Image Segmentation of the Continuous Action of Spiking in Volleyball Based on Spatial Neighborhood Information

To obtain the neighborhood correlation images on the spiking action of volleyball players, a 3×3 dynamic window was opened at the position of spiking in two time-phase images. The correlation coefficient, slope, and intercept are denoted by χ, κ, and ζ, respectively. The covariance of the brightness in all bands of the window is denoted by BBC. The standard deviations of the brightness of all bands in the windows corresponding to the previous and current time phases are denoted by r₁ and r₂, respectively. The brightness of the l-th pixel in the windows corresponding to the previous and current time phases are denoted by YU_l,1 and YU_l,2, respectively; the total number of pixels in all bands of the window is denoted by m; the mean brightness of all bands in the windows corresponding to the previous and current time phases are denoted by v₁ and v₂, respectively. Then, the χ, κ, and ζ of the neighborhood corresponding to each window can be computed by:

$BBC=\frac{\sum\limits_{l=1}^{m}{\left( Y{{U}_{l,1}}-{{v}_{1}} \right)\left( Y{{U}_{l,2}}-{{v}_{2}} \right)}}{m-1}$     (1)

$\chi =\frac{cov}{{{r}_{1}}{{r}_{2}}}$     (2)

$\kappa =\frac{cov}{r_{1}^{2}}$     (3)

$\zeta =\frac{\sum\limits_{l=1}^{m}{Y{{U}_{l,2}}}-\kappa \sum\limits_{l=1}^{m}{Y{{U}_{l,1}}}}{m}$     (4)

The three values are assigned to the center position of the dynamic window. Then, it is possible to create the neighborhood correlation image around the center of the dynamic window.

This paper treats χ and κ as their difference from 1, and visualizes ζ as its difference from 0. In this way, the dynamic parts and constant parts of the continuous action image on volleyball spiking can be represented by bright pixels and dark pixels, respectively.

This paper optimizes the FCM to effectively segment the images on the continuous action of volleyball spiking. The optimized FCM employs the mean shift algorithm to pinpoint the densest sample information in the image feature space. Based on the neighborhood correlation image acquired in the preceding section, the weight of each neighborhood image block is solved. Then, the detailed information of continuous images is obtained by weighted fuzzy factor. Figure 3 shows the framework of the image segmentation model.

3.png

Figure 3. Framework of image segmentation model

Suppose the image on the continuous action of volleyball spiking A={a₁,a₂,...,a_m} contains m pixels. Let f be the bandwidth of the gray value domain of the image; l'(a) be the gradient of the kernel function, which satisfies h(a)=-l'(a). Then, the mean shift vector of the mean shift algorithm can be calculated by:

${{N}_{H}}\left( a \right)=\frac{\sum\limits_{i=1}^{m}{{{a}_{i}}h\left( {{\left\| \left( a-{{a}_{i}} \right)/f \right\|}^{2}} \right)}}{\sum\limits_{i=1}^{m}{h\left( {{\left\| \left( a-{{a}_{i}} \right)/f \right\|}^{2}} \right)}}-a$     (7)

The new shift center n_H(a) can be calculated by:

${{n}_{H}}\left( a \right)=\frac{\sum\limits_{i=1}^{m}{{{a}_{i}}h\left( {{\left\| \left( a-{{a}_{i}} \right)/f \right\|}^{2}} \right)}}{\sum\limits_{i=1}^{m}{h\left( {{\left\| \left( a-{{a}_{i}} \right)/f \right\|}^{2}} \right)}}$     (8)

For the image on the continuous action of volleyball spiking, the neighborhood image blocks corresponding to the neighborhood correlation images contain much information about the neighborhood, and mirror the features of the continuous action of spiking. For the image on the continuous action of volleyball spiking A={a₁,a₂,...,a_m}, every pixel a_i has t_i={T_iw,w∈M_i} neighborhood image blocks, where M_i is the s-order neighborhood of a_i. Due to the presence of many noises and abnormal points in video images, the pixels in the neighborhood image blocks formed by different neighborhood correlation images were assigned different weights.

Let a_w be the center pixel in the image t_i on the continuous action of volleyball spiking; a_w' be the other pixels in each neighborhood image block; m_i be the number of pixels in a neighborhood image block; t_i be the neighborhood image block corresponding to a_i. Then, the variance of pixels in t_i can be calculated by:

${{\xi }_{iw}}={{\left[ \frac{\sum\limits_{w'\in {{M}_{i}}\backslash \left\{ w \right\}}{{{\left( {{a}_{w'}}-{{a}_{w}} \right)}^{2}}}}{{{m}_{i}}-1} \right]}^{\frac{1}{2}}}$      (9)

By Gaussian kernel function, the variance is mapped to the kernel space:

${{\delta }_{iw}}=exp\left[ -\left( {{\xi }_{iw}}-\frac{\sum\limits_{w\in {{M}_{i}}}{{{ \quad \xi }_{iw}}}}{{{m}_{i}}} \right) \right]$     (10)

The weights of all pixels in the neighborhood image block are normalized by:

${{\eta }_{iw}}=\frac{{{\delta }_{iw}}}{\sum\limits_{w\in {{M}_{i}}}{{{\delta }_{iw}}}}$     (11)

The traditional image segmentation algorithm has a low segmentation accuracy, because video images contain some nonnegligible noises and abnormal points. To solve the problem, this paper relies on the weighted fuzzy factor to characterize the information in the image on the continuous action of volleyball spiking. Let v_lj be the fuzzy membership matrix; u_l be the cluster center; η_ij be the weight. For any image pixel a_i, the weighted fuzzy factor can be calculated by:

${{H}_{li}}=\sum\limits_{j\in MY\left( {{a}_{i}} \right),j\ne i}{{{\eta }_{ij}}{{\left( 1-{{v}_{lj}} \right)}^{n}}}{{\left\| {{a}_{i}}-{{u}_{l}} \right\|}^{2}}$      (12)

The value of η_ij covers two parts: spatial distance weight and spatial intensity weight. By fully utilizing the two weights, the spatial information in the image on the continuous action of volleyball spiking could be realized, which contributes to the robustness of the image segmentation algorithm.

The spatial distance weight of η_ij represents the location constraint of any image pixel a_i. Let c_ij^r be the Euclidean distance between pixel a_i and neighborhood pixel a_j. Then, the spatial distance weight can be calculated by:

${{\eta }_{rc}}=\frac{1}{{{c}^{r}}+1}$     (13)

Almost all the information of the center pixel can be represented by the s-order neighborhood of the pixels in the image on the continuous action of volleyball spiking. Therefore, this paper determines the spatial distance weight based on the local information of the image.

The spatial intensity weight of η_ij is defined as the gray value ratio of pixels in the neighborhood of any image pixel a_i. Let a_i and a_j denote a pixel in the image on the continuous action of volleyball spiking, and a neighborhood pixel of that pixel, respectively. Then, the s-order neighborhoods MY_i and MY_j of a_i and a_j can be selected to define the spatial intensity distance of pixel a_i as:

$c_{ij}^{I}=\frac{1}{{{s}^{2}}}\sum\limits_{l=1}^{{{s}^{2}}}{\frac{{{I}_{M{{Y}_{i}}}}\left( l \right)}{{{I}_{M{{Y}_{j}}}}\left( l \right)}}$     (14)

The spatial intensity weight is mapped to the natural logarithm space:

${{\eta }_{ic}}=1-log\left( c_{ij}^{I} \right)$     (15)

The weight of the weighted fuzzy factor can be defined as:

${{\eta }_{ij}}={{\eta }_{rc}}\cdot {{\eta }_{ic}}$     (16)

Let v_li be the membership matrix; n be the fuzzy parameter; η_is be the weight of the neighborhood image block corresponding to pixel a_i; SU_is be a pixel in the neighborhood image block corresponding to pixel a_i; u_l be the cluster center; H_li be the weighted fuzzy factor. Based on H_li, the objective function of our image segmentation algorithm can be defined as follows:

${{J}_{MSIFCM}}=\sum\limits_{i=1}^{m}{\sum\limits_{l=1}^{d}{v_{li}^{n}\sum\limits_{s\in {{M}_{i}}}{{{\eta }_{is}}{{\left\| S{{U}_{is}}-{{u}_{l}} \right\|}^{2}}}+{{H}_{li}}}}$     (17)

The objective function is minimized by the Lagrange multiplier method. Let MY_i be the set of neighborhood pixels of pixel a_i; v'_li be the smoothed membership function; ψ_ij∈ [0,1] be the neighborhood pixel weight coefficient controlling the influence of neighborhood pixels on the center pixel; v_li be the updated membership. Then, the membership can be updated by:

${{v}_{li}}={{\left\{ \sum\limits_{k=1}^{d}{\left[ {{\left( \frac{\sum\limits_{s\in {{M}_{i}}}{{{\eta }_{is}}{{\left( S{{U}_{is}}-{{u}_{l}} \right)}^{2}}+{{H}_{li}}}}{\sum\limits_{s\in {{M}_{i}}}{{{\eta }_{is}}{{\left( S{{U}_{is}}-{{u}_{j}} \right)}^{2}}+{{H}_{ji}}}} \right)}^{\frac{1}{n-1}}} \right]} \right\}}^{-1}}$      (18)

The cluster center can be updated by:

${{u}_{l}}=\frac{\sum\limits_{i=1}^{m}{v_{li}^{n}\sum\limits_{s\in {{M}_{i}}}{{{\eta }_{is}}S{{U}_{is}}}}}{\sum\limits_{i=1}^{m}{v_{li}^{n}\sum\limits_{s\in {{M}_{i}}}{{{\eta }_{is}}}}}$     (19)

The membership can be smoothened by:

$v_{li}^{'}=\sum\limits_{j\in {{M}_{i}}}{{{\psi }_{ij}}{{v}_{li}}}$     (20)

Let a_av and ξ_i be the mean and variance of the neighborhood pixels of pixel a_i, respectively; UC(a_i,a_j) be the control coefficient. Then, the weight coefficient of each neighborhood pixel can be calculated by:

${{\psi }_{ij}}=\left\{ \begin{align}  & UC\left( {{a}_{i}},{{a}_{j}} \right),\left| {{a}_{i}}-{{a}_{av}} \right|<{{\xi }_{i}} \\ & 0,\left| {{a}_{i}}-{{a}_{av}} \right|>{{\xi }_{i}} \\\end{align} \right.$     (21)

The segmentation effect is measured by Euclidean distance. The control coefficient UC(a_i,a_j) can be calculated by:

$UC\left( {{a}_{i}},{{a}_{j}} \right)=\frac{c\left( {{a}_{i}},{{a}_{j}} \right)}{\sum\limits_{k\in {{M}_{i}}}{c\left( {{a}_{i}},{{a}_{j}} \right)}}$     (22)

The distance from the center pixel a_i to a neighborhood pixel a_j is negatively correlated with the influence of a_j over a_i. Hence, the control coefficient UC(a_i,a_j) is defined as the Euclidean distance ratio between the two pixels. The closer the UC(a_i,a_j) is to one, the stronger the correlation between a_j and a_i. The further the UC(a_i,a_j) is from one, the weaker the correlation. Based on Euclidean distance, the above differentiation of pixel correlation effectively clarifies the degree of impacts of each pixel in the image on the continuous action of volleyball spiking over the weight coefficient, making the clustering of the continuous action of volleyball spiking more accurate.