New algorithms for geometric transformations of digital images and patterns

Nonlinear geometric transformations, such as the splitting-shooting method (SSM), the splitting-integrating method (SIM) and their combination (CSIM), and their advanced versions S~S~M, S~I~M, CS~I~M have been developed. This paper proposes new versions of such nonlinear transformations, S~S~^#M, S~I~^#M, CS~^#I~^#M, approximated by piecewise linear transformations to circumvent the necessity of finding the nonlinear solutions, and to obtain exact integration computationally. The absolute errors of pixel greyness are proven to be O(H), where H is the length of a pixel region. It is worth pointing out that the new algorithms in this paper do not produce any sequential errors as N≥N₀. Apart from this distinctive feature, the absolute error bound O(H) can be applied to all kinds of images with discontinuity, including fully isolated pixels.