Radial Hahn Moment Invariants for 2D and 3D Image Recognition

Recently, orthogonal moments have become efficient tools for two-dimensional and three-dimensional（2D and 3D） image not only in pattern recognition, image vision, but also in image processing and applications engineering. Yet, there is still a major difficulty in 3D rotation invariants. In this paper, we propose new sets of invariants for 2D and 3D rotation, scaling and translation based on orthogonal radial Hahn moments. We also present theoretical mathematics to derive them. Thus, this paper introduces in the first case new 2D radial Hahn moments based on polar representation of an object by one-dimensional orthogonal discrete Hahn polynomials, and a circular function. In the second case, we present new 3D radial Hahn moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Hahn polynomials and a spherical function. Further 2D and 3D invariants are derived from the proposed 2D and 3D radial Hahn moments respectively, which appear as the third case. In order to test the proposed approach, we have resolved three issues： the image reconstruction, the invariance of rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Hahn moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and Princeton shape benchmark（PSB） database for 3D image.

On a generalized homogeneous Hahn polynomial

We investigate a generalized homogeneous Hahn polynomial in some detail. This polynomial includes as special cases the homogeneous Hahn polynomial and the homogeneous Rogers-Szeg o polynomial.A generating function, which contains a known generating function as a special case, is given. We also give a finite series generating function. Some results on the asymptotic expansion for this polynomial are derived.Certain results on zeros are also obtained. We deduce several results on zeros of certain entire functions involving this generalized Hahn polynomial. As results, one of Zhang(2017)’s results as well as others is obtained. Finally,we derive several general results on q-congruences of the generalized q-Apéry polynomials, from which two qcongruences involving the generalized homogeneous Hahn polynomial are deduced.