Research And Application Of Poisson Equation In Digital Image

In recent years,seamless image composite has been widely concerned,because it has wide applications in graphic design,photographic effects and so on.Seamless composite includes a variety of editing methods such as the original image gradient composite,the mixed gradient composite,the local brightness change and so on,which can directly generate new images that are usually fused by two or more images.The seamless composite of original image to target image can be obtained by solving Poisson equation with Dirichlet boundary condition.Traditionally,Poisson equation usually needs numerical discretization by finite difference or finite element.However,the linear equations generated by the above methods are usually large and sparse.The equations need not only occupy large memory but also consume time.Based on this problem,the composite method of boundary element harmonic coordinates based on Poisson equation is proposed.This method only needs to use the pixel value linear interpolation of the boundary point to obtain the pixel value of the interior point of the composite region.Solving the harmonic coordinates of the boundary element is only solving the directional derivative of the boundary element.Although the coefficient matrix of linear equations is dense,its order is far below the order of equations formed by finite difference or finite element method.In this paper,the boundary element harmonic coordinates are constructed from the boundary element theory,including the mixed element boundary element harmonic coordinates and the linear element boundary element harmonic coordinates.In order to improve the interpolation accuracy,we also propose an encryption boundary element harmonic coordinate method,enabling it to automatically encrypt the boundary points.At the same time,we further put forward the boundary element harmonic coordinate composite method based on Poisson equation for image editing.In the process of editing,we use the subdivision of adaptive triangular mesh,convolution theorem,fast Fourier and inverse Fourier transform in order to improve our composite speed.By comparing with the traditional Poisson composite method and the mean coordinate composite method,it is concluded that the proposed method has faster solution speed in the low resolution or high resolution image,and the composite effect is consistent with the other two methods.In addition,unlike the harmonic coordinates obtained by the finite difference method,the harmonic coordinates obtained by the boundary element method have the expressed form.