Accuracy and numerical stability of high-order polar harmonic transforms

Invariant moments and transforms are a special kind of statistical measure designed to remain constant after some transformations, such as object rotation, scaling, translation or image illumination changes, in order to improve the reliability of a pattern recognition system. Polar harmonic transforms (PHTs) are one of such image descriptors based on a set of orthogonal projection bases, which are better rated in comparison with other moments and transforms because they possess less redundant information. PHTs suffer from geometric error and numerical integration error. Geometric error is caused when a square image is mapped into a unit circular disk, which cannot exactly match the square domain. Numerical integration error is caused when the integration is approximated by zeroth-order summation. We propose a computational framework based on numerical integration approach that reduces the geometric error and the numerical integration error simultaneously. The effect of the improved accuracy in the PHTs is analysed for rotation invariance, scale invariance and image noise. Various numerical experiments demonstrate that the effect of these errors is much more prominent in small images and, therefore the proposed method is suitable for applications involving small images, such as in optical character recognition and template-matching applications.