scholarly journals Image Segmentation Method for an Illumination Highlight Region of Interior Design Effects Based on the Partial Differential Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Lina Wang ◽  
Yaoming Liu ◽  
Zhike Qian
Keyword(s):  
Image Segmentation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Convex Hull ◽  
Saliency Map ◽  
Segmentation Method ◽  
Clustering Method ◽  
Partial Differential ◽  
Ellipse Model ◽  
The Impact

The saliency calculation model based on the principle of partial differential equations sometimes highlights areas with high contrast in the background, and the salient targets obtained occasionally have holes. The above problems can be solved by combining the improved convex hull calculation center saliency map. This paper designs a single-target color image segmentation algorithm based on partial differential equations. First, we calculate the basic saliency map according to the uniqueness of the color and the spatial distribution of the color; second, we then use the superpixel to improve the convex hull and calculate the central saliency map according to the principle; finally, the basic saliency map and the central saliency map are calculated. The weighted fusion is used to obtain the comprehensive saliency map, and the threshold method is used to segment the comprehensive saliency map to obtain the final target image. This paper designs an evaluation standard suitable for the segmentation of the illuminated highlight area of the effect image. It compares the experimental results of the segmentation method in this paper with the SLIC (Simple Linear Iterative Clustering) method and the traditional superpixel method to segment the illuminated highlight area. The segmentation method is applied to the image enhancement experiment. Based on the fuzzy means clustering algorithm, a fuzzy clustering objective function including brightness, color, and distance parameters is designed, which improves the weight of the brightness value in the clustering and improves the edge fit of the segmentation of the lighting highlight area of the rendering. The segmentation method produced by combining the clustering method with the superpixel biased clustering method can improve the output effect of the illuminated highlight area of the effect image after segmentation. We perform color equalization processing on the image to be segmented to reduce the impact of light, then set the closed value of the brightness information component, perform segmentation judgment, and expand the long and short axes of the ellipse model in the high-brightness area to further reduce the impact of light. The experimental results prove that the above method has a better segmentation effect than the traditional ellipse model and can accurately segment the gesture image. Compared with the existing mainstream saliency calculation models, this algorithm is closer to the true value image in terms of visual effects and has obvious advantages in terms of accuracy.

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

2017 ◽  
Vol 59 (2) ◽  
pp. 167-182 ◽  
Author(s):  
H. Y. ALFIFI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Partial Differential ◽  
Analytical Results ◽  
Transient Solutions ◽  
The Impact

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

Numerical Examination of Thermophysical Properties of Cobalt Ferroparticles over a Wavy Surface Saturated in Non-Darcian Porous Medium

2020 ◽  
Vol 45 (2) ◽  
pp. 109-120
Author(s):  
Irfan Mustafa ◽  
Abuzar Ghaffari ◽  
Tariq Javed ◽  
Javeria Nawaz Abbasi
Keyword(s):  
Porous Medium ◽  
Nusselt Number ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Magnetic Parameter ◽  
Volume Fraction ◽  
Wavy Surface ◽  
Partial Differential ◽  
Darcy Model ◽  
The Impact

AbstractIn this study, the effect of magnetic field on an incompressible ferrofluid flow along a vertical wavy surface saturated in a porous medium is investigated. Ferrofluid is made by incorporating magnetic particles, in this case cobalt, at the nanoscale level into a base fluid. For the study of porous medium two well-known models, namely, Darcy and non-Darcy, are used. The mathematical model in terms of governing partial differential equations which are based on conservation laws in mechanics according to the assumption is developed, and this model is converted into a dimensionless form by suitable transformations. Due to the complex non-linear partial differential equations, the numerical solution is calculated by using an implicit finite difference scheme. The impact of involved parameters, namely, magnetic parameter, nanoparticle volume fraction parameter, the amplitude of the wavy surface, and the Grashof number, on Nusselt and average Nusselt numbers are studied through graphs and tables.The results show that for large values of the magnetic parameter, both the Nusselt number and the average Nusselt number decrease in ferrofluid flow. The value of the Nusselt number in the Darcy model is higher than the value of the Nusselt number in the non-Darcy model.

scholarly journals Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2868
Author(s):  
Hussam Aljarrah ◽  
Mohammad Alaroud ◽  
Anuar Ishak ◽  
Maslina Darus
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Applied Mathematics ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

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