Advanced combinations of splitting–shooting–integrating methods for digital image transformations

An image consists of many discrete pixels with greyness of di erent levels, which can be quanti ed by greyness values. The greyness values at a pixel can also be represented by an integral as the mean of continuous greyness functions over a small pixel region. Based on such an idea, the discrete images can be produced by numerical integration; several e cient 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 [6] to facilitate images and patterns under a cycle transformations 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 [8] to be only O(1=N ). In [10], the convergence rates Op(1=N ) in probability and Op(1=N ) in probability using a local partition are discovered. The CSIM is well suited to binary images and the images with a few greyness levels due to its simplicity. However, for images with large (e.g., 256) multi-greyness levels, the CSIM still needs more CPU time since a rather large division number is needed. 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 ) of greyness solutions can be achieved. The new combinations are simple to carry out for image transformations because no solutions of nonlinear equations are involved in, either. The computational gures for real images of 256×256 with 256 greyness levels display that N =4 is good enough for real applications. This clearly shows validity and e ectiveness of the new algorithms in this paper. c © 1999 Elsevier Science B.V. All rights reserved. MSC: 65N15; 65D30; Minor 68A45