Image representation using separable two-dimensional continuous and discrete orthogonal moments

This paper addresses bivariate orthogonal polynomials, which are a tensor product of two different orthogonal polynomials in one variable. These bivariate orthogonal polynomials are used to define several new types of continuous and discrete orthogonal moments. Some elementary properties of the proposed continuous Chebyshev–Gegenbauer moments (CGM), Gegenbauer–Legendre moments (GLM), and Chebyshev–Legendre moments (CLM), as well as the discrete Tchebichef–Krawtchouk moments (TKM), Tchebichef–Hahn moments (THM), Krawtchouk–Hahn moments (KHM) are presented. We also detail the application of the corresponding moments describing the noise-free and noisy images. Specifically, the local information of an image can be flexibly emphasized by adjusting parameters in bivariate orthogonal polynomials. The global extraction capability is also demonstrated by reconstructing an image using these bivariate polynomials as the kernels for a reversible image transform. Comparisons with the known moments are performed, and the results show that the proposed moments are useful in the field of image analysis. Furthermore, the study investigates invariant pattern recognition using the proposed three moment invariants that are independent of rotation, scale and translation, and an example is given of using the proposed moment invariants as pattern features for a texture classification application. & 2011 Elsevier Ltd. All rights reserved.