High convergence rates of digital image transformation by numerical integration using spline functions

This paper introduces an intriguing topic in image processing where accuracy of images even in details is important and adopts an intriguing 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 certain region, and evaluated by numerical integration approximately. However, contrasted to the traditional integration, the integrand has different smooth nature in different subregions due to piecewise interpolation and approximation. New treatments of approximate integration and new discrete algorithms of images have been developed. The cycle conversion T−1T of image transformations is said if an image is distorted by a transformation T and then restored back to itself by the inverse transformation T−1. Combined algorithms of discrete techniques proposed in [1–3] only have the convergence rates O(1/N) and O(1/N2) of sequential greyness errors, where N is the division number for a pixel split into N2 subpixels. This paper reports new combination algorithms using spline functions to achieve the high convergence rates O(1/N3) and O(1/N4) of digital image transformations under the cycle conversion. Both error analysis and numerical experiments have been provided to verify the high convergence rates. High convergence rates of discrete algorithms are important in saving CPU time, particularly to multi-greyness images. Moreover, the computational figures for real images of 256 × 256 with 256 greyness levels, in which N = 2 is good enough for practical requirements, display validity, and effectiveness of the new algorithms in this paper.