Probabilistic analysis on the splitting-shooting method for image transformations

摘要

This paper explores a new topic in image processing where accuracy of images even in details is crucial, and adopts a new methodology dealing with discrete topics by continuous mathematics and numerical approximation. The key idea is that a pixel of images at different levels can be quantified by a greyness value, which can then be regarded as the mean of an integral of continuous functions over a small pixel region, and evaluated by numerical integration approximately. Hence, new treatments of approximate integration and new discrete algorithms of images have been developed. This paper also integrates different mathematics disciplines: numerical analysis, geometry, probability and statistics to discrete images that can be applied to many areas in computer sciences: image processing, computer graphics, computer vision, geometric added designs, and pattern recognition. In this paper, new error analysis in terms of probability theory is explored for the popular splitting-shooting method (SSM) and the combination (CSIM) of the splitting-shooting-integrating methods proposed in [35-37], and convergence rates in probability of image greyness are proven to be O_p(1/N~(1.5)) higher than O(1/N) reported in [45] by strict error analysis. Moreover, a new partial refinement technique of pixel partition is also proposed in this paper, to achieve the convergence rate O_p (1/N~2) in probability for SSM. By the new study of this paper, the SSM can be applied to real images with 256 greyness levels. The numerical and graphical experiments are also provided to confirm the theoretical analysis made. Both the strict error bounds and the probabilistic error bounds with explicit constants are also derived for general α-norms as α ≥ 1, and the countable probability inequalities for several sums of random variables are developed from probability theory, which are more suited to numerical computations.