Analysis of discrete techniques for image transformation

Let an image be distorted by a nonlinear transformationT, and then be restored back to its original by the inverse transformationT ^–1. Such a cycle conversion,T ^–1 T, of digital images can be facilitated by the combination (CSIM) given in 6] using the splitting-shooting method and the splitting-integrating method forT andT ^–1 respectively. Since there is no need to solve nonlinear equations, CSIM has been widely applied to specific areas of image processing and pattern recognition, even those with complicated transformations, e.g., the harmonic transformation 12]; however, no strict error analysis has been provided so far. In this paper, a priori error estimates and convergence rates are derived for pixel greyness obtained from CSIM; also the analytical results are extended to the harmonic transformation. Numerical and graphical experiments are provided to support the theoretical analysis.Some preliminary results of this paper were presented at the 40th anniversary meeting of SIAM, Los Angeles, July 20–24, 1992. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds pour la Formation de Chercheurs et l'Aide à la Recherche of Quebec by the Ministère de l'Enseignement Supérieur et de la Science (Action Structurante), and the Science Council of Republic of China under Grant No. NSC-85-2121-M-008.