The oil painting retrieval technology based on the reaction diffusion equation has attracted widespread attention in the fields of oil painting processing and pattern recognition. The description and extraction of oil painting information and the classification method of oil paintings are two important processes in content-based oil painting retrieval. Inspired by the restoration and decomposition functional model of equal oil painting, we propose a reaction diffusion equation model. The new model contains two reaction diffusion equations with different principal parts. One principal part is total variation diffusion, which is used to remove noise. The other main part is thermal diffusion, which is used to modify the source term of the denoising reaction-diffusion equation to achieve the effect of protecting the texture of the oil painting. The interaction of the two reaction-diffusion equations finally achieves denoising while maintaining the boundaries and textures. Under the framework of the above reaction diffusion equation model, we introduce Laplace flow to replace the original total variation flow, so that the new denoising reaction diffusion equation combines the isotropic diffusion and total variation flow of the thermal reaction diffusion equation to achieve the effect of adaptive theoretical research. Using regularization methods and methods, we, respectively, get the well-posedness of the two model solutions, which provides the necessary preparation for numerical calculations. Based on the statistical theory and classification principles of support vector machines, combined with the characteristics of oil painting classification, the research and analysis are carried out from the three important aspects of kernel function, training algorithm, and multiclass classifier algorithm that affect the classification effect and speed. Numerical experiments show that the given filter model has a better processing effect on images with different types and different degrees of noise pollution. On this basis, an oil painting classification system based on texture features is designed, combined with an improved gray-level cooccurrence matrix algorithm and a multiclass support vector machine classification model, to extract, train, and classify oil paintings. Experiments with three types of oil paintings prove that the system can achieve a good oil painting classification effect. Different from the original model, the new model is based on the framework of reaction-diffusion equations. In addition, the new model has good effects in removing step effects, maintaining boundaries and denoising, especially in maintaining texture.
Generally speaking, oil painting texture processing technology takes digital oil painting as the object, including oil painting texture acquisition, oil painting texture analysis, and oil painting texture understanding. Oil painting analysis is a link between oil painting acquisition and oil painting understanding, which directly affects the degree of computer comprehension . The problems of oil painting restoration and oil painting segmentation belong to the research category of oil painting analysis and have always been two important and widely studied issues in oil painting processing . With the increasing demand and breakthroughs in physical technology, people pay close attention to and explore the essence of oil painting processing and try to use strict mathematical theory to classify and improve the existing oil painting processing models . Currently, digital oil painting processing technology has three main tools: random theory, wavelet analysis theory, and partial differential reaction diffusion equation theory. Among them, the stochastic model is directly applied to oil painting based on Bayesian estimation and Markov random field, and many reasonable models have been established, while wavelet analysis is based on Fourier analysis to transform oil painting information into the frequency domain and then build a model on the basis of it, which indirectly acts on oil painting texture recognition .
The noise and detailed texture of the oil painting are in the high-frequency area of the oil painting, and most of the information in the oil painting is stored in the edge part. In the process of removing the oil painting noise, the texture detail information of the oil painting will be mistakenly regarded as the noise information of the oil painting and be filtered out, resulting in the loss of image information and blurring of the oil painting . Obtaining high-resolution oil paintings often requires high-cost and complex systems. In order to obtain finer details of the imaging scene under given system parameters, image interpolation methods can be used . The usual nearest neighbor interpolation method, bilinear interpolation method, bicubic interpolation method, cubic convolution interpolation method, etc. enlarge the original oil painting, which improves the visual effect of the oil painting to a certain extent, but the interpolation precision of these algorithms is not high enough. The visual effect of oil painting is not improved enough . The model parameters and missing pixels are estimated through nonlinear optimization methods, and the model parameters and missing pixels are estimated at the same time using nonlinear optimization methods. Therefore, the protection of oil painting details and texture information and the removal of oil painting noise information have become an irreconcilable contradiction. The goal of oil painting denoising is to keep the edge and texture details as much as possible while removing noise . However, traditional oil painting denoising algorithms cannot balance the above two contradictions. It is imperative to find an algorithm that can well balance the contradiction between noise elimination and edge preservation information. The oil painting denoising algorithm based on the reaction diffusion equation can selectively smooth the oil painting and better balance the contradiction between the two . This paper uses a combination of theoretical analysis and simulation experiments, combined with fractional calculus and fidelity terms, to study the oil painting denoising model based on the reaction-diffusion equation.
By analyzing the dependence of the oil painting boundary on the position and structure of the oil painting, we construct a boundary mapping function with strong antinoise ability. The corresponding reaction diffusion equation not only has the characteristics of anisotropic diffusion but also conducts heat in the homogeneous area inside the oil painting. The average curvature diffusion is carried out in the near-boundary area, and finally, the effect of adaptive denoising is achieved. In terms of theoretical research, we first use the fixed-point method to prove the well-posedness of the new model, and then study the asymptotic state of the solution, and the conclusion shows that the new model limit state of the recovery result is the local mean value of the initial oil painting. Compared with the TV model, the new model conducts thermal diffusion in the internal homogeneous region, effectively avoiding the step effect. Compared with the PM model, the new model is well-posed, so the solution is more stable, and due to the diffusion mode and diffusion speed of the new model, the model has stronger adaptive ability, better denoising in the internal homogeneous area, and more precise maintenance of the border near the border. This paper studies a variety of texture features in the oil painting content description and extraction method. Several numerical experiments verify the robustness of the proposed index against changes in translation, scale, and rotation. The application of image denoising shows the effectiveness of the proposed index. Starting from the characteristics of texture and the practical application of oil painting content retrieval technology, the gray-level cooccurrence matrix algorithm in the statistical method is analyzed emphatically. Aiming at the shortcomings of this method that it has a large amount of redundant calculations and requires a large amount of storage space, several existing improved algorithms based on gray-level cooccurrence matrix algorithms are further studied, including the sum-and-difference statistical method algorithms.
2. Related Work
Many researchers have done further research along this route. Xing et al.  studied statistical information, and for the first time experimented and proposed that the deficit moment, contrast, and entropy have the greatest recognition ability. The two-dimensional cooccurrence matrix proposed by the predecessors discarded the color information, and Zhang et al.  proposed a three-dimensional form of the cooccurrence matrix to solve this problem. Based on the psychological research on the visual perception of texture by the human eye, Yao et al.  proposed six texture attributes that can constitute a texture visual model, which are contrast, granularity, directionality, line type, roughness, and uniformity. Zhang et al.  found that the gradient cooccurrence matrix method, which combines the boundary operator and the gray distribution, can better describe the texture characteristics. This method first convolves the five special convolution kernels in pairs and then convolves the oil painting on the template obtained from the convolution and extracts texture features. Chi et al.  proposed a two-stage self-supervised texture segmentation method. The method first obtains the initial segmentation map through unsupervised automatic regression clustering and then performs self-supervised wavelet classification on this basis. Wen et al.  introduced a statistical texture analysis method based on texture element mode, which can provide spatial structure information of oil painting texture and has the monotonic invariance of pixel gray value. Zhao et al.  proposed a texture analysis method based on BVLC moments and BDIP moments. BVLC can display rough and smooth characteristics, and BDIP can extract troughs and edges very well. They are processed directly on the color space, which can effectively combine color features. The frequency domain analysis method of texture has also been developed rapidly.
As a remedy, nonlinear technology has successfully coordinated the processing of oil painting denoising and edge enhancement and has become a very effective technical method. Among them, nonlinear diffusion technology has been a hot topic in recent years, including the reaction diffusion equation method and geometric method, among which good results have been achieved in oil painting enhancement, oil painting segmentation and edge detection, texture generation, and mathematical morphology. The high-quality results concerned are computer vision and pattern recognition . The method based on the Gabor filter conforms to the characteristics of the human visual perception system and the physiological vision of the human eye and is an important development direction for texture oil painting analysis. Gabor wavelet is a joint space-frequency method, and its texture model is based on a narrow-band texture field model, which can achieve local optima in both the frequency domain and the spatial domain. After the theoretical structure of wavelet transform was established, many scholars began to study how to use wavelet transform to represent texture features. Some scholars use statistics (variance and average) extracted from wavelet subbands as texture features, which have good retrieval results . It includes the use of wavelet frame transform package theory to analyze the texture of oil paintings to obtain the characteristics of translation invariance and stability. In order to explore the characteristics of medium waves, scholars use tree-shaped wavelet transform to further improve the accuracy of classification. The combination of wavelet transform and other technologies achieves better performance.
The model uses directional diffusion with protective edge characteristics instead of isotropic diffusion with Gaussian smooth kernel, laying the theoretical foundation of nonlinear diffusion method for oil painting processing, and opening up a new field of oil painting processing application research. Based on the same framework, the introduction of total variation further demonstrates the importance of PDE models and theories in oil painting processing. The use of nonlinear diffusion methods to obtain continuous models makes mesh selection and isotropic diffusion easier to deal with; nonlinear diffusion methods are also allowed to merge or separate known filtering methods and form new models; further PDE models make algorithm analysis and synthesis more natural and flexible to achieve high accuracy and stability [19–21]. Gaussian filtering, median filtering, and their improvements all have the advantage of a small amount of calculation. This is indispensable in the real-time processing of computer vision. For the need to restore high-quality oil paintings, nonlinear anisotropic diffusion model is an important method [22, 23].
3. Construction of an Adaptive Extraction Model of Oil Painting Texture Features Based on Reaction-Diffusion Equations
3.1. Distribution of the Solution Set of the Reaction-Diffusion Equation
There are two important frameworks for oil painting restoration and decomposition models based on reaction-diffusion equations. The variational model is based on energy functionals, and the reaction-diffusion equation model is based on fluid diffusion theory. First, we review the basic frameworks of denoising models based on variational methods and some classic models.
The value on the main diagonal of the gray-level cooccurrence matrix is used to measure the smoothness of the texture. The closer the gray pair value is to the main diagonal value, the smoother the texture is. The eigenvalues DIS, CON, INV, and IDM reflect the relationship between the value of the gray pair and the value of the main diagonal, so they can express the smoothness of the texture. DIS and CON are inversely proportional to INV and IDM, respectively. The CON value measures the size of the texture change.
Figure 1 shows the spatial distribution of the solution set of the reaction diffusion equation. According to the known training samples, the characteristic parameters are selected, and the discriminant function is established to classify each object. This method must have prior knowledge of the classification area to establish the discriminant function and obtain the training classifier.