In the application of digital model of animation scene, image restoration technology and image denoising technology are the basic tasks of practical operation, which are closely related, but there exist also essential differences. The reason is that both of them want to obtain the original image from the degraded noise image or damaged image, but generally speaking, as there is no sufficient constraint information to accurately recover the original image, both of them are unwell-posed inverse problems. Therefore, on the basis of understanding the basic content and application research status of variational partial differential equations (PDEs), this paper discusses the application value of variational PDEs in image denoising and restoration according to the image processing requirements in the digital model of animation scenes.

#### Keywords

- variational
- partial differential equation
- animation scene
- digital model
- images
- denoising

#### MSC 2010

- 62G08

In the development of digital model construction of animation scene, the related digital image processing has a long history of development. Previous image processing theories mainly involve Fourier analysis, filtering theory and other content; the processing methods used are mostly simple and single heuristic methods and thus the practical application has a strong limit, and cannot be used to calculate part of the characteristics and performance of the algorithm for in-depth discussion. But as more and more researchers became involved in the field of digital image processing, the technology gradually shifted from alchemy to modern chemistry. Partial differential equation (PDE) method, the most critical content in mathematical analysis, is closely related to the physical world. For example, common heat conduction equation and wave equation belong to PDE [1, 2]. In early studies, PDE was mainly used to represent physical and mechanical phenomena, while in today’s research and discussion, PDE used in various fields, such as finance, biology, image processing and so on. According to the conclusion of Koenderink et al.‘s study, the Gaussian convolution of the image is equal to the solution of the thermal diffusion equation. Thus, the relation between PDE dedrying and classical filter dedrying is constructed, and the result of the fusion with the concept of scale space proposed by Witkin is the basic content of PDE image processing. At the same time, Hummel also put forward the maximum and minimum value criterion of scale space according to the above content, which can be regarded as the mathematical explanation of causality. Guided by this, Perona and Malik et al. first proposed a denoising model based on nonlinear diffusion in the 1990s. However, after entering the 21st century, Tschumperle et al. analysed and verified that the diffusion tensor in the divergence operator framework could not directly present the local diffusion behaviour of the model by using the divergence mode of TV denoising model. Therefore, they finally proposed a denoising model with trace operator as the core. Finally, a highly efficient geometric adaptive denoising model based on the expression of trace operator is obtained by extending the framework of divergence group proposed by Weickert to the framework of trace operator. It should be noted that although this method can improve the drying level and edge retention level, there are still some problems that need to be improved in practical application. Therefore, on the basis of understanding the variational PDE, this paper makes a comparative analysis of the creation of the digital model of the animation scene and the image denoising and restoration technology, and finally makes clear the application advantages of variational PDE in the comparative analysis of the experimental effect [3, 4].

First, variational expression. Assuming it has been clear that the average value of Gaussian white noise

_{noisy} − I||^{2}

_{σ}

^{2} _{1}_{2} ≥ 0

^{*}, ξ^{*} and

_{1}_{2} _{1}_{2}

_{1}_{2}

^{*}of isoluminance line. The specific formula is as follows:

_{1}_{2}

^{*},ξ^{*} on the basis of the decomposition of characteristic values.

From the combination of the hierarchical diagram analysis of the expression of variational, divergence and trace operators as shown in Figure 1, it can be observed that three expression models for image denoising and restoration are proposed in order to optimise the application effect of the digital model of the animation scene in this paper. Figure 2 shows the repair effect. In the variational restoration model, the results obtained by the restoration of this image are shown in Figure 3.

According to the analysis of Figure 3, (a) represents the original TV model algorithm, (b) represents the adaptive threshold model algorithm and (c) represents the comprehensive adaptive threshold and multi-scale repair algorithm. It can be seen from the variation of the curve value that the actual repair time of the original algorithm will get higher and higher as the number of iterations increases. The same is true for the adaptive threshold, the comprehensive adaptive threshold and the multi-scale repair algorithm, but compared with the comprehensive adaptive threshold and the multi-scale repair algorithm, the number of iterations and repair time are less [7].

In the model based on divergence operator expression, the time required by CDD and FCDD algorithm in image restoration and the final rendering effect are compared and analysed, as shown in the Table 1.

The number and time of iterations required by CDD and FCDD models

Number of iterations | 19980 | 824 | 47012 | 3972 | 47841 | 550 | 13822 | 598 |

Repair time | 94.3 s | 7.3 s | 1433.3 s | 125.0 s | 987.6 s | 11.9 s | 307 s | 13.3 s |

Number of restored pixels | 150 | 1659 | 894 | 992 |

CDD, curvature-driven; FCDD, fast curvature driven.

In the model based on trace operator expression, the simulated image and the real image are used for comparative experimental analysis, and the final results of the overview model, the geometric adaptive repair model and the TV model are compared and studied. Through the observation test in this paper, on the edge of the algorithm to the damaged area larger vertical and inclined to repair ability, the damaged area is to initialise white before repair, and the results of the TV to repair the false edge, geometric adaptive model repair results cannot be damaged area of the inner region surrounding the spread of information, This is bound to form a false edge in the damaged edge area. From the observation of the repair results of the model outlined in this paper, we can see that the information is fully diffused to the damaged area and the sharpness of the repair edge can be ensured [8, 9].

To sum up, combining the research and analysis of the variational PDE image denoising and restoration technology, it can be seen that the requirement of image detailed processing is extremely high in the application of the creation of digital model of animation scene based on variational PDEs. Therefore, although some achievements have been made in the practical development, there are still many challenges. So, in the future, researchers, with the continuous optimisation and promotion of network technology, should make use of variational PDEs to do image denoising and restoration according to the requirements of image processing on the basis of a comprehensive understanding of the creation requirements of digital models of animation scenes.

#### The number and time of iterations required by CDD and FCDD models

Number of iterations | 19980 | 824 | 47012 | 3972 | 47841 | 550 | 13822 | 598 |

Repair time | 94.3 s | 7.3 s | 1433.3 s | 125.0 s | 987.6 s | 11.9 s | 307 s | 13.3 s |

Number of restored pixels | 150 | 1659 | 894 | 992 |

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