# Trajectory Tracking Method of Volleyball Player’s Arm Hitting Image Based on D-P Algorithm

Authors: Hao Qin
Journal: Scientific Programming
 (2021)

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## Abstract

Aiming at the problems of poor image tracking effect, low precision, and long time in the process of image tracking of volleyball player’s arm hitting, a volleyball player's arm hitting image tracking method based on D-P algorithm is proposed. This paper analyzes the basic concept, basic principle, and basic equation of D-P algorithm and collects the arm stroke trajectory image of volleyball players under the three-dimensional visual model. Using wavelet multiscale decomposition method, the arm stroke trajectory of volleyball players is filtered, and the edge contour feature points of the arm stroke image of volleyball players are extracted. Using the gray histogram feature extraction method, the gray information of volleyball player's arm hitting trajectory image is enhanced. Combined with pixel adaptive enhancement technology, the key action feature points of volleyball player's arm hitting image trajectory are located. Based on D-P algorithm, the volleyball player's arm hitting image trajectory is adjusted and modified to realize the correct tracking of volleyball player's arm hitting image trajectory. The experimental results show that the trajectory tracking effect of volleyball player's arm hitting image is better, which can effectively improve the tracking accuracy and shorten the tracking time.


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## Body

## 1. Introduction

At present, with the rapid development of computer technology, image processing technology, and image recognition technology, human motion trajectory recognition technology has been widely used in different industries such as entertainment and user interface, and its application in the field of sports training is becoming more and more extensive [1–3]. Applying image processing technology to sports training is the key to improving the correction ability of sports movements and the effect of sports training [4]. In volleyball competition, whether the athlete's arm movement is scientific and reasonable determines the quality of spiking. The volleyball hitting movement is complex and the technical difficulty is high. It is necessary to carry out real-time analysis and normative correction of the technical movement to improve the ability of sports planning [5]. In order to master the spiking technique better, it is necessary to track the arm trajectory of volleyball players. Volleyball arm stroke is the key to scoring. Tracking the image trajectory of volleyball arm stroke, combined with image processing technology for trajectory analysis, is of great significance to improve the accuracy of volleyball arm stroke [6, 7]. In this case, how to effectively recognize the arm trajectory of volleyball players has become the main problem to be solved in this field.At present, scholars in related fields have made some progress in motion trajectory tracking. Piao and Kim [8] proposed a trajectory tracking method based on a backward semi-Lagrangian method. The backward semi-Lagrangian method is used to track the trajectory required to solve the guidance center model. The discrete system numerical solution of Cauchy problem is designed, and the required trigger point is found by interpolation method. Based on the estimated value of the improved physical quantity, the interpolation solution is calculated at the starting point to solve the trajectory tracking required by the guidance center model. This method has certain reliability. Zhang et al. [9] proposed a semiglobal finite time trajectory tracking method for disturbed nonlinear systems based on high-order sliding mode. Through the high-order sliding mode integral finite time disturbance feedforward decoupling process, a nonrecursive design framework is proposed to simplify the gain adjustment mechanism. An inherently nonsmooth control law is constructed from the system information. By proposing a semiglobal tracking control target, the trajectory tracking without restrictive nonlinear growth constraints is realized. This method is simple and effective. However, the above methods have the problems of poor trajectory tracking effect, low accuracy, and long time.Aiming at the above problems, a trajectory tracking method of volleyball player's arm hitting image based on D-P algorithm is proposed. The edge contour feature points of the trajectory are extracted by filtering the arm stroke trajectory image of volleyball players. The gray information of the track image is enhanced to locate the key action feature points of the track. Based on D-P algorithm, the trajectory of volleyball player's arm hitting image is adjusted and corrected to realize correct trajectory tracking. The tracking effect of this method is good, which can effectively improve the tracking accuracy and shorten the tracking time.


## 2. Basic Theory of D-P Algorithm

### 2.1. Basic Concepts of D-P Algorithm

Dynamic programming (D-P) algorithm is an optimization method applied to solve multistage decision-making problems [10–12]. At present, D-P algorithm has been widely used in resource theory, chemical engineering, variational method, economics, and optimal control theory. It is used to solve some optimal problems. This kind of problem refers to such a kind of activity, which can be divided into several interrelated stages, and decisions need to be made in each stage. This decision requires that the whole process achieve the best activity effect [13, 14]. Here, the selection of decision-making in each stage is not determined arbitrarily. Its determination depends on the current state, and it affects its subsequent state. Assuming that it is divided into 10 stages, when the decisions in these 10 stages are determined, these decisions form a decision sequence. An activity route of the whole process can be determined through the decision sequence, which solves this problem. The multistage decision-making problem is shown in Figure 1.Figure 1
Multistage decision-making problem.As can be seen from Figure1, these interrelated stages can facilitate the solution of the problem. In the face of different processes, the problem can be divided into different stages, which is not invariable. Usually a variable is used to describe the stage. This variable is called the stage variable. In most cases, this variable is discrete, denoted by k. The first stage is from point A to point B, the second stage is from point B to point C, the third stage is from point C to point D, and the fourth stage is from point D to point E. Point A in Figure 1 is usually called a state, and the variables used to describe each state are called state variables [15]. This state variable does not have to be a number; it can also be a set of numbers, and even a vector can be used to describe the state. In order to facilitate the understanding of the state variable in the k stage when solving the problem, it is often represented by xk. One state of the first stage is represented by a, and three states of the second stage are represented by B1, B2, and B3. Similarly, C1, C2, C3, and C4 represent the state of stage 4, respectively, and D1, D2, and e represent the state of stages 4 and 5, respectively. When the state of a certain stage is known, it is necessary to choose from this state of this stage to the state of the next stage, and this choice is a decision. For ease of understanding, ukxk is usually used to represent the decision variable, where xk represents the k state of the x stage. The same as state variables, a number, a group of numbers, or even a vector can also be used as decision variables to describe decision-making. The strategy is for decision-making. The strategy can be said to be a group of decision sequences at various stages. In most problems, the transition from one state to another state is not random. It follows a certain criterion, which is called the transfer equation [16]. According to the state transition equation, when the state variable xk of the k stage and its decision variable uk are known, state xk+1 of the k+1 stage can be obtained through calculation.


### 2.2. Basic Principle of  D-P Algorithm

The basic idea of D-P algorithm is to find the point on the curve with the maximum distance from the connecting line between the points at both ends of the curve according to a certain tracing route and judge whether the maximum distance is greater than the given threshold. The principle of D-P algorithm is shown in Figure2.Figure 2
D-P algorithm principle.AB is the given curve. In order to find the inflection point on the curve, traverse the curve AB froma to B, calculate the distance from the point on the curve to the connecting line of AB, and find out the maximum value of the distance, which is obviously point C1. If the distance is greater than the set threshold, the curve is divided into two sections with C1 as the boundary, that is, curves AC1 and C1B. In these two sections of curves, the inflection points are found according to the same method as above. When setting the appropriate maximum distance threshold, two inflection points, that is, C1 and C2, can be found. It can be seen from the search process that the determined inflection points do not appear in sequence according to the contour curve. Because the feature points do not appear in sequence according to the contour curve, after obtaining the feature points, in order to fit the curve, the feature points should be stored in the form of binary tree, and each feature point should be obtained in the first order when storing and in the middle order when using.(1)
Optimization principle: the optimal solution of a multilevel decision problem must include the optimal solution of its subproblems. Decision-making is the choice made at each stage, and multilevel selection constitutes a decision strategy [17]. In the shortest path problem in Figure 1, the shortest path from A to E and its subpath from the intermediate node to the end point e must also be the shortest path from the current intermediate node to the end point E. In this problem, decision-making is the selection of the next location at each point from point A. The current location is called the state. The path from the selection made in each state to the final point is a decision strategy.(2)
State transition principle: D-P algorithm cannot be directly applied to the target state, so the premise of realizing D-P algorithm is to discretize the target state space and change the continuous state into discrete state [18–20]. At this time, the target space will be divided into units of size Δ×Δ. Similarly, the speed space of the target will also be divided into units of Δx×Δy, where Δx=Δy=Δv defines 1 resolution unit/frame. In order to distinguish it from the continuous state x,x˙,y,y˙, the discrete state is represented as x¯,x˙¯,y¯,y˙¯. For state x¯,x˙¯,y¯,y˙¯ of the k frame, the relationship between the k+1 frame and the k frame is expressed as(1)xk+1∈x˙¯+x¯−2Δ,x˙¯+x¯Δ,x˙k+1∈x˙¯−1Δv,x˙¯Δv,yk+1∈y˙¯+y¯−2Δ,y˙¯+y¯Δ,y˙k+1∈y˙¯−1Δv,y˙¯Δv.When the target is maneuvering, the number of state transitionsq will be severely affected. Increased target mobility will lead to an increase in state transitions q, because only by increasing the number of state transitions can the expansion of the target state brought about by changes in target speed be satisfied. Assuming that each state at time k is known as x¯,x˙¯,y¯,y˙¯, there are q possible valid states xk+1 corresponding to it at time k+1.(3)
Reverse order recurrence principle: reverse order recurrence is the specific implementation method of D-P algorithm to solve the optimal strategy; that is, it deduces forward from the final state and records the feasible decisions and corresponding costs in the previous stage for the next decision to evaluate the overall decision cost [21]. In this way, the final feasible decision set is obtained by recursion in reverse order until the initial state, and the optimal strategy is selected according to the recorded cost value, and then the whole decision order is traced back. In the shortest path problem, the overall cost is the path length, and the decision cost is the distance weight between two nodes.The idea of using D-P algorithm to solve multistage decision-making problem is to regard a shortest path of the final required solution as an optimal strategy of multilevel decision-making [22, 23]. The D-P algorithm divides the multilevel decision-making problem into a combination of multiple subproblems and then obtains the optimal strategy through the optimality principle and inverse recurrence principle of the D-P algorithm. The D-P algorithm has the following advantages:(1)
It can reduce the amount of calculation: Using D-P algorithm to solve this problem requires 31 addition times and 19 comparison times, which greatly reduces the amount of calculation. When the number of segments is large and the selection of multiple segments is large, the advantage of D-P algorithm is more prominent; and D-P algorithm is convenient to use mainframe computer to calculate the iterative method.(2)
The calculation results are rich: Using D-P algorithm for calculation, we can not only get the shortest path and corresponding distance from A to E but also get the shortest path and corresponding distance from any point to E. It can be seen that the whole optimal decision can be obtained based on D-P algorithm, so D-P algorithm is an effective method to solve the multistage decision-making problem.


### 2.3. Basic Equation of D-P Algorithm

The D-P algorithm problem can be boiled down to selecting aK stage strategy ux1,ux2,…,uxK so that the value function reaches the optimal expression as(2)fkxk=maxu1,…,ukvx1;u1,…,uk.In formula (2), fkxk represents the optimal value function of the entire strategy [24]. The setting function can be expressed as(3)vx1;u1,u2,…,uk=∑j=1kwjxj,uj.In formula (3), wjxj,uj represents the cost of the j stage. The optimality principle of the D-P algorithm has the following relationship:(4)fkxk=maxuk∈Uwkxk,uk+fk−1xk−1,k=2,3,…,M.The initial cost is assumed to be(5)f1x1=w1x1,u1.Formula (4) and formula (5) are the equations used to calculate the strategy value function, which are called the basic equations of D-P algorithm. The recursive form of formula (6) can be obtained by combining the first two formulas:(6)fkxk=maxuk∈Uhkxk.In formula (6), hkxk represents the stage value function, and there is(7)hkxk=wkxk,uk+maxuk−1∈Uhk−1xk−1.The initial condition is assumed to be(8)h1x1=w1x1,u1.Formula (7) is the general form of the basic equation of D-P algorithm, and the optimal strategy can be expressed as(9)x⌢k=x⌢k:∑k=1Khkxkmax.In formula (9), x⌢k=x⌢1,x⌢2,…,x⌢K is the state corresponding to each stage of decision-making in the optimal strategy recorded by the system in the recursive process.


## 2.1. Basic Concepts of D-P Algorithm

Dynamic programming (D-P) algorithm is an optimization method applied to solve multistage decision-making problems [10–12]. At present, D-P algorithm has been widely used in resource theory, chemical engineering, variational method, economics, and optimal control theory. It is used to solve some optimal problems. This kind of problem refers to such a kind of activity, which can be divided into several interrelated stages, and decisions need to be made in each stage. This decision requires that the whole process achieve the best activity effect [13, 14]. Here, the selection of decision-making in each stage is not determined arbitrarily. Its determination depends on the current state, and it affects its subsequent state. Assuming that it is divided into 10 stages, when the decisions in these 10 stages are determined, these decisions form a decision sequence. An activity route of the whole process can be determined through the decision sequence, which solves this problem. The multistage decision-making problem is shown in Figure 1.Figure 1
Multistage decision-making problem.As can be seen from Figure1, these interrelated stages can facilitate the solution of the problem. In the face of different processes, the problem can be divided into different stages, which is not invariable. Usually a variable is used to describe the stage. This variable is called the stage variable. In most cases, this variable is discrete, denoted by k. The first stage is from point A to point B, the second stage is from point B to point C, the third stage is from point C to point D, and the fourth stage is from point D to point E. Point A in Figure 1 is usually called a state, and the variables used to describe each state are called state variables [15]. This state variable does not have to be a number; it can also be a set of numbers, and even a vector can be used to describe the state. In order to facilitate the understanding of the state variable in the k stage when solving the problem, it is often represented by xk. One state of the first stage is represented by a, and three states of the second stage are represented by B1, B2, and B3. Similarly, C1, C2, C3, and C4 represent the state of stage 4, respectively, and D1, D2, and e represent the state of stages 4 and 5, respectively. When the state of a certain stage is known, it is necessary to choose from this state of this stage to the state of the next stage, and this choice is a decision. For ease of understanding, ukxk is usually used to represent the decision variable, where xk represents the k state of the x stage. The same as state variables, a number, a group of numbers, or even a vector can also be used as decision variables to describe decision-making. The strategy is for decision-making. The strategy can be said to be a group of decision sequences at various stages. In most problems, the transition from one state to another state is not random. It follows a certain criterion, which is called the transfer equation [16]. According to the state transition equation, when the state variable xk of the k stage and its decision variable uk are known, state xk+1 of the k+1 stage can be obtained through calculation.


## 2.2. Basic Principle of  D-P Algorithm

The basic idea of D-P algorithm is to find the point on the curve with the maximum distance from the connecting line between the points at both ends of the curve according to a certain tracing route and judge whether the maximum distance is greater than the given threshold. The principle of D-P algorithm is shown in Figure2.Figure 2
D-P algorithm principle.AB is the given curve. In order to find the inflection point on the curve, traverse the curve AB froma to B, calculate the distance from the point on the curve to the connecting line of AB, and find out the maximum value of the distance, which is obviously point C1. If the distance is greater than the set threshold, the curve is divided into two sections with C1 as the boundary, that is, curves AC1 and C1B. In these two sections of curves, the inflection points are found according to the same method as above. When setting the appropriate maximum distance threshold, two inflection points, that is, C1 and C2, can be found. It can be seen from the search process that the determined inflection points do not appear in sequence according to the contour curve. Because the feature points do not appear in sequence according to the contour curve, after obtaining the feature points, in order to fit the curve, the feature points should be stored in the form of binary tree, and each feature point should be obtained in the first order when storing and in the middle order when using.(1)
Optimization principle: the optimal solution of a multilevel decision problem must include the optimal solution of its subproblems. Decision-making is the choice made at each stage, and multilevel selection constitutes a decision strategy [17]. In the shortest path problem in Figure 1, the shortest path from A to E and its subpath from the intermediate node to the end point e must also be the shortest path from the current intermediate node to the end point E. In this problem, decision-making is the selection of the next location at each point from point A. The current location is called the state. The path from the selection made in each state to the final point is a decision strategy.(2)
State transition principle: D-P algorithm cannot be directly applied to the target state, so the premise of realizing D-P algorithm is to discretize the target state space and change the continuous state into discrete state [18–20]. At this time, the target space will be divided into units of size Δ×Δ. Similarly, the speed space of the target will also be divided into units of Δx×Δy, where Δx=Δy=Δv defines 1 resolution unit/frame. In order to distinguish it from the continuous state x,x˙,y,y˙, the discrete state is represented as x¯,x˙¯,y¯,y˙¯. For state x¯,x˙¯,y¯,y˙¯ of the k frame, the relationship between the k+1 frame and the k frame is expressed as(1)xk+1∈x˙¯+x¯−2Δ,x˙¯+x¯Δ,x˙k+1∈x˙¯−1Δv,x˙¯Δv,yk+1∈y˙¯+y¯−2Δ,y˙¯+y¯Δ,y˙k+1∈y˙¯−1Δv,y˙¯Δv.When the target is maneuvering, the number of state transitionsq will be severely affected. Increased target mobility will lead to an increase in state transitions q, because only by increasing the number of state transitions can the expansion of the target state brought about by changes in target speed be satisfied. Assuming that each state at time k is known as x¯,x˙¯,y¯,y˙¯, there are q possible valid states xk+1 corresponding to it at time k+1.(3)
Reverse order recurrence principle: reverse order recurrence is the specific implementation method of D-P algorithm to solve the optimal strategy; that is, it deduces forward from the final state and records the feasible decisions and corresponding costs in the previous stage for the next decision to evaluate the overall decision cost [21]. In this way, the final feasible decision set is obtained by recursion in reverse order until the initial state, and the optimal strategy is selected according to the recorded cost value, and then the whole decision order is traced back. In the shortest path problem, the overall cost is the path length, and the decision cost is the distance weight between two nodes.The idea of using D-P algorithm to solve multistage decision-making problem is to regard a shortest path of the final required solution as an optimal strategy of multilevel decision-making [22, 23]. The D-P algorithm divides the multilevel decision-making problem into a combination of multiple subproblems and then obtains the optimal strategy through the optimality principle and inverse recurrence principle of the D-P algorithm. The D-P algorithm has the following advantages:(1)
It can reduce the amount of calculation: Using D-P algorithm to solve this problem requires 31 addition times and 19 comparison times, which greatly reduces the amount of calculation. When the number of segments is large and the selection of multiple segments is large, the advantage of D-P algorithm is more prominent; and D-P algorithm is convenient to use mainframe computer to calculate the iterative method.(2)
The calculation results are rich: Using D-P algorithm for calculation, we can not only get the shortest path and corresponding distance from A to E but also get the shortest path and corresponding distance from any point to E. It can be seen that the whole optimal decision can be obtained based on D-P algorithm, so D-P algorithm is an effective method to solve the multistage decision-making problem.


## 2.3. Basic Equation of D-P Algorithm

The D-P algorithm problem can be boiled down to selecting aK stage strategy ux1,ux2,…,uxK so that the value function reaches the optimal expression as(2)fkxk=maxu1,…,ukvx1;u1,…,uk.In formula (2), fkxk represents the optimal value function of the entire strategy [24]. The setting function can be expressed as(3)vx1;u1,u2,…,uk=∑j=1kwjxj,uj.In formula (3), wjxj,uj represents the cost of the j stage. The optimality principle of the D-P algorithm has the following relationship:(4)fkxk=maxuk∈Uwkxk,uk+fk−1xk−1,k=2,3,…,M.The initial cost is assumed to be(5)f1x1=w1x1,u1.Formula (4) and formula (5) are the equations used to calculate the strategy value function, which are called the basic equations of D-P algorithm. The recursive form of formula (6) can be obtained by combining the first two formulas:(6)fkxk=maxuk∈Uhkxk.In formula (6), hkxk represents the stage value function, and there is(7)hkxk=wkxk,uk+maxuk−1∈Uhk−1xk−1.The initial condition is assumed to be(8)h1x1=w1x1,u1.Formula (7) is the general form of the basic equation of D-P algorithm, and the optimal strategy can be expressed as(9)x⌢k=x⌢k:∑k=1Khkxkmax.In formula (9), x⌢k=x⌢1,x⌢2,…,x⌢K is the state corresponding to each stage of decision-making in the optimal strategy recorded by the system in the recursive process.


## 3. Tracking Method of Arm Hitting Image Trajectory of Volleyball Players

### 3.1. Collect the Arm Stroke Trajectory Image of Volleyball Players

In order to realize the volleyball player's arm stroke image trajectory tracking based on the D-P algorithm, firstly, it is necessary to collect the digital features of the volleyball player's arm stroke trajectory, use the digital imaging equipment to track the volleyball player's arm stroke trajectory visually, and combine the computer three-dimensional vision acquisition method [25, 26] for image acquisition and feature analysis.The computer three-dimensional vision acquisition frame difference that defines the volleyball player's arm shot trajectory isg′, the discrete sampling rate is ∇x=1,−1, and the block pheromone of the edge contour of the single frame volleyball player's arm shot trajectory tracking image is(10)minFx=f1x,f2x,…,fmxT,s.t.g′≤0,i=1,2,…,q,hj=0,j=1,2,…,p.The pheromone obtained by formula (10) is combined into an image, and in the imaging sequence acquisition, it is assumed that ∇x=1,−1T is the high-frequency part y=∇xg′,∇yg′ of the tracking image of the volleyball player's arm hitting the ball. In the computer three-dimensional imaging space, using the spatial invariant feature decomposition method [27, 28], the obtained binary image of the volleyball player's arm shot trajectory is(11)minx,kλx⊗k−y22+x1x2+βk1.According to the binary image results, the texture information transmission model of action image in volleyball player's arm stroke trajectory tracking is constructed, which is described as follows:(12)cx,y=∑WIxi,yi−Ixi+Δx,yi+Δy2.In formula (12), Δx,Δy is the probability density function of the position distribution of the volleyball player's arm shot trajectory image in the air, and xi,yi is the coordinate point of the volleyball player's arm shot trajectory tracking.According to the visual information feature transmission structure, the pixel space of the above coordinate points is reconstructed, and the volleyball background imageB and the foreground image I are scaled and information is fused to obtain the noise distribution model of the volleyball player's arm hitting trajectory. The volleyball player's arm hitting trajectory distribution module is divided into W/2×H/2 subblocks, and the image information fusion tracking equation is expressed as(13)x˙=VcosθcosϕV,y˙=Vsinθ,z˙=−VcosθsinϕV,θ˙=ωysinγ+ωzcosγ.In formula (13), x,y,z is the distribution feature quantity of the local information feature points of the volleyball player's arm hitting trajectory. Through the above steps, the volleyball player's arm shot trajectory image acquisition is thus realized.


### 3.2. Pretreatment of Volleyball Player's Arm Hitting Trajectory

Based on the image acquisition of volleyball player's arm stroke trajectory, the wavelet multiscale decomposition method is used to filter the volleyball player's arm stroke trajectory. Blind noise separation is carried out in the imaging area of the image, and the wavelet analysis method is used to obtain the multicolor difference kernel matrix of volleyball player's arm hitting trajectory tracking, which is marked as [29–31](14)fx1,x2=r1x11−x1N1−σ1x2N2=0,gx1,x2=r2x21−σ2x1N1−x2N2=0.In formula (14), r1 represents the state correlation estimated feature value of the volleyball player's arm trajectory, r2 represents the correlation coefficient, σ1 represents the feature matching degree, and N1 is the linear component of the volleyball player's arm trajectory. Set h as the edge pixel set of volleyball player's arm hitting trajectory, use the adaptive block feature matching method to segment the image contour of volleyball player's arm hitting trajectory [32–34], divide the chromatographic image into M×N subblocks Gm,n of 2×2, and obtain the feature distribution matrix of volleyball player's arm hitting trajectory tracking as follows:(15)Gm,n=gm,n1,1gm,n1,2gm,n2,1gm,n2,2,m=1,2,…,M;n=1,2,…,N.We have(16)gm,nu,v=Ikg2m−1+u,2n−1+v.In formula (16), u∈1,2;v∈1,2 represents the scale of the block fusion of the volleyball player's arm shot trajectory. The wavelet multiscale decomposition method is used to filter the volleyball player's arm shot trajectory, and the one-dimensional wavelet transform is used to obtain the regular feature distribution pixel set of the volleyball player's arm shot trajectory as [35, 36](17)xit=ωi1lk,…,ωinlkx1t−k,..…,xnt−kT.In formula (17), ωi1lk,…,ωinlk is the surface block feature quantity of the volleyball player's arm hitting trajectory.Through the above steps, the volleyball player's arm shot trajectory filtering process is thus realized.


### 3.3. Extracting the Trajectory Feature of the Volleyball Player's Arm Shot Image

Based on the wavelet multiscale decomposition method for filtering the arm stroke trajectory of volleyball players, the arm stroke trajectory of volleyball players is tracked. This paper proposes a trajectory tracking method of volleyball player's arm hitting image based on D-P algorithm, extracts the edge contour feature points of volleyball player's arm hitting image trajectory, and gives the mother wavelet function of volleyball player's arm hitting image trajectory filtering as follows:(18)Hz=P1∑K=1Hpkgi,jP2∑K=1Hpk∗gi,j.Fork adjacent points, the graph decomposition of volleyball player's arm hitting image trajectory is carried out in the basis function of mother wavelet. Taking pixel θe,pe as the parent feature point of the trajectory of the volleyball player's arm hitting image based on wavelet transform, the multiscale wavelet decomposition method is used to dynamically filter the trajectory of the volleyball player's arm hitting image, and the gray pixel set of the volleyball player's arm hitting image trajectory is obtained:(19)σZ;Dx=∑i>jdijZ−dXxi,xj2.In formula (19), dijZ is the Euclidean distance of the boundary pixels of the volleyball player's arm shot image trajectory, and dXxi,xj is the active contour component of the local area of the image in the 4 × 4 subgrid. The edge contour feature points of the volleyball player's arm hitting image trajectory are extracted, and the gray histogram feature extraction method is used to enhance the gray information of the volleyball player's arm hitting trajectory image. The center pixel set and edge pixel set of the volleyball player's arm hitting image trajectory in N×N window ωi are expressed as follows:(20)Iifx,y=I∗Gx,y,σi,Iivx,y=I∗stdfiltx,y,ωi,Sgifx,y=−logPifx,y.In formula (20), Gx,y,σi represents the similarity feature. Combined with wavelet multiscale decomposition method for multiscale feature optimization, the boundary geometric feature of volleyball player's arm hitting image trajectory tracking can be obtained, so as to realize the feature extraction of volleyball player's arm hitting image trajectory.


### 3.4. Tracking the Image Track of Volleyball Player's Arm Hitting

According to the above extracted trajectory characteristics of volleyball player's arm hitting image, the trajectory algorithm of volleyball player's arm hitting image is designed.σ0 is set as the edge information scale of the volleyball player's arm shot image trajectory, and using the RGB decomposition method, the pixel feature component Jx of the volleyball player's arm hitting image trajectory is obtained as follows:(21)Jx=Ix−Amaxtx,t0+A∗σ0.In formula (21), t0 represents the edge autocorrelation feature amount of the volleyball player's arm shot image trajectory, and tx represents the edge autocorrelation feature function. Combined with the pixel adaptive enhancement technology, the key action feature points of the volleyball player's arm shot image trajectory are located, and the output is(22)F=p˜x,y=px,yvxyx1/2.The D-P algorithm is used to track the trajectory of the volleyball player's arm hitting image. The optimization process of the D-P algorithm is as follows:(23)Eintvi=Sc2αid−vi−vi−12+βivi−1+2vi+vi+12.We have(24)d=1n∑i=0n−1vi−vi−1.The D-P algorithm is used for block fusion processing of volleyball player's arm stroke trajectory image, and the output volleyball player's arm stroke image trajectory tracking map is as follows:(25)ωi,j=EintvifRzZiexp−di,jh2.In formula (25), Zi represents the symbolic distance function extracted from the volleyball player's arm shot image trajectory feature. Through the above analysis, the D-P algorithm is used to adjust and correct the error of the tracked volleyball player's arm hitting image trajectory, so as to realize the correct tracking of the volleyball player's arm hitting image trajectory. The implementation process of the algorithm is shown in Figure 3.Figure 3
Flow chart of algorithm implementation.Through the above steps, the volleyball player's arm stroke trajectory image is collected under the three-dimensional visual model, the volleyball player's arm stroke trajectory is filtered by wavelet multiscale decomposition method, and the edge contour feature points of the volleyball player's arm stroke image trajectory are extracted. The gray histogram feature extraction method is used to enhance the gray information of the volleyball player's arm hitting trajectory image. Combined with the block feature matching technology, the key action feature points of the volleyball player's arm hitting trajectory image are located. D-P algorithm is used to adjust and modify the image trajectory of volleyball player's arm stroke, so as to realize the correct tracking of volleyball player's arm stroke image trajectory.


## 3.1. Collect the Arm Stroke Trajectory Image of Volleyball Players

In order to realize the volleyball player's arm stroke image trajectory tracking based on the D-P algorithm, firstly, it is necessary to collect the digital features of the volleyball player's arm stroke trajectory, use the digital imaging equipment to track the volleyball player's arm stroke trajectory visually, and combine the computer three-dimensional vision acquisition method [25, 26] for image acquisition and feature analysis.The computer three-dimensional vision acquisition frame difference that defines the volleyball player's arm shot trajectory isg′, the discrete sampling rate is ∇x=1,−1, and the block pheromone of the edge contour of the single frame volleyball player's arm shot trajectory tracking image is(10)minFx=f1x,f2x,…,fmxT,s.t.g′≤0,i=1,2,…,q,hj=0,j=1,2,…,p.The pheromone obtained by formula (10) is combined into an image, and in the imaging sequence acquisition, it is assumed that ∇x=1,−1T is the high-frequency part y=∇xg′,∇yg′ of the tracking image of the volleyball player's arm hitting the ball. In the computer three-dimensional imaging space, using the spatial invariant feature decomposition method [27, 28], the obtained binary image of the volleyball player's arm shot trajectory is(11)minx,kλx⊗k−y22+x1x2+βk1.According to the binary image results, the texture information transmission model of action image in volleyball player's arm stroke trajectory tracking is constructed, which is described as follows:(12)cx,y=∑WIxi,yi−Ixi+Δx,yi+Δy2.In formula (12), Δx,Δy is the probability density function of the position distribution of the volleyball player's arm shot trajectory image in the air, and xi,yi is the coordinate point of the volleyball player's arm shot trajectory tracking.According to the visual information feature transmission structure, the pixel space of the above coordinate points is reconstructed, and the volleyball background imageB and the foreground image I are scaled and information is fused to obtain the noise distribution model of the volleyball player's arm hitting trajectory. The volleyball player's arm hitting trajectory distribution module is divided into W/2×H/2 subblocks, and the image information fusion tracking equation is expressed as(13)x˙=VcosθcosϕV,y˙=Vsinθ,z˙=−VcosθsinϕV,θ˙=ωysinγ+ωzcosγ.In formula (13), x,y,z is the distribution feature quantity of the local information feature points of the volleyball player's arm hitting trajectory. Through the above steps, the volleyball player's arm shot trajectory image acquisition is thus realized.


## 3.2. Pretreatment of Volleyball Player's Arm Hitting Trajectory

Based on the image acquisition of volleyball player's arm stroke trajectory, the wavelet multiscale decomposition method is used to filter the volleyball player's arm stroke trajectory. Blind noise separation is carried out in the imaging area of the image, and the wavelet analysis method is used to obtain the multicolor difference kernel matrix of volleyball player's arm hitting trajectory tracking, which is marked as [29–31](14)fx1,x2=r1x11−x1N1−σ1x2N2=0,gx1,x2=r2x21−σ2x1N1−x2N2=0.In formula (14), r1 represents the state correlation estimated feature value of the volleyball player's arm trajectory, r2 represents the correlation coefficient, σ1 represents the feature matching degree, and N1 is the linear component of the volleyball player's arm trajectory. Set h as the edge pixel set of volleyball player's arm hitting trajectory, use the adaptive block feature matching method to segment the image contour of volleyball player's arm hitting trajectory [32–34], divide the chromatographic image into M×N subblocks Gm,n of 2×2, and obtain the feature distribution matrix of volleyball player's arm hitting trajectory tracking as follows:(15)Gm,n=gm,n1,1gm,n1,2gm,n2,1gm,n2,2,m=1,2,…,M;n=1,2,…,N.We have(16)gm,nu,v=Ikg2m−1+u,2n−1+v.In formula (16), u∈1,2;v∈1,2 represents the scale of the block fusion of the volleyball player's arm shot trajectory. The wavelet multiscale decomposition method is used to filter the volleyball player's arm shot trajectory, and the one-dimensional wavelet transform is used to obtain the regular feature distribution pixel set of the volleyball player's arm shot trajectory as [35, 36](17)xit=ωi1lk,…,ωinlkx1t−k,..…,xnt−kT.In formula (17), ωi1lk,…,ωinlk is the surface block feature quantity of the volleyball player's arm hitting trajectory.Through the above steps, the volleyball player's arm shot trajectory filtering process is thus realized.


## 3.3. Extracting the Trajectory Feature of the Volleyball Player's Arm Shot Image

Based on the wavelet multiscale decomposition method for filtering the arm stroke trajectory of volleyball players, the arm stroke trajectory of volleyball players is tracked. This paper proposes a trajectory tracking method of volleyball player's arm hitting image based on D-P algorithm, extracts the edge contour feature points of volleyball player's arm hitting image trajectory, and gives the mother wavelet function of volleyball player's arm hitting image trajectory filtering as follows:(18)Hz=P1∑K=1Hpkgi,jP2∑K=1Hpk∗gi,j.Fork adjacent points, the graph decomposition of volleyball player's arm hitting image trajectory is carried out in the basis function of mother wavelet. Taking pixel θe,pe as the parent feature point of the trajectory of the volleyball player's arm hitting image based on wavelet transform, the multiscale wavelet decomposition method is used to dynamically filter the trajectory of the volleyball player's arm hitting image, and the gray pixel set of the volleyball player's arm hitting image trajectory is obtained:(19)σZ;Dx=∑i>jdijZ−dXxi,xj2.In formula (19), dijZ is the Euclidean distance of the boundary pixels of the volleyball player's arm shot image trajectory, and dXxi,xj is the active contour component of the local area of the image in the 4 × 4 subgrid. The edge contour feature points of the volleyball player's arm hitting image trajectory are extracted, and the gray histogram feature extraction method is used to enhance the gray information of the volleyball player's arm hitting trajectory image. The center pixel set and edge pixel set of the volleyball player's arm hitting image trajectory in N×N window ωi are expressed as follows:(20)Iifx,y=I∗Gx,y,σi,Iivx,y=I∗stdfiltx,y,ωi,Sgifx,y=−logPifx,y.In formula (20), Gx,y,σi represents the similarity feature. Combined with wavelet multiscale decomposition method for multiscale feature optimization, the boundary geometric feature of volleyball player's arm hitting image trajectory tracking can be obtained, so as to realize the feature extraction of volleyball player's arm hitting image trajectory.


## 3.4. Tracking the Image Track of Volleyball Player's Arm Hitting

According to the above extracted trajectory characteristics of volleyball player's arm hitting image, the trajectory algorithm of volleyball player's arm hitting image is designed.σ0 is set as the edge information scale of the volleyball player's arm shot image trajectory, and using the RGB decomposition method, the pixel feature component Jx of the volleyball player's arm hitting image trajectory is obtained as follows:(21)Jx=Ix−Amaxtx,t0+A∗σ0.In formula (21), t0 represents the edge autocorrelation feature amount of the volleyball player's arm shot image trajectory, and tx represents the edge autocorrelation feature function. Combined with the pixel adaptive enhancement technology, the key action feature points of the volleyball player's arm shot image trajectory are located, and the output is(22)F=p˜x,y=px,yvxyx1/2.The D-P algorithm is used to track the trajectory of the volleyball player's arm hitting image. The optimization process of the D-P algorithm is as follows:(23)Eintvi=Sc2αid−vi−vi−12+βivi−1+2vi+vi+12.We have(24)d=1n∑i=0n−1vi−vi−1.The D-P algorithm is used for block fusion processing of volleyball player's arm stroke trajectory image, and the output volleyball player's arm stroke image trajectory tracking map is as follows:(25)ωi,j=EintvifRzZiexp−di,jh2.In formula (25), Zi represents the symbolic distance function extracted from the volleyball player's arm shot image trajectory feature. Through the above analysis, the D-P algorithm is used to adjust and correct the error of the tracked volleyball player's arm hitting image trajectory, so as to realize the correct tracking of the volleyball player's arm hitting image trajectory. The implementation process of the algorithm is shown in Figure 3.Figure 3
Flow chart of algorithm implementation.Through the above steps, the volleyball player's arm stroke trajectory image is collected under the three-dimensional visual model, the volleyball player's arm stroke trajectory is filtered by wavelet multiscale decomposition method, and the edge contour feature points of the volleyball player's arm stroke image trajectory are extracted. The gray histogram feature extraction method is used to enhance the gray information of the volleyball player's arm hitting trajectory image. Combined with the block feature matching technology, the key action feature points of the volleyball player's arm hitting trajectory image are located. D-P algorithm is used to adjust and modify the image trajectory of volleyball player's arm stroke, so as to realize the correct tracking of volleyball player's arm stroke image trajectory.


## 4. Experimental Simulation and Analysis

### 4.1. Setting the Experimental Environment

In order to verify the effectiveness of the trajectory tracking method of volleyball player's arm hitting image based on D-P algorithm, a simulation experiment is carried out in MATLAB. The volleyball player's arm shot trajectory image acquisition and scanning frequency is 16 kHz, and functions such as receive in MATLAB are used to collect the volleyball player's arm shot trajectory image. The feature matching of the volleyball player's arm hitting trajectory is carried out in a 5 × 5 block mode, and the three-dimensional visual collection sample set of volleyball is 1000 images. According to the above simulation environment and parameter settings, the simulation experiment of volleyball player's arm stroke image trajectory tracking is carried out. Firstly, the computer three-dimensional vision acquisition of volleyball player's arm stroke trajectory image is carried out, and the original image acquisition results are given in Figure4.Figure 4
Acquisition results of arm hitting image of original volleyball player.


### 4.2. Trajectory Tracking Effect of Volleyball Player's Arm Hitting Image

In order to verify the trajectory tracking effect of volleyball player's arm hitting image, the trajectory tracking of volleyball player's arm hitting image is realized. It is necessary to filter the acquisition results of the original volleyball player's arm hitting image, extract the trajectory features of the volleyball player's arm hitting image, and locate the key action feature points. Using the proposed method, the trajectory tracking of volleyball player's arm hitting image is realized, and the trajectory tracking effect of volleyball player's arm hitting image of the proposed method is shown in Figure5.Figure 5
Trajectory tracking effect of volleyball player's arm hitting image based on the proposed method.According to Figure5, the proposed method can effectively track the volleyball player's arm hitting image trajectory, capture the volleyball player's arm hitting image trajectory, and locate the key action feature points of the volleyball player's arm hitting image trajectory. It can be seen that the trajectory tracking of volleyball player's arm hitting image based on the proposed method is better.


### 4.3. Trajectory Tracking Accuracy of Volleyball Player's Arm Hitting Image

In order to further verify the trajectory tracking accuracy of the volleyball player's arm hitting image of the proposed method, the root mean square error (RMSE) of position is taken as the evaluation index. The smaller the RMSE value, the higher the tracking accuracy of the method. The calculation method is as follows:(26)RMSE=1CK∑c=1C∑k=1Kp˜x,kc−px,kc2+p˜y,kc−py,kc2.In formula (26), C represents the number of experiments, K represents the number of processed frames, and p˜x,kc and p˜y,kc represent the coordinates of the target in the k frame of the c experiment. Using the method of [8], the methods of [9], and the proposed method, the arm stroke image trajectory of volleyball players is tracked, and the comparison results of tracking accuracy of arm stroke image trajectory of volleyball players with different methods are given in Figure 6.Figure 6
Comparison results of trajectory tracking accuracy of volleyball player's arm shot images with different methods.It can be seen from Figure6 that with the gradual increase of the number of experiments, the RMSE value of arm stroke image trajectory tracking of volleyball players with different methods increases. When the number of experiments is 40, the RMSE value of volleyball player's arm stroke image trajectory tracking in the method of reference [8] is 0.46, the RMSE value of volleyball player's arm stroke image trajectory tracking in the method of reference [9] is 0.57, while the RMSE value of volleyball player's arm stroke image trajectory tracking in the proposed method is only 0.2. Therefore, the RMSE value of the proposed method is small, which can effectively improve the trajectory tracking accuracy of the volleyball player's arm hitting image.


### 4.4. Trajectory Tracking Time of Volleyball Player's Arm Hitting Image

On this basis, the trajectory tracking time of volleyball player's arm hitting image of the proposed method is verified. The trajectory of volleyball player's arm hitting image is tracked by using the method of [8], the method of [9], and the proposed method, respectively. The comparison results of trajectory tracking time of volleyball player's arm hitting image of different methods are given in Figure 7.Figure 7
Comparison results of volleyball player's arm shot image trajectory tracking time with different methods.It can be seen from Figure7 that, with the increasing number of experiments, the trajectory tracking time of arm hitting image of volleyball players with different methods increases. When the number of experiments reaches 40, the trajectory tracking time of volleyball player's arm stroke image in the method of [8] is 43.2 s, and the trajectory tracking time of volleyball player's arm stroke image in the method of [9] is 55.6 s, while the trajectory tracking time of volleyball player's arm stroke image in the proposed method is only 24.3 s. It can be seen that the trajectory tracking time of volleyball player's arm hitting image is short.


## 4.1. Setting the Experimental Environment

In order to verify the effectiveness of the trajectory tracking method of volleyball player's arm hitting image based on D-P algorithm, a simulation experiment is carried out in MATLAB. The volleyball player's arm shot trajectory image acquisition and scanning frequency is 16 kHz, and functions such as receive in MATLAB are used to collect the volleyball player's arm shot trajectory image. The feature matching of the volleyball player's arm hitting trajectory is carried out in a 5 × 5 block mode, and the three-dimensional visual collection sample set of volleyball is 1000 images. According to the above simulation environment and parameter settings, the simulation experiment of volleyball player's arm stroke image trajectory tracking is carried out. Firstly, the computer three-dimensional vision acquisition of volleyball player's arm stroke trajectory image is carried out, and the original image acquisition results are given in Figure4.Figure 4
Acquisition results of arm hitting image of original volleyball player.


## 4.2. Trajectory Tracking Effect of Volleyball Player's Arm Hitting Image

In order to verify the trajectory tracking effect of volleyball player's arm hitting image, the trajectory tracking of volleyball player's arm hitting image is realized. It is necessary to filter the acquisition results of the original volleyball player's arm hitting image, extract the trajectory features of the volleyball player's arm hitting image, and locate the key action feature points. Using the proposed method, the trajectory tracking of volleyball player's arm hitting image is realized, and the trajectory tracking effect of volleyball player's arm hitting image of the proposed method is shown in Figure5.Figure 5
Trajectory tracking effect of volleyball player's arm hitting image based on the proposed method.According to Figure5, the proposed method can effectively track the volleyball player's arm hitting image trajectory, capture the volleyball player's arm hitting image trajectory, and locate the key action feature points of the volleyball player's arm hitting image trajectory. It can be seen that the trajectory tracking of volleyball player's arm hitting image based on the proposed method is better.


## 4.3. Trajectory Tracking Accuracy of Volleyball Player's Arm Hitting Image

In order to further verify the trajectory tracking accuracy of the volleyball player's arm hitting image of the proposed method, the root mean square error (RMSE) of position is taken as the evaluation index. The smaller the RMSE value, the higher the tracking accuracy of the method. The calculation method is as follows:(26)RMSE=1CK∑c=1C∑k=1Kp˜x,kc−px,kc2+p˜y,kc−py,kc2.In formula (26), C represents the number of experiments, K represents the number of processed frames, and p˜x,kc and p˜y,kc represent the coordinates of the target in the k frame of the c experiment. Using the method of [8], the methods of [9], and the proposed method, the arm stroke image trajectory of volleyball players is tracked, and the comparison results of tracking accuracy of arm stroke image trajectory of volleyball players with different methods are given in Figure 6.Figure 6
Comparison results of trajectory tracking accuracy of volleyball player's arm shot images with different methods.It can be seen from Figure6 that with the gradual increase of the number of experiments, the RMSE value of arm stroke image trajectory tracking of volleyball players with different methods increases. When the number of experiments is 40, the RMSE value of volleyball player's arm stroke image trajectory tracking in the method of reference [8] is 0.46, the RMSE value of volleyball player's arm stroke image trajectory tracking in the method of reference [9] is 0.57, while the RMSE value of volleyball player's arm stroke image trajectory tracking in the proposed method is only 0.2. Therefore, the RMSE value of the proposed method is small, which can effectively improve the trajectory tracking accuracy of the volleyball player's arm hitting image.


## 4.4. Trajectory Tracking Time of Volleyball Player's Arm Hitting Image

On this basis, the trajectory tracking time of volleyball player's arm hitting image of the proposed method is verified. The trajectory of volleyball player's arm hitting image is tracked by using the method of [8], the method of [9], and the proposed method, respectively. The comparison results of trajectory tracking time of volleyball player's arm hitting image of different methods are given in Figure 7.Figure 7
Comparison results of volleyball player's arm shot image trajectory tracking time with different methods.It can be seen from Figure7 that, with the increasing number of experiments, the trajectory tracking time of arm hitting image of volleyball players with different methods increases. When the number of experiments reaches 40, the trajectory tracking time of volleyball player's arm stroke image in the method of [8] is 43.2 s, and the trajectory tracking time of volleyball player's arm stroke image in the method of [9] is 55.6 s, while the trajectory tracking time of volleyball player's arm stroke image in the proposed method is only 24.3 s. It can be seen that the trajectory tracking time of volleyball player's arm hitting image is short.


## 5. Conclusion

With the continuous innovation of technical and tactical level of competitive volleyball and the continuous improvement of computer performance, higher requirements are put forward for badminton players. By studying the role and trajectory of volleyball players' arms in the hitting process of volleyball, on the one hand, it can provide scientific and effective exercise methods for volleyball lovers; on the other hand, it is helpful to the improvement and development of volleyball players' technical and tactical theory. Therefore, the image trajectory tracking method of volleyball player's arm hitting based on D-P algorithm is studied in this paper. The effectiveness of this method is verified by experiments; that is, it can improve the trajectory tracking accuracy of volleyball player's arm hitting image and shorten the trajectory tracking time of volleyball player's arm hitting image.However, the complexity of algorithm and background and the influence of noise on the tracking effect are not considered in the process of tracking the trajectory of volleyball player's arm hitting image. Therefore, in the next research, we can take Gaussian white noise as the background and expand the algorithm to further verify the trajectory tracking effect of the proposed method.



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*Source: 4848036-2021-12-03.xml*