Comparison at high spatial frequencies of two-pass and one-pass geometric transformation algorithms

Abstract

Two-pass image geometric transformation algorithms, in which an image is resampled first in one dimension, forming an intermediate image, then in the resulting orthogonal dimension, have many computational advantages over traditional, one-pass algorithms. For example, interpolation and anti-aliasing are easier to implement, being 1-dimensional operations; computer memory requirements are greatly reduced, with access to image data in external memory regularized; while pipelined parallel computation is greatly simplified. An apparent drawback of the two-pass algorithm which has tended to limit its universal adoption is a reported corruption at high spatial frequencies due to apparent undersampling, in certain cases, in the necessary intermediate image. This experimental study set out to resolve the question of possible corruption by computing the mean-square error when a sinusoidal grating test image is rotated, either by an efficient two-pass algorithm or by a traditional one-pass algorithm. It was found that the method used for interpolation has a major effect on the accuracy of the result, poorer methods accentuating differences between the two algorithms. A totally unexpected and fortuitous result is that, by using near-perfect interpolation (e.g., by the FFT), the two-pass algorithm is almost as accurate as one pleases, for rotations up to 45°, to very close to the Nyquist limit (as also is the one-pass algorithm, with near-perfect interpolation). For rotations of φ > 45°, the two-pass algorithm breaks down before the Nyquist limit, but these can be replaced by rotations of 90° - φ and transposition. Thus, the supposed drawback of the two-pass algorithm can be nullified by near-perfect interpolation, at least in the case of rotation, while a major bonus is the greater ease with which interpolation by the FFT may be implemented, in the two-pass case, leading to the possibility of highly faithful geometric transformation in practice, aided by the increasing availability of fast DSP and FFT microcircuits.