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To facilitate images under the nonlinear geometric transformation T and its inverse transformation T-1, we have developed numerical algorithms in . A cycle conversion T-1T of image transformations is said if an image is distorted by a transformation T and then restored back to itself. The combination (CSIM) of splitting-shooting-integrating methods was first proposed in Li for T-1T. In this paper other two combinations, CIIM and CI#IM, of splitting integrating methods for T-1T are provided. Combination CSIM has been successfully applied to many topics in image processing and pattern recognition (see ). Since combination CSIM causes large greyness errors, it well suited to a few greyness level images, but needs a huge computation work for 256 greyness level images of enlarged transformations (see ). We may instead choose combination CIIM which involves nonlinear solutions. However, the improved combination CI#IM may bypass the nonlinear solutions completely. Hence, both CIIM and CI#IM can be applied to q(q≥256) greyness level images of any enlarged transformations. On the other hand, the combined algorithms, CSIM, CIIM, and CI#IM, are applied to several important topics of image processing and pattern recognition: binary images, multi-greyness level images, image condensing, illumination, affine transformations, prospective and projection, wrapping images, handwriting characters, image concealment, the transformations with arbitrary shapes, and face transformation. This paper may also be regarded as a review of our recent research papers.