TAILIEUCHUNG - Art of Surface Interpolation-Chapter 2 The ABOS method

Tham khảo tài liệu 'art of surface interpolation-chapter 2 the abos method', công nghệ thông tin, kỹ thuật lập trình phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 2 The ABOS method The goal of this chapter is to design an interpolation approximation method which is sufficiently flexible and robust enough for solving large problems provides results comparable with the Kriging method respectively with the Radial basis function method or Minimum curvature method and which does not have disadvantages and limitations of these methods presented in the first chapter. The method was called ABOS Approximation Based On Smoothing and despite the fact that it should be used as an approximation method according to its name it can also be used for solving interpolation problems as it will be explained in this chapter. The definition of the interpolation function and notations The interpolation function is determined by a matrix P of real numbers whose elements z-coordinates are assigned to nodes of a regular rectangular grid covering the domain D see the next figure . Figure Regular rectangular grid for defining the interpolation function. The value of the interpolation function at any point x0 y0 within the grid can be evaluated from the equation of the bilinear polynomial f x y a-xy b-x c-y d which is defined by coordinates of corner points of the grid rectangle containing the point x0 y0 . The following notation is used in the next text x1 x2 minimum and maximum of x-coordinates of points XYZ y1 y2 minimum and maximum of y-coordinates of points XYZ z1 z2 minimum and maximum of z-coordinates of points XYZ il j1 size of the grid number of columns and rows of the matrix P Pij elements of the matrix P i 1 . i1 j 1 . j1 DP auxiliary matrix with the same size as the matrix P Z vector of z-coordinates of points XYZ DZ auxiliary vector with the same size as the vector Z 13 NB matrix of the nearest points - integer matrix with the same size as the matrix P containing for each node of the grid the order index of the nearest point XYZ K matrix of distances - integer matrix with the same size as the matrix P containing for each .