A Comparative Analysis of Radial-Tchebichef Moments and Zernike Moments

moment descriptors are commonly used in applications such as image classification, pattern recognition and identification. A radial-polar representation of image coordinate space is particularly useful in the above applications, since it facilitates the derivation of rotation invariants of any arbitrary order. Zernike moments and radial-Tchebichef moments fall into the category of moments that are defined using radial-polar coordinates. The discrete orthogonal nature of the kernel of radial-Tchebichef moments provides notable advantages over continuous Zernike moments. The paper presents a detailed analysis to prove that radial-Tchebichef moments have superior features compared to Zernike moments and are computationally less complex. This paper also presents a novel framework for accurately computing moments with kernel defined using polar coordinates, that is particularly suitable for discrete orthogonal rotation invariants. The method preserves the separability property of the kernel, which can be effectively used in computing both forward and inverse moment transforms. Thus the proposed framework yields a simple and fast implementation of radial-Tchebichef moments for image reconstruction, and invariants for pattern recognition. The efficiency of the proposed method is demonstrated through a series of experimental results. The most appropriate mathematical structure for computing radial Tchebichef moments is a set of discrete concentric rings, where each ring represents a fixed integer value of radial distance r from the centre of the image. We can subdivide the coordinate space into /2 concentric rings Rr , r = 0,1,…,(/2)−1. Depending on the number of points inside a ring, each ring can be further subdivided into mr regions. An example for = 10 is shown in Figure 1. On ring Rr, the angle θ varies from 0 to 2π in mr discrete intervals such that θ k = r m k π 2 , k = 0, 1, 2,…, mr−1. (1) Eq. (1) is valid only under the assumption that there exists a one-to-one mapping between pixels on the image, and points that are distributed uniformly around concentric circles. If we denote the image intensity value at location (r, θk) by f(r, k), then the radial-Tchebichef moments of order p and repetition q are defined using the equation () ∑ ∑ − = − = − = 1 2 / 0 1 0 2 , 1) (r m k m qk j r p pq r r k r f e m r t T π , (a) (b) Figure 1: (a) A 10x10 pixel space subdivided into 5 …