Orthogonal Rotation-Invariant Moments for Digital Image Processing

Orthogonal rotation-invariant moments (ORIMs), such as Zernike moments, are introduced and defined on a continuous unit disk and have been proven powerful tools in optics applications. These moments have also been digitized for applications in digital image processing. Unfortunately, digitization compromises the orthogonality of the moments and, therefore, digital ORIMs are incapable of representing subtle details in images and cannot accurately reconstruct images. Typical approaches to alleviate the digitization artifact can be divided into two categories: (1) careful selection of a set of pixels as close approximation to the unit disk and using numerical integration to determine the ORIM values, and (2) representing pixels using circular shapes such that they resemble that of the unit disk and then calculating ORIMs in polar space. These improvements still fall short of preserving the orthogonality of the ORIMs. In this paper, in contrast to the previous methods, we propose a different approach of using numerical optimization techniques to improve the orthogonality. We prove that with the improved orthogonality, image reconstruction becomes more accurate. Our simulation results also show that the optimized digital ORIMs can accurately reconstruct images and can represent subtle image details.