New splitting algorithms for geometric transformations of digital images and their error analysis

Abstract

For digital images and patterns under the nonlinear geometric transformation, T: (n, h) ? (x, y), this study develops the splitting algorithms (i.e., the pixel-division algorithms) that divide a 2D pixel into N 3 N subpixels, where N is a positive integer chosen as N 5 2 k (k ! 0) in practical computations. When the true intensity values of pixels are known, this method makes it easy to compute the true intensity errors. As true intensity values are often unknown, the proposed approaches can compute the sequential intensity errors based on the differences between the two approximate intensity values at N and N/2. This article proposes the new splitting-shooting method, new splitting integrating method, and their combination. These methods approximate results show that the true errors of pixel intensity are O(H), where H is the pixel size. Note that the algorithms in this article do not produce any sequential errors as N ! N 0 , where N 0 (!2) is an integer independent of N and H. This is a distinctive feature compared to our previous papers on this subject. The other distinct feature of this article is that the true error bound O(H) is well suited to images with all kinds of discontinuous intensity, including scattered pixels.