Educational reform informatisation based on fractional differential equation

In 1990, Perona and Malik proposed a nonlinear anisotropic diffusion equation, namely the P-M model, in order to maintain image edge information. The definitions are as follows: (1) {∂u(x,y,t)∂t=div(g(|∇u|)|∇u|),u(x,y,0)=u0(x,y), (x,y)∈Ω \left\{ {\matrix{ {{{\partial u\left( {x,y,t} \right)} \over {\partial t}} = div\left( {g\left( {\left| {\nabla u} \right|} \right)\left| {\nabla u} \right|} \right),} \hfill\cr {u\left( {x,y,0} \right) = {u_0}\left( {x,y} \right),} \hfill\cr } } \right.\quad \left( {x,y} \right) \in \Omega In Eq. (1), ∇ is the gradient operator; div is the divergence operator; ∇u is the image gradient modulus value; u₀ represents the initial noisy image; Ω is the area contained in the experimental image; and the diffusion coefficient g(|∇u|) is a function of |∇u|, the expressions of anisotropic diffusion coefficient g(|∇u|) 2 proposed by Perona and Malik are as follows: (2) g1(|∇u|)=exp((−|∇u|/K)2) {g_1}\left( {\left| {\nabla u} \right|} \right) = \exp \left( {{{\left( { - \left| {\nabla u} \right|/K} \right)}^2}} \right) (3) g2(|∇u|)=11+(−|∇u|/K)2 {g_2}\left( {\left| {\nabla u} \right|} \right) = {1 \over {1 + {{\left( { - \left| {\nabla u} \right|/K} \right)}^2}}} wherein K is the gradient threshold. The P-M model diffuses the noisy image by the size of the gradient. The |∇u| > K, g(|∇u|) → 0 diffusion stops; the |∇u| < K, g(|∇u|) → 1, corresponds to smooth filtering.

In 2010, a new edge detection operator, namely the differential curvature operator, was proposed. The expression is as follows: (4) D=||uηη|−|uξξ|| D = \left| {\left| {{u_{\eta \eta }}} \right| - \left| {{u_{\xi \xi }}} \right|} \right|

In Eq. (4), U ξ and U ηη, respectively, represent the second derivatives of the image in the horizontal and gradient directions. The unit vectors of horizontal and gradient directions are as follows: (5) ξ=[−uy,ux]ux2+uy2 and η=[ux,uy]ux2+uy2 \xi = {{\left[ { - {u_y},{u_x}} \right]} \over {\sqrt {u_{^x}^2 + u_y^2} }}\quad {\rm{and}}\quad \eta = {{\left[ {{u_x},{u_y}} \right]} \over {\sqrt {u_{^x}^2 + u_y^2} }} (6) uξξ=uxxuy2−2uxyuxuy+uyyux2ux2+uy2,uηη=uxxux2−2uxyuxuy+uyyuy2ux2+uy2 \matrix{ {{u_{\xi \xi }} = {{{u_{xx}}u_y^2 - 2{u_{xy}}{u_x}{u_y} + {u_{yy}}u_x^2} \over {u_x^2 + u_y^2}},} \hfill \cr {{u_{\eta \eta }} = {{{u_{xx}}u_x^2 - 2{u_{xy}}{u_x}{u_y} + {u_{yy}}u_y^2} \over {u_x^2 + u_y^2}}} \hfill \cr } Among them (7) ux(i,j)=ui+1,j−ui−1,j2,uy(i,j)=ui,j+1−ui,j−12,uxx(i,j)=ui+1,j−2ui,j+ui−1,j,uyy(i,j)=ui,j+1−2ui,j+ui,j−1,uxy(i,j)=(ui−1,j−1++ui+1,j+1)−(ui−1,j+1++ui+1,j−1)4 \matrix{ {{u_x}\left( {i,j} \right) = {{{u_{i + 1,j}} - {u_{i - 1,j}}} \over 2},} \hfill \cr {{u_y}\left( {i,j} \right) = {{{u_{i,j + 1}} - {u_{i,j - 1}}} \over 2},} \hfill \cr {{u_{xx}}\left( {i,j} \right) = {u_{i + 1,j}} - 2{u_{i,j}} + {u_{i - 1,j}},} \hfill \cr {{u_{yy}}\left( {i,j} \right) = {u_{i,j + 1}} - 2{u_{i,j}} + {u_{i,j - 1}},} \hfill \cr {{u_{xy}}\left( {i,j} \right) = {{\left( {{u_{i - 1,j - 1}} + + {u_{i + 1,j + 1}}} \right) - \left( {{u_{i - 1,j + 1}} + + {u_{i + 1,j - 1}}} \right)} \over 4}} \hfill \cr } Type (7): u_i,j represents the value of the function u at the node (I, j), u_i+1,j, u_i−1,j, u_i,j+1, u_i,j−1, something like the definition of u_i,j, u_i,j represents the value of the first partial derivative of u with respect to x at the node (I, j), u_y (i, j), u_xx (i, j), u_yy (i, j), u_xy (i, j), something like the definition of u_x (i, j).

According to Eq. (4), in the edge region of the image, if U ξ is smaller, uηη is larger, then the differential curvature D value will be larger. In the flat region of the image, if U ξ and uηη are both small, then D value is small; At the isolated noise points of the image, if U ξ and uηη are both large, almost equal, then D value is small. So depending on the difference curvature D value, better distinguish the flat areas, edges and noise points in the image [6, 7].

Based on the properties of differential curvature and fractional-order operator, the diffusion coefficients are proposed as follows: (8) g(∇u,d,v)=11+(Gσ|⋅(∇u+d)|vk)2 g\left( {\nabla u,d,v} \right) = {1 \over {1 + {{\left( {{{{G_\sigma }{{\left| { \cdot \left( {\nabla u + d} \right)} \right|}^v}} \over k}} \right)}^2}}} The new model can be obtained as: (9) {∂u(x,y,t)∂t=div(g(∇u,d,v)∇u),u(x,y,0)=u0(x,y),(x,y)∈Ω \left\{ {\matrix{ {{{\partial u\left( {x,y,t} \right)} \over {\partial t}} = div\left( {g\left( {\nabla u,d,v} \right)\nabla u} \right),} \hfill \cr {u\left( {x,y,0} \right) = {u_0}\left( {x,y} \right),} \hfill \cr } } \right.\left( {x,y} \right) \in \Omega Type (9): d is the normalised differential curvature, d=D−minDmaxD−minD d = {{D - \min D} \over {\max D - \min D}} , Gσ(x,y)=12πσ2exp(−x2+y22σ2) {G_\sigma }\left( {x,y} \right) = {1 \over {2\pi {\sigma ^2}}}\exp \left( { - {{{x^2} + {y^2}} \over {2{\sigma ^2}}}} \right) is the Gaussian kernel function, k is the threshold value, taking the constant.

In this paper, the order of fractions is an adaptive process related to local variance: (10) v=1+σx,y2−min(σx,y2)10×max(σx,y2) v = 1 + {{\sigma _{x,y}^2 - \min \left( {\sigma _{x,y}^2} \right)} \over {10 \times {\rm{max}}\left( {\sigma _{x,y}^2} \right)}} Type (10): σx,y2=1m1×m2∑xi,xj∈H(uxi′,yj2−μxi′,yj2) \sigma _{x,y}^2 = {1 \over {{m_1} \times {m_2}}}\sum\limits_{{x_i},{x_j} \in H} \left( {u_{x_i^\prime,{y_j}}^2 - \mu _{x_i^\prime,{y_j}}^2} \right) , μxi′,yj=1m1×m2∑xi′,xj′∈Huxi′,yj′ {\mu _{x_i^\prime,{y_j}}} = {1 \over {{m_1} \times {m_2}}}\sum\limits_{x_i^\prime,x_j^\prime \in H} {u_{x_i^\prime,y_j^\prime}} . Where, μxi′,yj {\mu _{x_i^\prime,{y_j}}} and σx,y2 \sigma _{x,y}^2 are the local mean and local variance of the noisy image u, respectively; H is a rectangular window of m₁ × m₂, centred on (x, y).

The time interval is set as Δt, the space step size is 1, (I, j) represents the discrete points in the image and the iterative formula after the dispersion of the model in this paper is as follows: (11) ui,jn+1=ui,jn+Δt4∑l=14(g((|∇ui,jn|)ldi,jn,v))(|∇ui,jn|)l \matrix{ {u_{i,j}^{n + 1} = u_{i,j}^n + {{\Delta t} \over 4}\sum\limits_{l = 1}^4 \left( {g\left( {{{\left( {\left| {\nabla u_{i,j}^n} \right|} \right)}^l}d_{i,j}^n,v} \right)} \right)} \hfill \cr {{{\left( {\left| {\nabla u_{i,j}^n} \right|} \right)}^l}} \hfill \cr } |∇ui,jn| \left| {\nabla u_{i,j}^n} \right| , (l=1, 2, 3, 4) represent the gradient of the four neighbourhood directions of upper, lower, left and right, respectively, namely: (12) {|∇ui,j1|=ui,j−1−ui,j|∇ui,j2|=ui,j+1−ui,j|∇ui,j3|=ui−1,j−ui,j|∇ui,j4|=ui+1,j−ui,j \left\{ {\matrix{ {\left| {\nabla u_{i,j}^1} \right| = {u_{i,j - 1}} - {u_{i,j}}} \hfill \cr {\left| {\nabla u_{i,j}^2} \right| = {u_{i,j + 1}} - {u_{i,j}}}\hfill \cr {\left| {\nabla u_{i,j}^3} \right| = {u_{i - 1,j}} - {u_{i,j}}} \hfill \cr {\left| {\nabla u_{i,j}^4} \right| = {u_{i + 1,j}} - {u_{i,j}}} \hfill \cr } } \right. The mean fractional order differential equation combined with multivariate mixed criterion is used to complete the information processing process [8, 9]. The image information to be processed is set as a matrix, which is named W, and the dimension of the information vector is set as C_i (i = 1, 2,...,n), and n is the number of vectors. For image expansion and initialisation processing, it can be known that its mean region is J_i, then: (13) ∑i=1wJi=1 \sum\limits_{i = 1}^w {J_i} = 1 Set the fuzzy centre as Z and the information number as L, then: (14) Zi=((∑i=1wJi)lCi)(∑i=1wJi)l {Z_i} = {{\left( {{{\left( {\sum\limits_{i = 1}^w {J_i}} \right)}^l}{C_i}} \right)} \over {{{\left( {\sum\limits_{i = 1}^w {J_i}} \right)}^l}}} The image blur processing centre is obtained through Eq. (14), the image information is processed with this as the reference point, and the processing results are obtained [10]. Setting the processing information as Y and H as the frame length of image information, then: (15) Y={(1H2(Ci,Zi))1l−1}{∑i=1w(1H2(Ci,Zi))1l−1} Y = {{\left\{ {{{\left( {{1 \over {{H^2}\left( {{C_i},{Z_i}} \right)}}} \right)}^{{1 \over {l - 1}}}}} \right\}} \over {\left\{ {\sum\limits_{i = 1}^w {{\left( {{1 \over {{H^2}\left( {{C_i},{Z_i}} \right)}}} \right)}^{{1 \over {l - 1}}}}} \right\}}}

(1) First, the second-order derivative U ξ and U ηη of the horizontal and gradient directions of the image are calculated using Eq. (6); (2) by substituting the values of U ξ and U ηη into Eq. (4), the differential curvature D of the image can be obtained, and then D can be obtained through normalisation processing; (3) the values of four directional gradients can be obtained from Eq. (12), and the adaptive fractional order can be obtained from Eq. (10); (4) substitute the values obtained in Steps (2) and (3) into Eq. (8) to obtain the value of diffusion coefficient; (5) finally, the image containing noise is processed according to Eq. (11). Repeat the above steps until better denoising image effect is achieved.

Image information contains a lot of information which needs to be processed in the construction of university education informatisation. The original power method of image information processing ability is poor, so the information collection work has become the basis to improve the processing efficiency, and set a comprehensive information collection method to help improve the information processing ability. In order to collect and preserve all the information in the informationisation construction without causing the loss of information, the new data extraction technology is introduced in the part of information collection and acquisition.

In the process of information construction, information sources are abundant and forms are varied. When the above information is collected, the traditional single collection method cannot complete the collection of multiple forms of text and image. The single data and information acquisition network is transformed into a sensor acquisition network, and the image and voice information are collected by wireless induction sensor. Combined with the original information acquisition channel, the comprehensive construction information acquisition is realised.

For the collected information, fractional differential equation is used to complete the noise reduction. Common fractional DES contains many contents, so choosing the method suitable for image information is the key problem in this design. There are many kinds of common fractional-order DES. In order to ensure the feasibility of the calculation process, the content of fractional-order DES is compared to complete the selection of algorithms. The contents of commonly used fractional DES are shown in Table 2.

Using the above design to complete the image processing speed is too slow power work, on the power method design in addition to the algorithm setting, increasing the processing process part. The specific process is shown in Figure 2. The process in Figure 2 was used to complete the processing work, because the amount of processed data was too large. In the process of processing to ensure the normal operation of the algorithm, the 8-core embedded CPU is used to control the computing equipment. In the process of calculation, the consistency of the unit quantity of information should be guaranteed, the unified information format should be set, and the exclusive database should be built to complete the information storage after processing to avoid the loss of data and information. In order to solve the problem that the processing time of image information is too long in the construction of university education information, a new method is designed which takes into account the multi-component mixed criterion fractional differential equation. In order to ensure the feasibility of the method design, experimental links were set to study its application effect.

Fig. 2

In this paper, the effectiveness of image denoising is evaluated objectively by the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). The larger the PSNR value, the higher the denoising ability, and the higher the SSIM, the more similar the structure. The expressions are as follows: (16) PSNR=10×lg2551M×N∑i=1M∑j=1N(u0(i,j)−u(i,j))2 PSNR = 10 \times \lg {{255} \over {{1 \over {M \times N}}\sum\limits_{i = 1}^M \sum\limits_{j = 1}^N {{\left( {{u_0}\left( {i,j} \right) - u\left( {i,j} \right)} \right)}^2}}} (17) SSIM(u,u0)=L(u,u0)*C(u,u0)*S(u,u0) SSIM\left( {u,{u_0}} \right) = L\left( {u,{u_0}} \right)*C\left( {u,{u_0}} \right)*S\left( {u,{u_0}} \right) where M and N represent the length and width of the image, and M × N represents the number of pixels in the image, u₀ (i, j) represents the original image and u(i, j) the denoising image, Type (17): L(u,u0)=2μuμ0+C1μu2+μu02+C1C(u,u0)=2σuσ0+C2σu2+σu02+C2S(u,u0)=σu0u+C3σu+σu0+C3 \matrix{ {L\left( {u,{u_0}} \right) = {{2{\mu _u}{\mu _0} + {C_1}} \over {\mu _u^2 + \mu _{u0}^2 + {C_1}}}} \hfill \cr {C\left( {u,{u_0}} \right) = {{2{\sigma _u}{\sigma _0} + {C_2}} \over {\sigma _u^2 + \sigma _{u0}^2 + {C_2}}}} \hfill \cr {S\left( {u,{u_0}} \right) = {{{\sigma _{u0}}_u + {C_3}} \over {{\sigma _u} + {\sigma _{u0}} + {C_3}}}} \hfill \cr } where μ_u and μ₀ represent the mean values of images u and u₀, σu and σu0 are the standard deviations of images u and u0, σu2 \sigma _u^2 and σu02 \sigma _{u0}^2 represent the variance of images u and u₀, respectively,σ_u0u is the covariance of the graph with u, and C1, C2, C3 are constants respectively, C3=C22 {C_3} = {{{C_2}} \over 2} .

The collected information will be recycled by the above setting algorithm until the final value does not drop any more, and then the processing process is over. In the processing process, the colour data is preset to complete the convergence from the local to the centre, so as to improve the efficiency of large-scale data processing. In contrast, the fractional differential equation is more appropriate.

In order to verify the effectiveness of the method designed in this paper, which takes into account the multivariate mixed criterion fractional differential equation, the test environment is constructed to complete the test of its effect. In the process of testing, the study was completed in the form of comparison with the traditional power assist method.

(3)

The experimental results

Use the above settings to complete the test, the test results were presented in the form of a table. The test results of the design method in this paper were set as the experimental group, while the original method was set as the control group. The specific experimental results are shown in Figure 3.

According to the above experimental results, there is a huge difference between the experimental results obtained by using the original method and the method in this paper. With the increase of experimental samples, the processing time of the original method and the proposed method increases. However, through comparison, the processing time of the original method is significantly longer than that of the design method in this paper. When the sample size is 100,000, the processing time of the original method is 45 s, which is much higher than that of the design method in this paper. It can be seen that the auxiliary effect of the design method in this paper is better than that of the original method in the informationisation construction of university education, and the design method in this paper should be popularised and applied.

Fig. 3