Boundary Element Method And Application Of Heat Conduction Equation

Image inpainting has always been a key research object in the field of digital image processing because it is widely used in the restoration of cultural relics,old photos,and medical imaging.There are a variety of model algorithms for image inpainting.Their essence is to use the unbroken background information of the image and apply the algorithm to inpaint the damaged area.Using partial differential equations,especially heat conduction equations,to inpaint images is one of the inpainting methods.It uses the boundary information of the undamaged area and applies partial differential equations to diffuse the value of the boundary pixels into the damaged area to achieve the purpose of inpainting.Traditionally,the finite difference method is needed to solve the heat conduction equation.However,for higher resolution images,this method takes up more memory and the calculation speed is slower.In response to this problem,this paper proposes a boundary element algorithm based on the heat conduction equation.This method only needs to use the pixel values of the boundary points of the damaged area to interpolate to obtain the values of all points inside the damaged area.Since the boundary condition of the equation is Dirichlet boundary condition when processing images,the direction derivative of each boundary element needs to be solved.In this paper,starting from the boundary element method in real space,the explicit expression of the two-dimensional unsteady heat conduction equation is obtained.In order to deal with the integral term at the right end of the boundary element equation,we obtain a fast batch of numerical integration with parametric variables Processing calculation methods and we applied convolution theorem,fast Fourier transform and inverse Fourier transform speed up the process of solving.For the integral term of the three-dimensional integral at the right end of the boundary element equation,we use Laplace transform of the heat conduction equation in real space to derive the boundary integral equation of heat conduction equation in complex space,and reduce the three-dimensional integral term in the real space to the two-dimensional integral in the complex space.The numerical results of Stehfest inversion are the values in real space.This method further speeds up the process of solving and makes the memory footprint smaller.Finally,we give some examples of image restoration and analyze it to draw a conclusion.