Research And Application Of Two-dimensional Green Coordinates And Boundary Element Harmonic Coordinates

In recent years,in two-dimensional animation production and two-dimensional image editing,cage-based image deformation methods have received more and more attention.The so-called cage-based image deformation method refers to enclosing an image with a simple geometric frame and representing the points in the image as a linear combination of the geometric quantities associated with the geometric frame.The geometric frame is called a cage.The weight of the linear combination of geometric quantities is called the coordinates.The cage-based image deformation methods have the advantages of simplicity,quickness,and ease of operation.The two-dimensional mean coordinates obtained from the triangular barycentric coordinates are commonly used.The coordinates have an explicit representation and are simple to calculate,but The deformation effect is not good for the concave cage.The mean coordinates are obtained by the mean function theorem of the harmonic function.Similarly,the two-dimensional Green coordinates and the two-dimensional harmonic coordinates obtained by the properties of the two-dimensional harmonic function directly compensate for the defects of the mean coordinate,and each has its own characteristics.The two-dimensional Green coordinates have an explicit integral form and can achieve conformal deformation.Although there is no explicit format for the two-dimensional harmonic coordinates,the points in the image can be represented as a convex combination of the cage vertices.This paper starts with the two-dimensional Green coordinates,and simplifies the original complex explicit representation of the two-dimensional Green coordinates into a more intuitive explicit representation of the geometric quantities only related to the cage and the position of the deformation point.The explicit representation of the two-dimensional Green coordinates gradient simplifies the conformality of the two-dimensional Green coordinates and simplifies the explicit representation of the two-dimensional Green coordinates to the extension,and in the practical application of image deformation using the explicit representation of the two-dimensional Green coordinates gradient we obtained,using Newton iterative method for functions of several variables to solve the image damage problem and realize the recovery of the deformed image.Then we use the dimension reduction method boundary element method to approximate the two-dimensional harmonic coordinates,simplify the calculation,obtain a new coordinate we call the two-dimensional boundary element harmonic coordinates,and obtain the the explicit representation of two-dimensional boundary element harmonic coordinates and its gradient,and the proof that in the image deformation,the two-dimensional boundary element harmonic coordinates are completely equivalent to the two-dimensional harmonic coordinates and have a close relationship with the two-dimensional Green coordinates.Finally,an application example is given by applying two-dimensional Green coordinates and two-dimensional edge harmonic coordinates.