scholarly journals Adaptive Extraction of Oil Painting Texture Features Based on Reaction Diffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Qicai Huang
Keyword(s):  
Diffusion Equation ◽  
Total Variation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Support Vector ◽  
Oil Painting ◽  
New Model ◽  
Oil Paintings

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.

scholarly journals An extension of the principle of spatial averaging for inertial manifolds

1999 ◽  
Vol 66 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Hyukjin Kwean
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Specific Class ◽  
Spatial Averaging ◽  
The Laplace Operator

AbstractIn this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

The Model of the Pearl River Water Pollution Based on Reaction Diffusion Equation

2013 ◽  
Vol 464 ◽  
pp. 199-202
Author(s):  
Guan Jia Huang ◽  
Shao Mei Fang ◽  
Jian Xian ◽  
Jie Mei Jiang
Keyword(s):  
Water Pollution ◽  
Diffusion Equation ◽  
River Water ◽  
Reaction Diffusion ◽  
Pollution Sources ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Pearl River ◽  
The Pearl River

In this paper, we study the Pearl River cadmium spill. Based on the reaction diffusion equations, we can monitor the concentration of pollutants any time. We find that in 20 kilometers distance from pollution sources, the concentration of cadmium is high. At last, we improve the model.

Uniqueness, multiplicity and stability for positive solutions of a pair of reaction–diffusion equations

1996 ◽  
Vol 126 (4) ◽  
pp. 777-809 ◽  
Author(s):  
Yihong Du
Keyword(s):  
Diffusion Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Complete Understanding ◽  
Satisfactory Description

We study the number and stability of the positive solutions of a reaction–diffusion equation pair. When certain parameters in the equations are large, the equation pair can be viewed as singular or regular perturbations of some single (or essentially single) equation problems, for which the number and stability of their solutions can be well understood. With the help of these simpler equations, we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large. In particular, we obtain a fairly satisfactory description of the positive solution set of the equation pair.

scholarly journals On the maximization problem for solutions of reaction–diffusion equations with respect to their initial data

2020 ◽  
Vol 15 ◽  
pp. 71
Author(s):  
Grégoire Nadin ◽  
Ana Isis Toledo Marrero
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Adjoint Problem ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Maximization Problem ◽  
Biological Conservation ◽  
Open Problems

We consider in this paper the maximization problem for the quantity ∫ Ωu(t, x)dx with respect to u0 =: u(0, ⋅), where u is the solution of a given reaction diffusion equation. This problem is motivated by biological conservation questions. We show the existence of a maximizer and derive optimality conditions through an adjoint problem. We have to face regularity issues since non-smooth initial data could give a better result than smooth ones. We then derive an algorithm enabling to approximate the maximizer and discuss some open problems.

scholarly journals Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains

2015 ◽  
Vol 16 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Bao Quoc Tang
Keyword(s):  
Diffusion Equation ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Random Attractors ◽  
External Forces ◽  
Stochastic Reaction

The existence of a unique random attractors in [Formula: see text] for a stochastic reaction-diffusion equation with time-dependent external forces is proved. Due to the presence of both random and non-autonomous deterministic terms, we use a new theory of random attractors which is introduced in [B. Wang, J. Differential Equations 253 (2012) 1544–1583] instead of the usual one. The asymptotic compactness of solutions in [Formula: see text] is established by combining “tail estimate” technique and some new estimates on solutions. This work improves some recent results about the regularity of random attractors for stochastic reaction-diffusion equations.

Entire solutions of reaction—diffusion equations with balanced bistable nonlinearities

2006 ◽  
Vol 136 (6) ◽  
pp. 1207-1237 ◽  
Author(s):  
Xinfu Chen ◽  
Jong-Shenq Guo ◽  
Hirokazu Ninomiya
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Entire Solutions ◽  
Stable Equilibria ◽  
Travelling Fronts

This paper deals with entire solutions of a bistable reaction—diffusion equation for which the speed of the travelling wave connecting two constant stable equilibria is zero. Entire solutions which behave as two travelling fronts approaching, with super-slow speeds, from opposite directions and annihilating in a finite time are constructed by using a quasi-invariant manifold approach. Such solutions are shown to be unique up to space and time translations.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

scholarly journals Uniqueness of fast travelling fronts in reaction–diffusion equations with delay

2008 ◽  
Vol 464 (2098) ◽  
pp. 2591-2608 ◽  
Author(s):  
Maitere Aguerrea ◽  
Sergei Trofimchuk ◽  
Gabriel Valenzuela
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Delayed Reaction ◽  
Birth Function ◽  
Scale Of Banach Spaces ◽  
Large Velocity ◽  
Travelling Fronts ◽  
Equations With Delay

We consider positive travelling fronts, u ( t ,  x )= ϕ ( ν . x + ct ), ϕ (−∞)=0, ϕ (∞)= κ , of the equation u t ( t ,  x )=Δ u ( t ,  x )− u ( t ,  x )+ g ( u ( t − h ,  x )), x ∈ m . This equation is assumed to have exactly two non-negative equilibria: u 1 ≡0 and u 2 ≡ κ >0, but the birth function g ∈ C 2 ( ,  ) may be non-monotone on [0, κ ]. We are therefore interested in the so-called monostable case of the time-delayed reaction–diffusion equation. Our main result shows that for every fixed and sufficiently large velocity c , the positive travelling front ϕ ( ν . x + ct ) is unique (modulo translations). Note that ϕ may be non-monotone. To prove uniqueness, we introduce a small parameter ϵ =1/ c and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.

scholarly journals Exact solutions of coupled multispecies linear reaction-diffusion equations on a uniformly growing domain

2015 ◽  
Author(s):  
Matthew Simpson ◽  
Jesse Sharp ◽  
Liam Morrow ◽  
Ruth Baker
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Moving Boundary ◽  
Reaction Diffusion ◽  
Proliferation Rate ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Growing Domain ◽  
Linear Reaction

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction-diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction-diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction-diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The frst example calculation corresponds to a situation where the initially-confned population diffuses suffciently slowly that it isunable to reach the moving boundary at x=L(t). In contrast, the second example calculation corresponds to a situation where the initially-confned population is able to overcome the domain growth and reach the moving boundary at x=L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

scholarly journals An efficient convergent method for the delay reaction-diffusion equations

2021 ◽  
Author(s):  
Marzieh Heidari ◽  
Mehdi Ghovatmand ◽  
Mohammad Hadi Noori Skandari
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
System Of Equations ◽  
Equivalent System ◽  
Numerical Examples ◽  
Delay Function ◽  
Mild Conditions ◽  
Convergent Method

In this manuscript, we consider the delay reaction-diffusion equation and implement an efficient spectral collocation method to approximate the solution of this equation. We first replace the delay function in the delay reaction-diffusion equation and achieve an equivalent system of equations. We then utilize the Legendre-Gauss-Lobatto and two-dimensional interpolating polynomial to approximate the solution of obtained system. Moreover, we prove the convergent of method under some mild conditions. Finally, the capability and efficiency of the method is illustrated by providing several numerical examples and comparing them with others

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