Splitting-integrating methods for the image transformations T and T−1T

The splitting-integrating method (SIM) is developed to facilitate the image transformations T and T−1T, to reach the convergence rates O(1/N2) of sequential image greyness errors, where N is the division number of an image pixel split into N2 subpixels. Since solutions of two non-linear equations may be involved in SIM for T, an interpolatory technique is proposed to significantly reduce the number of the nonlinear solutions so as to be independent of N. This leads to a modification of the algorithm for T (denoted as SI*M). Furthermore, the algorithms SIM and SI*M join with SIM for the inverse transformation T−1, to form the combination algorithms CIIM and CI*M for T−1T, a cycle conversion of image transformations. Error analysis and numerical experiments are provided to confirm the convergence rate O(1/N2).