The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm

Combined with the analysis of the system structure diagram shown in Figure 1, it can be seen that the three-part animation system designed and analysed by using virtual reality technology is mainly divided into four parts: First, the perception module. This part refers to the acquisition of simulated information of virtual characters according to different virtual scenes, and the conversion of the selected props and the data of nearby characters into perceptual information, so as to provide services for the emotional module and behaviour module. Second one the emotional module [2]. This part refers to including the characters’ customised emotions in the plot description, and setting up personalised emotion transformation strategy, so as to improve the rendering effect of 3D animation characters while changing emotions in time. Third is the behaviour module. This part refers to the behaviour selection ability of animation characters. Their performance ability and reaction ability during sports are mainly affected by the design level of behaviour modules. Fourth is three animation generation module. This part is the core content of the operation of 3D animation system, which involves motion control, scene modelling, character modelling and other aspects of the content. From the overall perspective, the flow chart of 3D animation modelling is shown in Figure 2 [3].

Fig. 1

Fig. 2

Conjecture 5. After studying all the texture mapping processes, it can be seen that the image size can be designed as m × n, which can simplify the above formula. The specific process is as follows: [uv1]T=[1/m0001/n0001][100s010t0011][xyz]T=[1/m00s/m01/n0t/n0001][xyz1] \matrix{{{{\left[ {uv1} \right]}^T} = \left[ {\matrix{{1/m00} \hfill \cr {01/n0} \hfill \cr {001} \hfill \cr}} \right]\left[ {\matrix{{100s} \hfill \cr {010t} \hfill \cr {0011} \hfill \cr}} \right]{{\left[ {xyz} \right]}^T} = \left[ {\matrix{{1/m00s/m} \hfill \cr {01/n0t/n} \hfill \cr {0001} \hfill \cr}} \right]\left[ {xyz1} \right]} \hfill \cr}

In the above formula, s and t represent the unknown parameters, and (x, y, z) represent the vertex coordinates of the surface. At the same time, only one set of feature points is needed in the formula to calculate and obtain unknown parameters. In the process of texture acquisition, all textures are the minimum bounding box, and there are multiple tangential points on the edge of the image, so the corresponding feature points can be obtained quickly.

Assume that the texture coordinates (u’, v’) and the vertex coordinates (x’, y’, z’) of the surface belong to the same set of feature points; then using them into the above formula to obtain: [uv1]T=[1/m00(u′−x′)/m01/n0(v′−y′)/n0001]⋅[xyz1]T \matrix{{{{\left[ {uv1} \right]}^T} = \left[ {\matrix{{1/m00\left({u' - x'} \right)/m} \hfill \cr {01/n0\left({v' - y'} \right)/n} \hfill \cr {0001} \hfill \cr}} \right] \cdot} \hfill \cr {{{\left[ {xyz1} \right]}^T}} \hfill \cr}

Note 7. In the above formula, I_ij represents the light intensity obtained by the connection of all pixel points in a two-dimensional image. Such algorithms are mainly used in image processing without background noise, or under the condition of the same background noise or too high signal-to-noise ratio.

Second, the strongly weighted centroid algorithm: xc=∑j=y0−w0,y/2y0+w0,y/2∑i=x0−w0,x/2x0+w0,x/2xiIijw∑j=y0−w0,y/2y0+w0,y/2∑i=x0−w0,x/2x0+w0,x/2Iijw {x_c} = {{\sum\limits_{j = {y_0} - {w_0},y/2}^{{y_0} + {w_0},y/2} \sum\limits_{i = {x_0} - {w_0},x/2}^{{x_0} + {w_0},x/2} {x_i}{I_{ij}}w} \over {\sum\limits_{j = {y_0} - {w_0},y/2}^{{y_0} + {w_0},y/2} \sum\limits_{i = {x_0} - {w_0},x/2}^{{x_0} + {w_0},x/2} {I_{ij}}w}}

In this kind of algorithm, three forms of weighting function are mainly involved, as shown below: yc=∑i=x0−W0, x / 2 x0+W0, x / 2 ∑j=y0−W0, y / 2 y0+W0, y / 2 yjIijw∑i=x0−W0, x / 2 x0+W0, x / 2 ∑j=y0−W0, y / 2 y0+W0, y / 2 Iijw {y_c} = {{\sum\limits_{i = {x_0} - {W_0},{\kern 1pt} x{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}}^{{x_0} + {W_0},{\kern 1pt} x{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}} \sum\limits_{j = {y_0} - {W_0},{\kern 1pt} y{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}}^{{y_0} + {W_0},{\kern 1pt} y{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}} {y_j}{I_{ij}}w} \over {\sum\limits_{i = {x_0} - {W_0},{\kern 1pt} x{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}}^{{x_0} + {W_0},{\kern 1pt} x{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}} \sum\limits_{j = {y_0} - {W_0},{\kern 1pt} y{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}}^{{y_0} + {W_0},{\kern 1pt} y{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}} {I_{ij}}w}}

Open Problem 8. Where, A and P represent strength values. Generally speaking, the above algorithm is mainly used to analyse and detect the centroid of light spot.

Third, distance centroid algorithm. xc=∑ijxiIijWij∑ijIijWijyc*=∑ijyijIijWij∑ijIijWijWij=1s=1(xi−xc)2+(yj−yc)2 \matrix{{{x_c} = {{\sum\limits_{ij} {x_i}{I_{ij}}{W_{ij}}} \over {\sum\limits_{ij} {I_{ij}}{W_{ij}}}}} \hfill \cr {y_c^* = {{\sum\limits_{ij} {y_{ij}}{I_{ij}}{W_{ij}}} \over {\sum\limits_{ij} {I_{ij}}{W_{ij}}}}} \hfill \cr {{W_{ij}} = {1 \over s} = {1 \over {\sqrt {{{\left({{x_i} - {x_c}} \right)}^2} + {{\left({{y_j} - {y_c}} \right)}^2}}}}} \hfill \cr}

In the above formula, (x_i, y_j) represents the coordinates of the detected pixel, (x_c, y_c) represents the central coordinates of the spot, ( xc* x_c^* , yc* y_c^* ) refers to the centroid coordinates of the spot obtained by calculation, and I_ij refers to the value of the current pixel.

The image centroid is the centre of gravity of the image grey level. Assume that the image has two directions of I and j, then m and n represent the number of pixels in the two directions of I and j, respectively, and g (I, j) represents the grey value of the region where the pixel point (I, j) is located. Then the coordinate expression of the corresponding image centroid position is as follows: x=∑j=1n∑i=1mg(i,j)×i∑j=1n∑i=1mg(i,j)y=∑j=1n∑i=1mg(i,j)×j∑j=1n∑i=1mg(i,j) \matrix{{x = {{\sum\limits_{j = 1}^n \sum\limits_{i = 1}^m g\left({i,j} \right) \times i} \over {\sum\limits_{j = 1}^n \sum\limits_{i = 1}^m g\left({i,j} \right)}}} \hfill \cr {y = {{\sum\limits_{j = 1}^n \sum\limits_{i = 1}^m g\left({i,j} \right) \times j} \over {\sum\limits_{j = 1}^n \sum\limits_{i = 1}^m g\left({i,j} \right)}}} \hfill \cr}

Fourth is the traditional centroid algorithm. Assuming that the coordinate of the pixel in row I’s column j is (x_i, y_j) and the grey value is G (x_i, y_j), then the centroid of its star can be expressed as: x^=∑i=i1i2∑j=j1j2xiG(xi,yj)∑i=i1i2∑j=j1j2G(xi,yi) \hat x = {{\sum\limits_{i = {i_1}}^{{i_2}} \sum\limits_{j = {j_1}}^{{j_2}} {x_i}G\left({{x_i},{y_j}} \right)} \over {\sum\limits_{i = {i_1}}^{{i_2}} \sum\limits_{j = {j_1}}^{{j_2}} G\left({{x_i},{y_i}} \right)}}

In previous research on the positioning accuracy of the algorithm, let, and because the grey value of the star pixel has noise, it can be obtained as follows:

In the above formula, S I represents the grey value of the signal, N and I represents the grey value of the noise. By integrating the above formulas, we can get: x^=∑i=i1i2xi(Si+Ni)∑i=i1i2(Si+Ni)=∑i=i1i2xiSi∑i=i1i2Si(1−∑i=i1i2Ni∑i=i1i2(Si+Ni))+∑i=i1i2xiNi∑i=i1i2(Si+Ni)=x¯(1+η1)+η2 \hat x = {{\sum\limits_{i = {i_1}}^{{i_2}} {x_i}\left({{S_i} + {N_i}} \right)} \over {\sum\limits_{i = {i_1}}^{{i_2}} \left({{S_i} + {N_i}} \right)}} = {{\sum\limits_{i = {i_1}}^{{i_2}} {x_i}{S_i}} \over {\sum\limits_{i = {i_1}}^{{i_2}} {S_i}}}\left({1 - {{\sum\limits_{i = {i_1}}^{{i_2}} {N_i}} \over {\sum\limits_{i = {i_1}}^{{i_2}} \left({{S_i} + {N_i}} \right)}}} \right) + {{\sum\limits_{i = {i_1}}^{{i_2}} {x_i}{N_i}} \over {\sum\limits_{i = {i_1}}^{{i_2}} \left({{S_i} + {N_i}} \right)}} = \bar x\left({1 + {\eta _1}} \right) + {\eta _2}

And the above formula meets the following conditions: x¯=∑i=i1i2xiSi∑i=i1i2Si,η1=−∑i=i1i2Ni∑i=i1i2(Si+Ni),η2=∑i=i1i2xiNi∑i=i1i2(Si+Ni) \bar x = {{\sum\limits_{i = {i_1}}^{{i_2}} {x_i}{S_i}} \over {\sum\limits_{i = {i_1}}^{{i_2}} {S_i}}},{\eta _1} = - {{\sum\limits_{i = {i_1}}^{{i_2}} {N_i}} \over {\sum\limits_{i = {i_1}}^{{i_2}} \left({{S_i} + {N_i}} \right)}},{\eta _2} = {{\sum\limits_{i = {i_1}}^{{i_2}} {x_i}{N_i}} \over {\sum\limits_{i = {i_1}}^{{i_2}} \left({{S_i} + {N_i}} \right)}}

When calculating median filtering, the specific formula is as follows: g(m,n)=Median{f(m−k,n−1),(k,l)∈W} g\left({m,n} \right) = Median\left\{{f\left({m - k,n - 1} \right),\left({k,l} \right) \in W} \right\}

The corresponding weighted median filter is: yi=Weighted_Med(xi−1,xi,xi+1) {y_i} = Weighted\_Med\left({{x_{i - 1}},{x_i},{x_{i + 1}}} \right)

From a physical perspective, the centroid refers to a three-dimensional manifold W in an Euclidean space. The coordinates of its centroid C can be calculated by using the weighted average coordinates of all internal points. The specific formula is as follows: c=∫Ωρ(x)xdσ∫Ωρ(x)dσ c = {{\int_\Omega \rho \left(x \right)xd\sigma} \over {\int_\Omega \rho \left(x \right)d\sigma}}

In the above formula, P represents the density distribution function in the three-dimensional manifold, so it can be clear that its centroid is consistent with the minimised moment of inertia. The specific formula is as follows: c=argminy∫Ωρ(x)‖x−y‖2dσ c = \mathop {\arg \min}\limits_y \int_\Omega \rho \left(x \right){\left\| {x - y} \right\|^2}d\sigma

Therefore, in the algorithm analysis, the corresponding definition formula can be obtained by replacing the Euclidean distance with the internal distance of the three-dimensional manifold: c=argminy∫Ωρ(x)dg2(x,y)dσ c = \mathop {\arg \min}\limits_y \int_\Omega \rho \left(x \right)d_g^2\left({x,y} \right)d\sigma

In the above formula, c represents the region where the centre of mass is located, p (x) represents the density of point x and dg (x, y) represents the internal distance between point x and point y. The corresponding energy function is: E(y)=∫Ωρ(x)dg2(x,y)dσ E\left(y \right) = \int_\Omega \rho \left(x \right)d_g^2\left({x,y} \right)d\sigma

Its gradient formula can be expressed as: ∂E∂y=2∫Ωρ(x)dg(x,y)∂dg(x,y)∂ydσ {{\partial E} \over {\partial y}} = 2\int_\Omega \rho \left(x \right){d_g}\left({x,y} \right){{\partial {d_g}\left({x,y} \right)} \over {\partial y}}d\sigma

When calculating and analysing the geometric internal distance, the heat conduction equation can be used to study. Suppose that the solution of the equation representing heat conduction, where T refers to the time passed, and X and and respectively represent two points in the space, then the corresponding equation is: ∂μ∂t=12Δμ {{\partial \mu} \over {\partial t}} = {1 \over 2}\Delta \mu

Combined with the theoretical analysis proposed by Varadhan, it can be known that there is the following relationship between the distance d at two points in Euclidean space and the heat transferred through time t: μ(t,x,y)=(2π)−k2ed2(x,y)−2t,t→0. \mu \left({t,x,y} \right) = {\left({2\pi} \right)^{{{- k} \over 2}}}{e^{{{{d^2}\left({x,y} \right)} \over {- 2t}}}},t \to 0.

In the above formula, K represents the dimension of the region.

Next, we need to calculate and analyse the discretisation of 3D manifolds and the discretisation of differential operators. Taking the latter as an example, if f is a scalar function defined inside the geometry, then in the discrete case, f can be expressed as f = (f₁, f₂. . . . F_n), and f_i represents the value of the function at the i-th vertex. It can be seen that the gradient of function f can be calculated by using the function value of tetrahedral vertices as shown below: (∇f)i=13Vi∑jSjfjnj {(\nabla f)_i} = {1 \over {3{V_i}}}\sum\limits_j {S_j}{f_j}{n_j}

In the above formula, V_i represents the area of tetrahedron I, f_j represents the function value of vertex j and S_j and n_j represent the area of the triangle corresponding to the tetrahedron and vertex and its unit normal vector, respectively. In essence, discretisation refers to the divergence of the vector field. Suppose g = (g₁,g₂,...,g_m) represents the piecewise linear vector field within the tetrahedron grid, where m represents the number of tetrahedron in the grid, and the divergence formula for the vertex i is: (∇⋅g)i=∑jSj3Vjnj⋅Vj⋅gj⋅ {\left({\nabla \cdot g} \right)_i} = \sum\limits_j {{{S_j}} \over {3{V_j}}}{n_j} \cdot {V_j} \cdot {g_j} \cdot