolutions. Nevertheless, the circular harmonic functions of different orders n can be used to analyze the image and decompose the keyframes in the complex image. In this way, it can be concluded that there are different two-dimensional models in the image. Such as edge model (n=1), straight-line model (n=2), vertex model (n=3), orthogonal intersection model (n=4) and so on. The decomposition of the Lagrangian-Gaussian decomposition function has a certain degree of maneuverability. Assuming n = 1, then the structure rotated by an angle around a given center is equal to the product of the circular harmonic wavelet coefficient and the complex factor. This maneuverability makes the Lagrangian-Gaussian function play an important role in analyzing image features from many aspects.

Based on the above analysis, we began to extract features from the keyframes of the Olympic sports video files. Since the audio information in the video file is too noisy and is of little value in nature, this article only focuses on the visual information presented in the video file. We need to extract and analyze the keyframes to determine the category of the sports video file. We analyze the features such as brightness, edge, and texture of keyframes to extract multiple feature vectors [4]. We use it as the input vector of the video classifier to classify Olympic sports videos. Next, the method of extracting each feature vector will be introduced.

Video keyframe brightness characteristics

The brightness feature is an important attribute that describes the keyframes in the video file, and the distribution graph of the brightness feature can be embodied as the most widely used low-level feature. On the one hand, according to probability theory, we believe that the probability distribution function of the brightness feature can be uniquely represented by using the corresponding torque. On the other hand, it is usually necessary to reflect the important part of all the information in the first three brightness characteristic moments. With this in mind, we can use a set of numbers related to its torque to characterize the brightness distribution of the keyframes in the video file. This includes the mean, standard deviation, and skewness. It is almost impossible to use only the above three parameters to describe the brightness difference between different parts of a natural image. To solve this problem, this paper divides the extracted video keyframes into smaller non-overlapping image blocks. We then use the average, standard deviation, and skewness to represent the brightness information of each block [5]. If n is used to represent the number of image blocks, then the luminance component information of each keyframe can be represented by a vector with 3n elements.

Video keyframe edge features

For sports video files, the low-order shape features of keyframes are the key information to distinguish such files. Generally speaking, we can express the image by defining the main edges of any keyframe. In addition, video edge features are of great significance for solving sports video classification tasks. Mainly because the lines on the court and the players have significant edge characteristics. According to the MPEG standard, this article mainly uses the edge feature to construct the direction histogram of the image to represent the shape information corresponding to the keyframe [6]. The edge direction histogram is obtained by calculating the phase of the main edge points and will not change. In addition, to detect the main edges of the video keyframes, the expansion coefficients of the Lagrangian Gaussian function with n=1 and k=0 at high resolution are calculated respectively. We use the Laplacian zero-crossing map of the original keyframe image to cover the Lagrangian Gaussian image plane. The Laplacian zero-crossing map is obtained by using the Laplacian operator to process the keyframes and selecting the zero-crossing points in the image. In addition, the phase of each edge point is estimated by calculating the sparse phase of the LG function. Finally, we quantify the abscissa range of [0,π] into 36 scales. Represent the edge information in the keyframe by drawing the edge direction histogram.

Video keyframe texture features

In the case of sports video editing, the texture feature of the video quality is relatively reliable. In fact, in this case, the textures of different venues in the Olympic Games are very useful for classifying keyframes. In this work, we analyze the texture features of sports videos by using LG functions determined by the order (n = 1, k = 0) and (n = 3, k = 0) respect. These two order cases are calculated in three different resolution modes. In addition, because the LG function expansion is more complicated, the generalized Gaussian density function is used to describe the real and imaginary parts of the edge density respectively. Generally speaking, any generalized Gaussian density function can be represented by two parameters. The first parameter is directly related to the width of the key frame texture peak, and the other parameter is related to the rate of decrease of the peak. Therefore, this paper uses a vector composed of 24 elements to represent the keyframe texture information. In addition, we also use 12 Gaussian density parameters to represent the edge density of the Lagrangian-Gaussian coefficient.

Classification method of Olympic video files

This paper uses support vector machine SVM to classify the keyframes in sports video files. At present, two types of support vector machines are used, and the appropriate type is selected for training and distinguishing keyframes. In this case, we can choose two different strategies: 1) one-to-many; 2) one-to-one.

Given a training set S = UiSi. Where i represents a category i. It is a collection of keyframes belonging to category i. The training of each category of SVM of the first strategy depends on data set si and another data set ni = {sj | ji}. The elements in ni are composed of elements other than si. It is worth noting that the number of elements in si should be as close as possible to the number of elements in ni. After a given data set, we need to train the SVM model according to each feature of the video key frame. Similarly, the second strategy also needs to train the model based on features. In this case, the sets used to train SVM are si and ni. Where ni = sj · ∀ji. For these two strategies, the number of sets we choose to train is C-1. Where C represents the number of categories, such as the athletes in the video as the classification basis. Through the above method, then each video category can be represented by SVMt,ki . Among them, i represents a generic category, t represents other categories, and k represents a generic feature. By using the training data set, the system automatically searches for the parameters of the support vector machine in the process of training the SVM model, and we obtain the best results based on the true positives and false positives of these parameters [7]. There are two kinds of kernels of support vector machine used in this paper, which are polynomial function kernel and Gaussian radial basis function kernel. The two kernel parameters used in this article are shown in Table 1. The automatic selection process of SVM parameters depends on the current cost function cf (x), which is defined as follows: cf(x)=a1tp(x)tpMAX+a2fp(x)tpMAX

SVM parameter set under different kernels

Kernel Kernel Palmers
Polynomial kernel 1 4 7 11 15 19
Gaussian radial basis kernel 1 3 7 15 30 60

Among them, a1 = 1, a2 = −1,tp and fp respectively represent the number of true positives and false positives in order to determine the SVM parameters. tpMAX and fpMAX respectively represent the maximum value under true and false positive conditions.

We substitute each parameter given in Table 1 into the cost function for verification and evaluation. The parameter that can maximize cf (x) will be selected. This article uses three characteristics, and the content includes six data parameters. In the case of the two kernels, a total of ninety SVM sets need to be trained. The statistics for the two cores are shown in Figures 1 and 2.

Figure 1

SVM statistical results based on polynomial kernel

Figure 2

SVM statistical results based on Gaussian radial basis function kernel

Finally, based on the above analysis, we found that the general key frame classification process in the Olympic sports video test set obeys the following two points:

For each category i, change the value of t and k, we need to calculate the number of times the extracted key frame classification belongs to that category, so that we get a vector composed of C elements.

For the generalized key frames extracted from the Olympic video, we can classify them according to the following formula.

i={ i'|r(i')=max[ r ] }
Simulation experiment and result analysis

The data set used for experimental testing and evaluation in this article comes from the composition of video clips taken during the 2016 Olympic Games (the main video data set comes from network downloads). The data set mainly includes 6 categories, which are track and field, basketball, badminton, swimming, weightlifting and shooting. The entire data set includes approximately 25 hours. We need to further divide it into the following three subsets: The training set is mainly used to train the video classifier [8]. The content includes about 9 hours, including 2,000 key frames.

The verification set is mainly used to verify and optimize the parameter settings of the classifier. The content includes about 6 hours, including 1,400 key frames; The test set is mainly used to evaluate the performance of the classifier. The content includes about 10 hours, including 2200 key frames.

The comparison result of the experiment based on SVM-based sports video classifier is shown in Figure 3. On the one hand, Figure 3 shows the true and false positive values obtained for different categories. The content is the proportion of the video file classified into the correct category or the wrong category. On the whole, the Olympic video classification algorithm based on SVM can achieve good results. Because given any sports category, its corresponding true positive value is much larger than the false positive value. In addition, it can be observed that the proportion of true positives is always higher than 70%, and the proportion of false positives is always lower than 10%. This shows that the proposed algorithm has higher efficiency. On the other hand, in Figure 3, the algorithm proposed in this paper is compared with the k-neighbor algorithm [9]. On the one hand, it can be seen that the proportion of classification errors of the two algorithms is close, but the algorithm proposed in this paper is relatively lower. On the other hand, the proportion of correct video classification by the algorithm proposed in this paper is significantly higher than that of the k-neighbor algorithm. When the video files are the same and the proportion of classification errors is close, the higher proportion of correct video classification shows that the algorithm proposed in this paper has strong key frame recognition ability. This shows that the proposed algorithm is better than the comparison algorithm.

Figure 3

Olympic Games based on SVM and k-neighbor algorithm

Conclusion

This paper proposes for the first time a combination of Lagrangian-Gaussian function and support vector machine technology to solve the classification problem faced by a large number of Olympic video files. First extract the key frames in the video file. By using feature vectors to describe the corresponding key frames, it supports the construction of feature vector sets as the input parameters of the SVM classifier. Then construct a cost function for the support vector machine to realize the automatic selection of parameters. Finally, two schemes of true positive and false positive are used to evaluate the proposed algorithm. The results show that compared with the classic k-neighbor algorithm, the video classification algorithm based on Lagrangian function and SVM technology can achieve better results.

Figure 1

SVM statistical results based on polynomial kernel
SVM statistical results based on polynomial kernel

Figure 2

SVM statistical results based on Gaussian radial basis function kernel
SVM statistical results based on Gaussian radial basis function kernel

Figure 3

Olympic Games based on SVM and k-neighbor algorithm
Olympic Games based on SVM and k-neighbor algorithm

SVM parameter set under different kernels

Kernel Kernel Palmers
Polynomial kernel 1 4 7 11 15 19
Gaussian radial basis kernel 1 3 7 15 30 60

Hu, H. X., Wen, G., Yu, W., Cao, J., & Huang, T. Finite-time coordination behavior of multiple Euler–Lagrange systems in cooperation-competition networks. IEEE transactions on cybernetics., 2019; 49(8): 2967-2979 Hu H. X. Wen G. Yu W. Cao J. Huang T. Finite-time coordination behavior of multiple Euler–Lagrange systems in cooperation-competition networks IEEE transactions on cybernetics. 2019 49 8 2967 2979 10.1109/TCYB.2018.283614030762574 Search in Google Scholar

Nikolaidis, P., & Poullikkas, A. Enhanced Lagrange relaxation for the optimal unit commitment of identical generating units. IET Generation, Transmission & Distribution., 2020; 14(18): 3920-3928 Nikolaidis P. Poullikkas A. Enhanced Lagrange relaxation for the optimal unit commitment of identical generating units IET Generation, Transmission & Distribution. 2020 14 18 3920 3928 10.1049/iet-gtd.2020.0410 Search in Google Scholar

Sabermahani, S., Ordokhani, Y., & Yousefi, S. A. Fractional-order general Lagrange scaling functions and their applications. BIT Numerical Mathematics., 2020; 60(1): 101-128 Sabermahani S. Ordokhani Y. Yousefi S. A. Fractional-order general Lagrange scaling functions and their applications BIT Numerical Mathematics. 2020 60 1 101 128 10.1007/s10543-019-00769-0 Search in Google Scholar

Sun, Y., Chen, L., & Qin, H. Distributed chattering-free containment control for multiple Euler–Lagrange systems. Journal of the Franklin Institute)., 2019; 356(12): 6478-6501 Sun Y. Chen L. Qin H. Distributed chattering-free containment control for multiple Euler–Lagrange systems Journal of the Franklin Institute). 2019 356 12 6478 6501 10.1016/j.jfranklin.2019.05.032 Search in Google Scholar

Sheng, Y., Lewis, F. L., Zeng, Z., & Huang, T. Lagrange stability and finite-time stabilization of fuzzy memristive neural networks with hybrid time-varying delays. IEEE transactions on cybernetics., 2019; 50(7): 2959-2970 Sheng Y. Lewis F. L. Zeng Z. Huang T. Lagrange stability and finite-time stabilization of fuzzy memristive neural networks with hybrid time-varying delays IEEE transactions on cybernetics. 2019 50 7 2959 2970 10.1109/TCYB.2019.291289031059467 Search in Google Scholar

Hu, X., Li, J. & Aram, Research on style control in planning and designing small towns. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 57-64 Hu X. Li J. Aram Research on style control in planning and designing small towns Applied Mathematics and Nonlinear Sciences. 2021 6 1 57 64 10.2478/amns.2020.2.00077 Search in Google Scholar

Evirgen, F., Uçar, S. & Özdemir, N. System Analysis of HIV Infection Model with CD4+T under Non-Singular Kernel Derivative. Applied Mathematics and Nonlinear Sciences., 2020; 5(1): 139-146 Evirgen F. Uçar S. Özdemir N. System Analysis of HIV Infection Model with CD4+T under Non-Singular Kernel Derivative Applied Mathematics and Nonlinear Sciences. 2020 5 1 139 146 10.2478/amns.2020.1.00013 Search in Google Scholar

Ge, M. F., Liu, Z. W., Wen, G., Yu, X., & Huang, T. Hierarchical controller-estimator for coordination of networked Euler–Lagrange systems. IEEE transactions on cybernetics., 2019; 50(6): 2450-2461 Ge M. F. Liu Z. W. Wen G. Yu X. Huang T. Hierarchical controller-estimator for coordination of networked Euler–Lagrange systems IEEE transactions on cybernetics. 2019 50 6 2450 2461 10.1109/TCYB.2019.291486131150351 Search in Google Scholar

Zwick, D., & Balachandar, S. A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements. The International Journal of High Performance Computing Applications., 2020; 34(3): 316-339 Zwick D. Balachandar S. A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements The International Journal of High Performance Computing Applications. 2020 34 3 316 339 10.1177/1094342019867756 Search in Google Scholar

Articles recommandés par Trend MD

Planifiez votre conférence à distance avec Sciendo