New discrete techniques for 3D image transformations

For facilitating images under nonlinear transformations, the splitting-shooting method, the splitting-integrating method, and their combinations are presented in [1], which do not require nonlinear solutions, but have the low convergence rate O(1/N) of sequential pixel (or voxel) greyness, where N is the division number to split a 3D image voxel into N3 subvoxels. In this paper, new partition techniques of 3D subregions are proposed to evaluate carefully overlaps between a cube and the distorted subvoxel region and then applied to the splitting-shooting method, the splitting-integrating method, and their combinations. Error analysis is made to prove that the convergence rate O(1/N2) can be achieved, and numerical experiments are carried out to confirm the theoretical analysis mode. A remarkable advantage of the new discrete techniques is that no solutions of nonlinear equations are required either. The discrete techniques in this paper are well suited not only to solid objects but also to surfaces and curves in three dimensions, thus to benefit the study in many topics in computer science, such as computer vision, image processing, and pattern recognition.