# Tennis Serve Trajectory Capture Algorithm Based on Wavelet Multiscale Decomposition

Authors: Songkun Yu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Human motion analysis is one of the important research directions in computer vision technology. Aiming at the problems of poor effect, accuracy, and efficiency in the current tennis serve trajectory capture process, a tennis serve trajectory capture algorithm based on wavelet multiscale decomposition is proposed. This article analyzes the related concepts and principles of wavelet transform and multiresolution analysis, uses the computer three-dimensional vision acquisition method to collect the tennis serve trajectory image, and uses the mean filter to smooth and denoise the collected tennis serve trajectory image. The wavelet multiscale decomposition algorithm is used to process the tennis serve trajectory image after smooth denoising, find the local maximum point through the modulus value and phase angle value, and obtain the initial tennis serve trajectory image. Using the local layer threshold method, the adaptive threshold is set, and the trajectory line under each scale is obtained to capture the trajectory line of tennis serve. It can be observed from the experimental and simulation results that the trajectory capture effect of the proposed algorithm is good, and it can effectively improve the accuracy and efficiency of trajectory capture.


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## Body

## 1. Introduction

With the development of image processing technology, it is more and more widely used in sports training [1]. Applying image processing technology in sports training is the key to improving the correction ability of sports action and the effect of sports training [2]. Tennis serve technology has always been one of the secret weapons for today’s outstanding athletes to defeat the enemy, and it occupies an absolutely important position in competitive tennis. High-speed and accurate serve is an important means for scoring to win and overwhelming opponents in momentum. The tennis serve is complicated and technically difficult to perform. Real-time analysis and standardized correction of technical movements are required to improve the ability of sports planning [3]. The trajectory line of tennis serve is the key to score. Effectively capturing the trajectory line of tennis serve and analyzing the trajectory combined with image processing technology is of great significance to correct the standardization of tennis serve and improve the ability of motion planning. Therefore, the research on the tennis serve trajectory capture has become a hot issue in this field.At present, scholars in related fields have studied serve trajectory capture. Li and Huamg [4] proposed a tennis serving behavior simulation algorithm based on video image processing and wireless sensor technology. Using video human motion analysis, the moving target is detected from the video sequence, the key parts of the human body are extracted, and the joints of the T-shaped arm are color coded. Use high-speed cameras to capture tennis serve videos. Use the coordinates of the positioning point in each frame instead of the knuckle to study the trajectory of the T-arm. The sparse representation is used to reconstruct the interference-free service graph, and the Gaussian mixture background modeling method is used to extract the motion foreground. The marker points are extracted through color features, and the binary operation is performed on the marker points, so as to realize the tennis serve trajectory capture. This method has certain reliability. Piao and Kim [5] proposed a serve trajectory tracking algorithm based on the backward half-Lagrangian method. The backward semi-Lagrangian method is used to track the serving trajectory required to solve the guidance center model. A completely explicit formula for the numerical solution of the discrete system of the Cauchy problem is designed, and the required trigger point is found by the interpolation method. The interpolation solution is calculated at the starting point to realize the serve trajectory tracking required to solve the guidance center model. This method saves a lot of computing time. However, the above methods still have the problems of poor track line capture and tracking effect, low accuracy, and efficiency. Tang [6] proposed a parabolic detection algorithm of tennis serve based on video image analysis technology. In this study, the author shows through experimental results that the proposed algorithm recognizes the trajectory of the parabola at various stages and the accuracy of the detection of the parabola is much larger in the 3-dimensional space of the tennis service.The existing algorithm for the detection of tennis serve trajectory ignores the estimation of the global motion that leads to low recognition and effectiveness. The main contribution of this article is to solve the above problems, where tennis serve trajectory capture algorithm based on wavelet multiscale decomposition is proposed. Using the computer three-dimensional vision acquisition method, the tennis serve trajectory image is collected, and the collected image is smoothed and denoised. The wavelet multiscale decomposition algorithm is used to process the smooth denoised image, find the local maximum point through the modulus and phase angle value, and obtain the image trajectory of the initial tennis serve. Set the adaptive threshold, calculate the trajectory line under each scale, and realize the trajectory line capture of tennis serve. The trajectory line capture effect of the algorithm is good and can effectively improve the accuracy and efficiency of trajectory line capture.The remainder of the article is organized as follows. Section2 discusses the wavelet multiscale decomposition algorithms with different transformation techniques. Section 3 describes the tennis serve trajectory line capture algorithm with different types. Section 4 discusses the experimental analysis and simulation study of the proposed algorithm. Finally, Section 5 presents the conclusion of the study.


## 2. Wavelet Multiscale Decomposition

### 2.1. Wavelet Transform

Wavelet transform is developed from Fourier transform, which can be viewed as the projection of a signal into a set of basis functions named wavelets. Such basis functions offer localization in the frequency domain. Compared with the Fourier transform, the wavelet transform emphasizes the local characterization of time-frequency [6]. The signal to be analyzed is decomposed at multiple scales, and the high-frequency part and the low-frequency part of the signal are analyzed, respectively, by selecting an appropriate time window. The wavelet transform can change adaptively, and the variable time window enables the wavelet transform to extract effective information from the signal and clearly show the trend component and detail component of the signal. Mathematically speaking, wavelet is a function that oscillates and decays with time, “small” refers to its attenuation and “wave” refers to its volatility, that is, its amplitude oscillates between positive and negative.


#### 2.1.1. Continuous Wavelet Transform

A continuous wavelet transform is a tool that provides an overcomplete representation of a signal by allowing the translation and scale parameters of the wavelets to vary continuously. The main advantage of using wavelet-based coding in image compression is that it provides significant improvements in picture quality at higher compression ratios over conventional techniques.If the function meet the following conditions(1)RQ=∫−∞∞α⌢β2βdβ<∞., then Qx is called a wavelet function. In formula (1), α⌢β is the Fourier transform of Qx. According to formula (1), α⌢β|β=0=0 can be derived, which is equivalent to the following formula in the time domain:(2)∫−∞∞Qxdx=0.That is,α⌢β must have a band-pass property, and Qx must be an oscillatory wave with alternating positive and negative, with an average value of zero. This is why Qx is called a wavelet. The wavelet mother function Qt is scaled and translated, and the expansion factor is χ, the translation factor is δ, and the function Qχ,δt after expansion and translation is the wavelet basis function. The expansion factor χ and the translation factor δ are continuously changing, so Qχ,δt is a continuous wavelet basis function, which is a family of functions obtained from the same generating function after stretching and translation. It is defined as follows:(3)Qχ,δt=χ−1/2Qt−δχ.SupposeYx is the square integrable function Yt∈W2E and Qx is the wavelet function, then the wavelet transform of Yx is as follows:(4)UTYχ,δ=Yt,Qχ,δt=χ−1/2∫−∞∞YtQt−δχdt.Becauseχ,δ changes continuously, what we get is continuous wavelet transform (CWT) [7].


#### 2.1.2. Discrete Wavelet Transform

In practical applications, in order to conveniently use a computer for analysis and processing, the signalYx must be discretized into a discrete time series. For continuous wavelet transform, its expansion and translation parameters χ and δ must also be discretized to convert them into discrete wavelet transform (DWT) [8]. The wavelet frame is a family of functions obtained by stretching and translating the basic wavelet Qx:(5)Qo,pt=χ−1/2Qχ−ot−pζ.They satisfy(6)ZY2≤∑o∑pY,Qo,p2≤XY2.In formulas (5) and (6), Qo,pt is called a wavelet frame, and its upper and lower bounds are Z,X, respectively. When Z=X, its expanded set is called a tight frame. If a continuous wavelet transform is discretized in scale and displacement, and the discretized function family satisfies the conditional formula (6), then the discrete wavelet transforms composed of this discrete wavelet system is given as follows:(7)PTYo,p=Y,Qo,pt.


### 2.2. Multiresolution Analysis

Multiresolution analysis (MRA), also known as multiscale analysis, is a theory based on the concept of function space [9]. MRA not only provides a simple method for the construction of wavelet basis but also provides a theoretical basis for the fast algorithm of the wavelet transform, which can be combined with the theory of digital filter. The tree structure of the three-layer multiresolution analysis is as Figure 1.Figure 1
Three-layer multiresolution analysis tree structure diagram.In this structure, the decomposition relationship is given as follows:(8)A=S3+D1+D2+D3.It can be seen from Figure1 and the decomposition relationship that the multiresolution analysis only further decomposes the low-frequency space, making the frequency resolution higher and higher. The ultimate goal of decomposition is to try to construct a wavelet base that is highly close to the W2E space in frequency. These wavelet bases with different resolutions are equivalent to band-pass filters with different bandwidths [10].


#### 2.2.1. Framework of Multiresolution Analysis

The concept of multiresolution is introduced from the subdivision of function space, and the square-integrable functionYt∈W2E is regarded as a limit case of step-by-step approximation. Each level of approximation is the result of smoothing Yt with a low-pass smoothing function Ft. when approaching step by step, the smoothing function Ft also scales step by step, which is multiresolution, that is, the function Yt to be analyzed is approached step by step with different resolutions. MRA refers to a series of closed subspaces Goo∈R satisfying the following properties:(i)
Monotonicity: for anyo∈R, there is Go⊂Go−1, namely,(9)⋯⊂G2⊂G1⊂G0⊂G−1⊂G−2⊂⋯.(ii)
Approximation:(10)∩o∈RGo=0,∪o=−∞∞Go=W2E.(iii)
Scalability:(11)Yt∈Go⇔Yo2t∈G0.The scalability reflects the consistency of the transformation of scale, the change of approximation orthogonal wavelet function and the change of space.(iv)
Translation invariance: for anyh∈R, there are(12)Yt∈G0⇒Yt−h∈G0.(v)
The existence of orthogonal basis: there isF∈G0, so that Ft−hh∈R constitutes the orthogonal basis of G0, namely,(13)G0=spanhFt−h,∫RFt−hdt.IfFt−h is the orthogonal basis of space G0, then according to formula (13), Fo,h=2−o/2F2−ot−h must be the orthonormal basis of subspace Go. It can be seen from the definition of multiresolution that all closed subspaces Goo∈R are scale spaces stretched by the translation sequence stretched by the same scale function F, and Ft is called the scale function of multiresolution analysis. It is defined by multiresolution analysis as follows:(14)G0=G1⊕H1=G2⊕H2⊕H1=G3⊕H3⊕H2⊕H1=⋯.For any functionYt∈G0, it can be decomposed into detail part H1 and large-scale approximation part G1, and then, the large-scale approximation part G1 can be further decomposed. In this way, the approximation part and detail part on any scale can be obtained by repetition, which is the framework of multiresolution analysis.


#### 2.2.2. Decomposition and Reconstruction of Mallat Algorithm

The Mallat algorithm is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves or wavelets. Mallat algorithm vividly illustrates the multiresolution characteristics of wavelet from the concept of space. With the change of scale from large to small, different features of the image can be observed from coarse to fine on each scale [11].The basic idea of Mallat algorithm is as follows: assuming thatCoY is the discrete approximation value of energy limited signal Yt∈W2E at resolution 2−o, CoY can be further decomposed into the sum of discrete approximation value Co+1Y of Yt at resolution 2−o+1 and detail Vo+1Y between resolution 2−o+1 and 2−o. Let F−o,pt and Q−o,pt be the scale function and wavelet function of function Yt under 2−o resolution approximation [12], then its discrete approximation CoYt and detail part VoYt can be expressed as(15)CoYt=∑p=−∞∞Bo,pF−o,pt,VoYt=∑p=−∞∞No,pQ−o,pt.In formula (15), Bo,p and No,p are the approximate component decomposition coefficient and the detail component decomposition coefficient under 2−o resolution, respectively. According to the decomposition idea of Mallat algorithm, CoYt can be decomposed into the sum of approximate component Co+1Yt and detail component Vo+1Yt:(16)CoYt=Co+1Yt+Vo+1Yt.From formulas (15) and (16), the decomposition iteration formulas of Bo,p and No,p can be obtained as follows:(17)Bo,p=∑p=−∞∞qp−mBo+1,p,No,p=∑p=−∞∞wp−mno+1,p.In formula (17), qp−m=F−o,p,F−o+1,m and wp−m=F−o,p,Q−o+1,m, and at the same time, substituting formula (17) into formula (16) is called signal decomposition [13], which constitutes the tower decomposition of the Mallet algorithm, as shown in Figure 2.Figure 2
Tower decomposition of Mallet algorithm.In addition, formulas (15) and (16) can get the synthetic iterative formula of Bo,p and No,p:(18)Bo,p=∑p=−∞∞qp−mBo+1,m+∑p=−∞∞wp−mno+1,m.Formula (18) is called signal reconstruction, which constitutes the tower reconstruction of the Mallet algorithm, as shown in Figure 3.Figure 3
Tower reconstruction of Mallet algorithm.


### 2.3. Two-Dimensional Wavelet Transform and Two-Dimensional Multiresolution Analysis

A digital image is a two-dimensional signal. For example, for an image, any pointε,ϕ has an image signal gray value Yε,ϕ corresponding to it. When the point coordinate ε,ϕ changes continuously, a continuously changing two-dimensional signal Yε,ϕ is determined. Therefore, in order to apply wavelet transform to digital image processing, we must first expand wavelet transform from one-dimensional to two-dimensional signal.LetYε1,ε2∈W2E denotes a two-dimensional signal, ε1,ε2 denote the abscissa and ordinate, respectively, Qε1,ε2 denotes the two-dimensional basic wavelet, then the two-dimensional continuous wavelet is defined as follows:(19)Qf;g1,g2ε1,ε2=1fQε1−g1f,ε2−g2f.Then, the two-dimensional continuous wavelet transform is given as(20)UTYf;g1,g2=Yε1,ε2,Qf;g1,g2ε1,ε2.The two-dimensional wavelet transform is more complicated than the one-dimensional wavelet transform in that it also performs coordinate rotation while scaling, that is to say, the scale factorφ can be written as a matrix:(21)J=φγ.In formula (21), γ is the rotation factor, so the two-dimensional continuous wavelet transform can be expressed as(22)UTYJ,d=Yε,QJ,dε.The specific expression is(23)UTYf,θ;g1,g2=1f∫∫Yε1,ε2=Qcosθε1−g1−sinθε2−g2f,sinθε1−g1−cosθε2−g2fdε1dε2.From the previous formula, it can be seen that the two-dimensional wavelet transform has the ability of rotation. It not only has the function of amplification but also has the property of “polarization” or “direction.” The best “polarization” direction can be selected for analysis [14]. The two-dimensional signal Yε,ϕ is transformed into a function with 4 variables f,θ,g1,g2, so the information must be redundant. It can be discretized in the multidimensional space formed by f,θ,g1,g2 like a one-dimensional signal to reduce the redundancy of information. After discretization, the two-dimensional discrete wavelet transform is obtained as follows:(24)UTYo,n=Yε,Qo,nε=J0−o∫∫Yε1,ε2Qf11oε1+f12oε2−n1,f21oε1+f22oε2−n2dε1dε2.In formula (24), J0 is a nonsingular 2×2 matrix taken, and n1,n2 is the coordinate number of each point extracted along the ε1,ε2 direction. Two-dimensional multiresolution analysis is a generalization of one-dimensional multiresolution analysis MRA. The following relationships still exist in two-dimensional multi-resolution analysis:(25)Go−1ε1,ε2=Goε1,ε2⊕Hoε1,ε2,Ko−1Yε1,ε2=KoYε1,ε2+NoYε1,ε2.In formula (25), Hoε1,ε2 is only a complement subspace, not an orthogonal complement. The basis function of Go is Fopxp∈R, and the basis function of Ho is Qopxp∈R. If the two-dimensional space Goε1,ε2 is separable, it can be decomposed into the tensor product of two one-dimensional spaces Goε1 and Goε2, and we can get(26)Go−1ε1,ε2=Goε1⊗Goε2.ForKo−1Yε1,ε2∈Go−1ε1,ε2 can be decomposed as follows:(27)Go−1Yε1,ε2=GoYε1,ε2+NoYε1,ε2.


## 2.1. Wavelet Transform

Wavelet transform is developed from Fourier transform, which can be viewed as the projection of a signal into a set of basis functions named wavelets. Such basis functions offer localization in the frequency domain. Compared with the Fourier transform, the wavelet transform emphasizes the local characterization of time-frequency [6]. The signal to be analyzed is decomposed at multiple scales, and the high-frequency part and the low-frequency part of the signal are analyzed, respectively, by selecting an appropriate time window. The wavelet transform can change adaptively, and the variable time window enables the wavelet transform to extract effective information from the signal and clearly show the trend component and detail component of the signal. Mathematically speaking, wavelet is a function that oscillates and decays with time, “small” refers to its attenuation and “wave” refers to its volatility, that is, its amplitude oscillates between positive and negative.


### 2.1.1. Continuous Wavelet Transform

A continuous wavelet transform is a tool that provides an overcomplete representation of a signal by allowing the translation and scale parameters of the wavelets to vary continuously. The main advantage of using wavelet-based coding in image compression is that it provides significant improvements in picture quality at higher compression ratios over conventional techniques.If the function meet the following conditions(1)RQ=∫−∞∞α⌢β2βdβ<∞., then Qx is called a wavelet function. In formula (1), α⌢β is the Fourier transform of Qx. According to formula (1), α⌢β|β=0=0 can be derived, which is equivalent to the following formula in the time domain:(2)∫−∞∞Qxdx=0.That is,α⌢β must have a band-pass property, and Qx must be an oscillatory wave with alternating positive and negative, with an average value of zero. This is why Qx is called a wavelet. The wavelet mother function Qt is scaled and translated, and the expansion factor is χ, the translation factor is δ, and the function Qχ,δt after expansion and translation is the wavelet basis function. The expansion factor χ and the translation factor δ are continuously changing, so Qχ,δt is a continuous wavelet basis function, which is a family of functions obtained from the same generating function after stretching and translation. It is defined as follows:(3)Qχ,δt=χ−1/2Qt−δχ.SupposeYx is the square integrable function Yt∈W2E and Qx is the wavelet function, then the wavelet transform of Yx is as follows:(4)UTYχ,δ=Yt,Qχ,δt=χ−1/2∫−∞∞YtQt−δχdt.Becauseχ,δ changes continuously, what we get is continuous wavelet transform (CWT) [7].


### 2.1.2. Discrete Wavelet Transform

In practical applications, in order to conveniently use a computer for analysis and processing, the signalYx must be discretized into a discrete time series. For continuous wavelet transform, its expansion and translation parameters χ and δ must also be discretized to convert them into discrete wavelet transform (DWT) [8]. The wavelet frame is a family of functions obtained by stretching and translating the basic wavelet Qx:(5)Qo,pt=χ−1/2Qχ−ot−pζ.They satisfy(6)ZY2≤∑o∑pY,Qo,p2≤XY2.In formulas (5) and (6), Qo,pt is called a wavelet frame, and its upper and lower bounds are Z,X, respectively. When Z=X, its expanded set is called a tight frame. If a continuous wavelet transform is discretized in scale and displacement, and the discretized function family satisfies the conditional formula (6), then the discrete wavelet transforms composed of this discrete wavelet system is given as follows:(7)PTYo,p=Y,Qo,pt.


## 2.1.1. Continuous Wavelet Transform

A continuous wavelet transform is a tool that provides an overcomplete representation of a signal by allowing the translation and scale parameters of the wavelets to vary continuously. The main advantage of using wavelet-based coding in image compression is that it provides significant improvements in picture quality at higher compression ratios over conventional techniques.If the function meet the following conditions(1)RQ=∫−∞∞α⌢β2βdβ<∞., then Qx is called a wavelet function. In formula (1), α⌢β is the Fourier transform of Qx. According to formula (1), α⌢β|β=0=0 can be derived, which is equivalent to the following formula in the time domain:(2)∫−∞∞Qxdx=0.That is,α⌢β must have a band-pass property, and Qx must be an oscillatory wave with alternating positive and negative, with an average value of zero. This is why Qx is called a wavelet. The wavelet mother function Qt is scaled and translated, and the expansion factor is χ, the translation factor is δ, and the function Qχ,δt after expansion and translation is the wavelet basis function. The expansion factor χ and the translation factor δ are continuously changing, so Qχ,δt is a continuous wavelet basis function, which is a family of functions obtained from the same generating function after stretching and translation. It is defined as follows:(3)Qχ,δt=χ−1/2Qt−δχ.SupposeYx is the square integrable function Yt∈W2E and Qx is the wavelet function, then the wavelet transform of Yx is as follows:(4)UTYχ,δ=Yt,Qχ,δt=χ−1/2∫−∞∞YtQt−δχdt.Becauseχ,δ changes continuously, what we get is continuous wavelet transform (CWT) [7].


## 2.1.2. Discrete Wavelet Transform

In practical applications, in order to conveniently use a computer for analysis and processing, the signalYx must be discretized into a discrete time series. For continuous wavelet transform, its expansion and translation parameters χ and δ must also be discretized to convert them into discrete wavelet transform (DWT) [8]. The wavelet frame is a family of functions obtained by stretching and translating the basic wavelet Qx:(5)Qo,pt=χ−1/2Qχ−ot−pζ.They satisfy(6)ZY2≤∑o∑pY,Qo,p2≤XY2.In formulas (5) and (6), Qo,pt is called a wavelet frame, and its upper and lower bounds are Z,X, respectively. When Z=X, its expanded set is called a tight frame. If a continuous wavelet transform is discretized in scale and displacement, and the discretized function family satisfies the conditional formula (6), then the discrete wavelet transforms composed of this discrete wavelet system is given as follows:(7)PTYo,p=Y,Qo,pt.


## 2.2. Multiresolution Analysis

Multiresolution analysis (MRA), also known as multiscale analysis, is a theory based on the concept of function space [9]. MRA not only provides a simple method for the construction of wavelet basis but also provides a theoretical basis for the fast algorithm of the wavelet transform, which can be combined with the theory of digital filter. The tree structure of the three-layer multiresolution analysis is as Figure 1.Figure 1
Three-layer multiresolution analysis tree structure diagram.In this structure, the decomposition relationship is given as follows:(8)A=S3+D1+D2+D3.It can be seen from Figure1 and the decomposition relationship that the multiresolution analysis only further decomposes the low-frequency space, making the frequency resolution higher and higher. The ultimate goal of decomposition is to try to construct a wavelet base that is highly close to the W2E space in frequency. These wavelet bases with different resolutions are equivalent to band-pass filters with different bandwidths [10].


### 2.2.1. Framework of Multiresolution Analysis

The concept of multiresolution is introduced from the subdivision of function space, and the square-integrable functionYt∈W2E is regarded as a limit case of step-by-step approximation. Each level of approximation is the result of smoothing Yt with a low-pass smoothing function Ft. when approaching step by step, the smoothing function Ft also scales step by step, which is multiresolution, that is, the function Yt to be analyzed is approached step by step with different resolutions. MRA refers to a series of closed subspaces Goo∈R satisfying the following properties:(i)
Monotonicity: for anyo∈R, there is Go⊂Go−1, namely,(9)⋯⊂G2⊂G1⊂G0⊂G−1⊂G−2⊂⋯.(ii)
Approximation:(10)∩o∈RGo=0,∪o=−∞∞Go=W2E.(iii)
Scalability:(11)Yt∈Go⇔Yo2t∈G0.The scalability reflects the consistency of the transformation of scale, the change of approximation orthogonal wavelet function and the change of space.(iv)
Translation invariance: for anyh∈R, there are(12)Yt∈G0⇒Yt−h∈G0.(v)
The existence of orthogonal basis: there isF∈G0, so that Ft−hh∈R constitutes the orthogonal basis of G0, namely,(13)G0=spanhFt−h,∫RFt−hdt.IfFt−h is the orthogonal basis of space G0, then according to formula (13), Fo,h=2−o/2F2−ot−h must be the orthonormal basis of subspace Go. It can be seen from the definition of multiresolution that all closed subspaces Goo∈R are scale spaces stretched by the translation sequence stretched by the same scale function F, and Ft is called the scale function of multiresolution analysis. It is defined by multiresolution analysis as follows:(14)G0=G1⊕H1=G2⊕H2⊕H1=G3⊕H3⊕H2⊕H1=⋯.For any functionYt∈G0, it can be decomposed into detail part H1 and large-scale approximation part G1, and then, the large-scale approximation part G1 can be further decomposed. In this way, the approximation part and detail part on any scale can be obtained by repetition, which is the framework of multiresolution analysis.


### 2.2.2. Decomposition and Reconstruction of Mallat Algorithm

The Mallat algorithm is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves or wavelets. Mallat algorithm vividly illustrates the multiresolution characteristics of wavelet from the concept of space. With the change of scale from large to small, different features of the image can be observed from coarse to fine on each scale [11].The basic idea of Mallat algorithm is as follows: assuming thatCoY is the discrete approximation value of energy limited signal Yt∈W2E at resolution 2−o, CoY can be further decomposed into the sum of discrete approximation value Co+1Y of Yt at resolution 2−o+1 and detail Vo+1Y between resolution 2−o+1 and 2−o. Let F−o,pt and Q−o,pt be the scale function and wavelet function of function Yt under 2−o resolution approximation [12], then its discrete approximation CoYt and detail part VoYt can be expressed as(15)CoYt=∑p=−∞∞Bo,pF−o,pt,VoYt=∑p=−∞∞No,pQ−o,pt.In formula (15), Bo,p and No,p are the approximate component decomposition coefficient and the detail component decomposition coefficient under 2−o resolution, respectively. According to the decomposition idea of Mallat algorithm, CoYt can be decomposed into the sum of approximate component Co+1Yt and detail component Vo+1Yt:(16)CoYt=Co+1Yt+Vo+1Yt.From formulas (15) and (16), the decomposition iteration formulas of Bo,p and No,p can be obtained as follows:(17)Bo,p=∑p=−∞∞qp−mBo+1,p,No,p=∑p=−∞∞wp−mno+1,p.In formula (17), qp−m=F−o,p,F−o+1,m and wp−m=F−o,p,Q−o+1,m, and at the same time, substituting formula (17) into formula (16) is called signal decomposition [13], which constitutes the tower decomposition of the Mallet algorithm, as shown in Figure 2.Figure 2
Tower decomposition of Mallet algorithm.In addition, formulas (15) and (16) can get the synthetic iterative formula of Bo,p and No,p:(18)Bo,p=∑p=−∞∞qp−mBo+1,m+∑p=−∞∞wp−mno+1,m.Formula (18) is called signal reconstruction, which constitutes the tower reconstruction of the Mallet algorithm, as shown in Figure 3.Figure 3
Tower reconstruction of Mallet algorithm.


## 2.2.1. Framework of Multiresolution Analysis

The concept of multiresolution is introduced from the subdivision of function space, and the square-integrable functionYt∈W2E is regarded as a limit case of step-by-step approximation. Each level of approximation is the result of smoothing Yt with a low-pass smoothing function Ft. when approaching step by step, the smoothing function Ft also scales step by step, which is multiresolution, that is, the function Yt to be analyzed is approached step by step with different resolutions. MRA refers to a series of closed subspaces Goo∈R satisfying the following properties:(i)
Monotonicity: for anyo∈R, there is Go⊂Go−1, namely,(9)⋯⊂G2⊂G1⊂G0⊂G−1⊂G−2⊂⋯.(ii)
Approximation:(10)∩o∈RGo=0,∪o=−∞∞Go=W2E.(iii)
Scalability:(11)Yt∈Go⇔Yo2t∈G0.The scalability reflects the consistency of the transformation of scale, the change of approximation orthogonal wavelet function and the change of space.(iv)
Translation invariance: for anyh∈R, there are(12)Yt∈G0⇒Yt−h∈G0.(v)
The existence of orthogonal basis: there isF∈G0, so that Ft−hh∈R constitutes the orthogonal basis of G0, namely,(13)G0=spanhFt−h,∫RFt−hdt.IfFt−h is the orthogonal basis of space G0, then according to formula (13), Fo,h=2−o/2F2−ot−h must be the orthonormal basis of subspace Go. It can be seen from the definition of multiresolution that all closed subspaces Goo∈R are scale spaces stretched by the translation sequence stretched by the same scale function F, and Ft is called the scale function of multiresolution analysis. It is defined by multiresolution analysis as follows:(14)G0=G1⊕H1=G2⊕H2⊕H1=G3⊕H3⊕H2⊕H1=⋯.For any functionYt∈G0, it can be decomposed into detail part H1 and large-scale approximation part G1, and then, the large-scale approximation part G1 can be further decomposed. In this way, the approximation part and detail part on any scale can be obtained by repetition, which is the framework of multiresolution analysis.


## 2.2.2. Decomposition and Reconstruction of Mallat Algorithm

The Mallat algorithm is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves or wavelets. Mallat algorithm vividly illustrates the multiresolution characteristics of wavelet from the concept of space. With the change of scale from large to small, different features of the image can be observed from coarse to fine on each scale [11].The basic idea of Mallat algorithm is as follows: assuming thatCoY is the discrete approximation value of energy limited signal Yt∈W2E at resolution 2−o, CoY can be further decomposed into the sum of discrete approximation value Co+1Y of Yt at resolution 2−o+1 and detail Vo+1Y between resolution 2−o+1 and 2−o. Let F−o,pt and Q−o,pt be the scale function and wavelet function of function Yt under 2−o resolution approximation [12], then its discrete approximation CoYt and detail part VoYt can be expressed as(15)CoYt=∑p=−∞∞Bo,pF−o,pt,VoYt=∑p=−∞∞No,pQ−o,pt.In formula (15), Bo,p and No,p are the approximate component decomposition coefficient and the detail component decomposition coefficient under 2−o resolution, respectively. According to the decomposition idea of Mallat algorithm, CoYt can be decomposed into the sum of approximate component Co+1Yt and detail component Vo+1Yt:(16)CoYt=Co+1Yt+Vo+1Yt.From formulas (15) and (16), the decomposition iteration formulas of Bo,p and No,p can be obtained as follows:(17)Bo,p=∑p=−∞∞qp−mBo+1,p,No,p=∑p=−∞∞wp−mno+1,p.In formula (17), qp−m=F−o,p,F−o+1,m and wp−m=F−o,p,Q−o+1,m, and at the same time, substituting formula (17) into formula (16) is called signal decomposition [13], which constitutes the tower decomposition of the Mallet algorithm, as shown in Figure 2.Figure 2
Tower decomposition of Mallet algorithm.In addition, formulas (15) and (16) can get the synthetic iterative formula of Bo,p and No,p:(18)Bo,p=∑p=−∞∞qp−mBo+1,m+∑p=−∞∞wp−mno+1,m.Formula (18) is called signal reconstruction, which constitutes the tower reconstruction of the Mallet algorithm, as shown in Figure 3.Figure 3
Tower reconstruction of Mallet algorithm.


## 2.3. Two-Dimensional Wavelet Transform and Two-Dimensional Multiresolution Analysis

A digital image is a two-dimensional signal. For example, for an image, any pointε,ϕ has an image signal gray value Yε,ϕ corresponding to it. When the point coordinate ε,ϕ changes continuously, a continuously changing two-dimensional signal Yε,ϕ is determined. Therefore, in order to apply wavelet transform to digital image processing, we must first expand wavelet transform from one-dimensional to two-dimensional signal.LetYε1,ε2∈W2E denotes a two-dimensional signal, ε1,ε2 denote the abscissa and ordinate, respectively, Qε1,ε2 denotes the two-dimensional basic wavelet, then the two-dimensional continuous wavelet is defined as follows:(19)Qf;g1,g2ε1,ε2=1fQε1−g1f,ε2−g2f.Then, the two-dimensional continuous wavelet transform is given as(20)UTYf;g1,g2=Yε1,ε2,Qf;g1,g2ε1,ε2.The two-dimensional wavelet transform is more complicated than the one-dimensional wavelet transform in that it also performs coordinate rotation while scaling, that is to say, the scale factorφ can be written as a matrix:(21)J=φγ.In formula (21), γ is the rotation factor, so the two-dimensional continuous wavelet transform can be expressed as(22)UTYJ,d=Yε,QJ,dε.The specific expression is(23)UTYf,θ;g1,g2=1f∫∫Yε1,ε2=Qcosθε1−g1−sinθε2−g2f,sinθε1−g1−cosθε2−g2fdε1dε2.From the previous formula, it can be seen that the two-dimensional wavelet transform has the ability of rotation. It not only has the function of amplification but also has the property of “polarization” or “direction.” The best “polarization” direction can be selected for analysis [14]. The two-dimensional signal Yε,ϕ is transformed into a function with 4 variables f,θ,g1,g2, so the information must be redundant. It can be discretized in the multidimensional space formed by f,θ,g1,g2 like a one-dimensional signal to reduce the redundancy of information. After discretization, the two-dimensional discrete wavelet transform is obtained as follows:(24)UTYo,n=Yε,Qo,nε=J0−o∫∫Yε1,ε2Qf11oε1+f12oε2−n1,f21oε1+f22oε2−n2dε1dε2.In formula (24), J0 is a nonsingular 2×2 matrix taken, and n1,n2 is the coordinate number of each point extracted along the ε1,ε2 direction. Two-dimensional multiresolution analysis is a generalization of one-dimensional multiresolution analysis MRA. The following relationships still exist in two-dimensional multi-resolution analysis:(25)Go−1ε1,ε2=Goε1,ε2⊕Hoε1,ε2,Ko−1Yε1,ε2=KoYε1,ε2+NoYε1,ε2.In formula (25), Hoε1,ε2 is only a complement subspace, not an orthogonal complement. The basis function of Go is Fopxp∈R, and the basis function of Ho is Qopxp∈R. If the two-dimensional space Goε1,ε2 is separable, it can be decomposed into the tensor product of two one-dimensional spaces Goε1 and Goε2, and we can get(26)Go−1ε1,ε2=Goε1⊗Goε2.ForKo−1Yε1,ε2∈Go−1ε1,ε2 can be decomposed as follows:(27)Go−1Yε1,ε2=GoYε1,ε2+NoYε1,ε2.


## 3. Tennis Serve Trajectory Line Capture Algorithm

In order to realize the tennis serve trajectory capture based on wavelet multiscale decomposition, the tennis serve trajectory image is collected, and the collected tennis serve trajectory image is denoised. The denoised image is processed by wavelet multiscale decomposition, and the local maximum points are found using modulus and phase angle values to obtain the possible image trajectory of the tennis serve. Using the local layer threshold method, set the adaptive threshold, calculate the trajectory line under each scale, and output the tennis serve trajectory line capture results. The algorithm implementation flow is as shown in Figure4.Figure 4
Algorithm implementation flow chart.


### 3.1. Image Collection of the Tennis Serve Trajectory

Using the computer three-dimensional vision acquisition method [15], the trajectory image of tennis serve is collected. Assuming that zc is the frame difference of the trajectory of tennis serve, and ls is the discrete sampling rate, the block pheromone of the trajectory of tennis serve is expressed as(28)cx,y=zcxi,yi−lsxi+Δx,yi+Δy2.In the computer three-dimensional imaging space, the binary image of tennis serve trajectory obtained using the spatial invariant feature decomposition method is as follows:(29)ωi,j=cx,yfRzZi.In formula (29), fRz is the probability density function of the position distribution of the tennis serve trajectory image in the air, and Zi is the coordinate point captured by the tennis serve trajectory. Through the above steps, the image collection of the tennis serve motion trajectory is completed.


### 3.2. Tennis Serve Trajectory Image Denoising

The image acquisition signal is easy to be disturbed, and the acquired image may contain some noise, which has a great impact on the subsequent track line capture. Therefore, considering the operation cost of image processing, this article uses mean filtering to smooth and denoise the image. Mean filtering is the most typical filtering technology in linear filtering in spatial domain, also known as linear smoothing filtering [16]. Take a field for each pixel in the noisy image and then calculate the average value of the gray value of each pixel in the field as the value of each pixel in the processed image. That is,(30)μ⌢i,j=1ν∑i,j∈ομi,j.In formula (30), μi,j represents a noisy image, ο represents a neighborhood, generally a 3×3 or 5×5 window, and ν represents the number of pixels in the domain. The sound filtering effect of mean filtering is related to the size of the selected field. The larger the size, the better the denoising effect, but at the same time, it should cause blurring of the image boundary and contour. In order to improve this blurring effect, a partially averaged mean filter is proposed:(31)μ⌢i,j=1ν∑i,j∈ομi,j,μi,j−1ν∑i,j∈ομi,j>ψμi,j,other.In formula (31), ψ is a nonnegative threshold. When the difference between the value of the pixel point and the average value of the area is greater than the threshold ψ, the mean filtering is performed on it; otherwise, it is not processed.


### 3.3. Tennis Serve Motion Image Wavelet Multiscale Decomposition Trajectory Line Capture

Perform wavelet multiscale decomposition on the tennis serve motion trajectory image after denoising, extend the one-dimensional mutation detection to the two-dimensional situation, setFε,ϕ as a smooth function, satisfying ∫∫R2Fε,ϕdεdϕ=1, and for Fε,ϕ, use the first derivative of ε,ϕ as the basic wavelet:(32)Qεε,ϕ=∂Fε,ϕ∂εQϕε,ϕ=∂Fε,ϕ∂ϕ.The dyadic wavelet transform of the imageωi,j at scale 2o is defined as(33)Mεωi,j=ωi,j×Qεi,jMϕωi,j=ωi,j×Qϕi,j.In formula (33), Mεωi,j,Mϕωi,j corresponds to the horizontal high-frequency trajectory information and vertical high-frequency trajectory information of the image under the scale 2o, respectively. It can be seen from formula (33) that the two components ω×Fi,j of the wavelet transform are proportional to the gradient component of the image function ωi,j smoothed by Fε,ϕ. The modulus and argument of wavelet transform at scale 2o are(34)ρεωi,j=Mεωi,j2+Mϕωi,j2ϑϕωi,j=arctanMϕωi,jMεωi,j.It can be known from the above formula that the sudden change point of the smoothed function corresponds to the local maximum point ofρεωi,j in the direction ϑϕωi,j of the gradient vector. Therefore, only by detecting the local maximum point of ρεωi,j along the direction ϑϕωi,j, the sudden change point of the image can be obtained.Using the nonmaximum suppression method, the wavelet coefficient modulus value is detected along the gradient direction, the local modulus maximum value is retained, and the nonmaximum value is deleted, and the initial image trajectoryτεωi,j is obtained. The obtained initial trajectory τεωi,j is affected by noise and may have false edges. Therefore, it is necessary to set a threshold to remove false edges. This article adopts the local hierarchical threshold method to set the threshold:(35)nowSH=SH×1−ϖ×κ.In formula (35), SH is the initial threshold, κ is the current decomposition layer number, and ϖ is a constant. Perform normalization processing on the possible trajectories θεωi,j of different scales and use the threshold nowSH to determine the final capture result of the tennis serve motion trajectory:(36)gji,j=1,τεωi,j>nowSH0,τεωi,j<nowSH.Through the above analysis, the trajectory of tennis serve is captured.


## 3.1. Image Collection of the Tennis Serve Trajectory

Using the computer three-dimensional vision acquisition method [15], the trajectory image of tennis serve is collected. Assuming that zc is the frame difference of the trajectory of tennis serve, and ls is the discrete sampling rate, the block pheromone of the trajectory of tennis serve is expressed as(28)cx,y=zcxi,yi−lsxi+Δx,yi+Δy2.In the computer three-dimensional imaging space, the binary image of tennis serve trajectory obtained using the spatial invariant feature decomposition method is as follows:(29)ωi,j=cx,yfRzZi.In formula (29), fRz is the probability density function of the position distribution of the tennis serve trajectory image in the air, and Zi is the coordinate point captured by the tennis serve trajectory. Through the above steps, the image collection of the tennis serve motion trajectory is completed.


## 3.2. Tennis Serve Trajectory Image Denoising

The image acquisition signal is easy to be disturbed, and the acquired image may contain some noise, which has a great impact on the subsequent track line capture. Therefore, considering the operation cost of image processing, this article uses mean filtering to smooth and denoise the image. Mean filtering is the most typical filtering technology in linear filtering in spatial domain, also known as linear smoothing filtering [16]. Take a field for each pixel in the noisy image and then calculate the average value of the gray value of each pixel in the field as the value of each pixel in the processed image. That is,(30)μ⌢i,j=1ν∑i,j∈ομi,j.In formula (30), μi,j represents a noisy image, ο represents a neighborhood, generally a 3×3 or 5×5 window, and ν represents the number of pixels in the domain. The sound filtering effect of mean filtering is related to the size of the selected field. The larger the size, the better the denoising effect, but at the same time, it should cause blurring of the image boundary and contour. In order to improve this blurring effect, a partially averaged mean filter is proposed:(31)μ⌢i,j=1ν∑i,j∈ομi,j,μi,j−1ν∑i,j∈ομi,j>ψμi,j,other.In formula (31), ψ is a nonnegative threshold. When the difference between the value of the pixel point and the average value of the area is greater than the threshold ψ, the mean filtering is performed on it; otherwise, it is not processed.


## 3.3. Tennis Serve Motion Image Wavelet Multiscale Decomposition Trajectory Line Capture

Perform wavelet multiscale decomposition on the tennis serve motion trajectory image after denoising, extend the one-dimensional mutation detection to the two-dimensional situation, setFε,ϕ as a smooth function, satisfying ∫∫R2Fε,ϕdεdϕ=1, and for Fε,ϕ, use the first derivative of ε,ϕ as the basic wavelet:(32)Qεε,ϕ=∂Fε,ϕ∂εQϕε,ϕ=∂Fε,ϕ∂ϕ.The dyadic wavelet transform of the imageωi,j at scale 2o is defined as(33)Mεωi,j=ωi,j×Qεi,jMϕωi,j=ωi,j×Qϕi,j.In formula (33), Mεωi,j,Mϕωi,j corresponds to the horizontal high-frequency trajectory information and vertical high-frequency trajectory information of the image under the scale 2o, respectively. It can be seen from formula (33) that the two components ω×Fi,j of the wavelet transform are proportional to the gradient component of the image function ωi,j smoothed by Fε,ϕ. The modulus and argument of wavelet transform at scale 2o are(34)ρεωi,j=Mεωi,j2+Mϕωi,j2ϑϕωi,j=arctanMϕωi,jMεωi,j.It can be known from the above formula that the sudden change point of the smoothed function corresponds to the local maximum point ofρεωi,j in the direction ϑϕωi,j of the gradient vector. Therefore, only by detecting the local maximum point of ρεωi,j along the direction ϑϕωi,j, the sudden change point of the image can be obtained.Using the nonmaximum suppression method, the wavelet coefficient modulus value is detected along the gradient direction, the local modulus maximum value is retained, and the nonmaximum value is deleted, and the initial image trajectoryτεωi,j is obtained. The obtained initial trajectory τεωi,j is affected by noise and may have false edges. Therefore, it is necessary to set a threshold to remove false edges. This article adopts the local hierarchical threshold method to set the threshold:(35)nowSH=SH×1−ϖ×κ.In formula (35), SH is the initial threshold, κ is the current decomposition layer number, and ϖ is a constant. Perform normalization processing on the possible trajectories θεωi,j of different scales and use the threshold nowSH to determine the final capture result of the tennis serve motion trajectory:(36)gji,j=1,τεωi,j>nowSH0,τεωi,j<nowSH.Through the above analysis, the trajectory of tennis serve is captured.


## 4. Experimental Analysis and Simulation

### 4.1. Setting the Experimental Environment

In order to verify the effectiveness of the tennis serve trajectory capture algorithm based on wavelet multiscale decomposition, MATLAB simulation software is used as the experimental platform for simulation experiments. Set the frequency of image acquisition and scanning of tennis serve trajectory line as 16 kHz, and the sample set of tennis serve image acquisition is 400 images. According to the above simulation environment and setting parameters, the computer three-dimensional vision of tennis serve trajectory image is collected, and the original image acquisition results are obtained, as shown in Figure5.Figure 5
Image collection result of original tennis serve action.


### 4.2. Tennis Serve Trajectory Line Capture Effect

In order to verify the capture effect of the tennis serve trajectory line of the proposed algorithm, the receive function in MATLAB simulation software is used to collect the tennis serve trajectory line image. The proposed algorithm is used to filter the image acquisition results of the original tennis service action, so as to realize the tennis service action trajectory capture, and the tennis service action trajectory capture effect of the proposed algorithm is as Figure6.Figure 6
Tennis serve trajectory capture effect of the proposed algorithm.Analysis of Figure6 shows that the proposed algorithm can effectively capture the trajectory of the tennis serve and can locate the key points of the tennis serve trajectory. It can be seen that the proposed algorithm has a better effect in capturing the trajectory of the tennis serve.


### 4.3. Accuracy of Tennis Serve Trajectory Line Capture

In order to verify the accuracy of the tennis serve trajectory line capture of the proposed algorithm, the trajectory capture accuracy is taken as the evaluation index. The greater the trajectory capture accuracy, the higher the accuracy of the tennis serve trajectory line capture of the method. The calculation method is as follows:(37)Zq=λzσz×100%.In formula (37), λz is expressed as capturing the correct tennis service action trajectory, and σz is expressed as capturing the tennis service action trajectory. The algorithm of Li and Huang [4], the algorithm of Piao and Kim [5], and the proposed algorithm are used to capture the tennis serve trajectory line, and the comparison results of the capture accuracy of tennis serve trajectory line of different methods are obtained, as shown in Figure 7.Figure 7
Comparison results of the accuracy rate of tennis serve movement trajectory capture by different methods.It can be seen from the analysis of Figure7 that when the collection sample set of tennis service action images is 400, the average accuracy of tennis service action trajectory line capture of the algorithm of Li and Huang [4] is 81.2%, the average accuracy of tennis service action trajectory line capture of the algorithm of Piao and Kim [5] is 86.1%, and the average accuracy of tennis service action trajectory line capture of the proposed algorithm is as high as 95%. It can be seen that the tennis serve trajectory line capture accuracy of the proposed algorithm is high, which shows that the tennis serve trajectory line capture accuracy of the proposed algorithm is high.


### 4.4. Tennis Serve Trajectory Line Capture Efficiency

On this basis, the trajectory capture efficiency of the proposed algorithm is verified. Taking the trajectory capture time as the evaluation index, the shorter the trajectory capture time, the higher the trajectory capture efficiency of the proposed method. The algorithm of Li and Huang [4], the algorithm of Piao and Kim [5], and the proposed algorithm are used to capture the trajectory of tennis serve, and the comparison results of capture time of tennis serve trajectory of different methods are obtained, as shown in Table 1.Table 1
Comparison results of different methods of tennis serve movement trajectory capture time.
Tennis serve action image collection sample set/pieceThe proposed algorithm/sThe algorithm of Li and Huang [4]/sThe algorithm of Piao and Kim [5]/s1005.27.611.12008.310.414.830010.215.619.540013.618.823.6According to the data in Table1, with the increase of tennis serve motion image acquisition sample set, the tennis serve motion trajectory capture time of different methods increases. When the collection sample set of tennis service action images is 400, the tennis service action trajectory line capture time of the algorithm of Li and Huang [4] is 18.8s, the tennis service action trajectory line capture time of the algorithm of Piao and Kim [5] is 23.6s, whereas the tennis service action trajectory line capture time of the proposed algorithm is only 13.6s. Therefore, the tennis service trajectory capture time of the proposed algorithm is short, which can effectively improve the tennis service trajectory capture efficiency.


## 4.1. Setting the Experimental Environment

In order to verify the effectiveness of the tennis serve trajectory capture algorithm based on wavelet multiscale decomposition, MATLAB simulation software is used as the experimental platform for simulation experiments. Set the frequency of image acquisition and scanning of tennis serve trajectory line as 16 kHz, and the sample set of tennis serve image acquisition is 400 images. According to the above simulation environment and setting parameters, the computer three-dimensional vision of tennis serve trajectory image is collected, and the original image acquisition results are obtained, as shown in Figure5.Figure 5
Image collection result of original tennis serve action.


## 4.2. Tennis Serve Trajectory Line Capture Effect

In order to verify the capture effect of the tennis serve trajectory line of the proposed algorithm, the receive function in MATLAB simulation software is used to collect the tennis serve trajectory line image. The proposed algorithm is used to filter the image acquisition results of the original tennis service action, so as to realize the tennis service action trajectory capture, and the tennis service action trajectory capture effect of the proposed algorithm is as Figure6.Figure 6
Tennis serve trajectory capture effect of the proposed algorithm.Analysis of Figure6 shows that the proposed algorithm can effectively capture the trajectory of the tennis serve and can locate the key points of the tennis serve trajectory. It can be seen that the proposed algorithm has a better effect in capturing the trajectory of the tennis serve.


## 4.3. Accuracy of Tennis Serve Trajectory Line Capture

In order to verify the accuracy of the tennis serve trajectory line capture of the proposed algorithm, the trajectory capture accuracy is taken as the evaluation index. The greater the trajectory capture accuracy, the higher the accuracy of the tennis serve trajectory line capture of the method. The calculation method is as follows:(37)Zq=λzσz×100%.In formula (37), λz is expressed as capturing the correct tennis service action trajectory, and σz is expressed as capturing the tennis service action trajectory. The algorithm of Li and Huang [4], the algorithm of Piao and Kim [5], and the proposed algorithm are used to capture the tennis serve trajectory line, and the comparison results of the capture accuracy of tennis serve trajectory line of different methods are obtained, as shown in Figure 7.Figure 7
Comparison results of the accuracy rate of tennis serve movement trajectory capture by different methods.It can be seen from the analysis of Figure7 that when the collection sample set of tennis service action images is 400, the average accuracy of tennis service action trajectory line capture of the algorithm of Li and Huang [4] is 81.2%, the average accuracy of tennis service action trajectory line capture of the algorithm of Piao and Kim [5] is 86.1%, and the average accuracy of tennis service action trajectory line capture of the proposed algorithm is as high as 95%. It can be seen that the tennis serve trajectory line capture accuracy of the proposed algorithm is high, which shows that the tennis serve trajectory line capture accuracy of the proposed algorithm is high.


## 4.4. Tennis Serve Trajectory Line Capture Efficiency

On this basis, the trajectory capture efficiency of the proposed algorithm is verified. Taking the trajectory capture time as the evaluation index, the shorter the trajectory capture time, the higher the trajectory capture efficiency of the proposed method. The algorithm of Li and Huang [4], the algorithm of Piao and Kim [5], and the proposed algorithm are used to capture the trajectory of tennis serve, and the comparison results of capture time of tennis serve trajectory of different methods are obtained, as shown in Table 1.Table 1
Comparison results of different methods of tennis serve movement trajectory capture time.
Tennis serve action image collection sample set/pieceThe proposed algorithm/sThe algorithm of Li and Huang [4]/sThe algorithm of Piao and Kim [5]/s1005.27.611.12008.310.414.830010.215.619.540013.618.823.6According to the data in Table1, with the increase of tennis serve motion image acquisition sample set, the tennis serve motion trajectory capture time of different methods increases. When the collection sample set of tennis service action images is 400, the tennis service action trajectory line capture time of the algorithm of Li and Huang [4] is 18.8s, the tennis service action trajectory line capture time of the algorithm of Piao and Kim [5] is 23.6s, whereas the tennis service action trajectory line capture time of the proposed algorithm is only 13.6s. Therefore, the tennis service trajectory capture time of the proposed algorithm is short, which can effectively improve the tennis service trajectory capture efficiency.


## 5. Conclusion

The tennis serve trajectory capture algorithm based on wavelet multi-scale decomposition studied in this article can give full play to the advantages of wavelet multiscale decomposition algorithm, effectively improve the accuracy and efficiency of tennis serve trajectory capture, and ensure the effect of tennis serve trajectory capture. The experimental results show that the proposed algorithm can identify the trajectory of tennis serve in different stages with high accuracy, which provides a reliable theoretical support for further research. However, this method does not consider the influence of the complexity of the algorithm on the trajectory capture of tennis service. Therefore, in our future work, the main purpose will be to reduce the complexity of the algorithm, simplify the algorithm, and reduce the amount of calculation, so as to effectively reduce the complexity of the algorithm. [17].



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*Source: 8290282-2022-04-13.xml*