Advanced Combinations of Splitting-Shooting-Integrationg Methods for Digital Image Transformations

Abstract

The greyness values at a pixel can also be represented by an integral as the mean of continuous greyness functions over asmall pixel region. Based on such an idea, the discrete images can ve produced by numerical integration; several efficient algorithms are developed to convert images under transformations. Among these algorithms, the combination of splitting-shooting-integrating methods (CSIM) is most promising because no solutions of nonlinear equations are required for the inverse transformation. The CSIM is proposed in [5] to facilitate images and patterns under a cycle transformation T^-1T, where T is a nonlinear transformation. When a pixel region in two dimensions is split into N^2 subpixels, convergence rates of pixel greyness by CSIM are proven in [7] to be only O(1/N). In [8], the convergence rates Op(1/N^1.5) in probability and Op(1/N^2) in probability using a local partition are discovered. In this paper, a partition technique for numerical integration is proposed to evaluate carefully any overlaps between the transformed subpixel regions and the standard square pixel regions. This technique is employed to evolve the CSIM such that the convergence rate O(1/N^2) of greyness solutions can be achieved. The computational figures for real images of 256 * 256 with 256 greyness levels display that N=4 is good enough for real applicatoins. This clearly shows validity and effectiveness of the new algorithms in this paper.