Pattern Recognition 36 (2003) 731 – 742 www.elsevier.com/locate/patcog A comparative analysis of algorithms for fast computation of Zernike moments Chee-Way Chong a ;∗ , P. Raveendran b , R. Mukundan c a Faculty of Engineering and Technology, Multimedia University, Jalan Air Keroh Lama, Melaka 75450, Malaysia b Department of Electrical Engineering, University of Malaya, Kuala Lumpur 50603, Malaysia c Department of Computer Science, University of Canterbury, Private Bag 4800, Christchurch, New Zealand Received 9 August 2001; received in revised form 20 March 2002; accepted 27 March 2002 Abstract This paper details a comparative analysis on time taken by the present and proposed methods to compute the Zernike moments, Zpq . The present method comprises of Direct, Belkasim’s, Prata’s, Kintner’s and Coecient methods. We propose a new technique, denoted as q-recursive method, specically for fast computation of Zernike moments. It uses radial polynomials of xed order p with a varying index q to compute Zernike moments. Fast computation is achieved because it uses polynomials of higher index q to derive the polynomials of lower index q and it does not use any factorial terms. Individual order of moments can be calculated independently without employing lower- or higher-order moments. This is especially useful in cases where only selected orders of Zernike moments are needed as pattern features. The performance of the present and proposed methods are experimentally analyzed by calculating Zernike moments of orders 0 to p and specic order p using binary and grayscale images. In both the cases, the q-recursive method takes the shortest time to compute Zernike moments. ? 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Kintner’s method; Prata’s method; Coecient method; Belkasim’s method; Zernike radial polynomials 1. Introduction Zernike moments are used in pattern recognition appli- cations [1–3] as invariant descriptors of the image shape. They have been proven to be superior to moment functions such as geometric moments in terms of their feature repre- sentation capabilities and robustness in the presence of im- age quantization error and noise [4 –7]. Their orthogonality property helps in achieving a near zero value of redundancy measure in a set of moment functions. Thus, moments of dierent orders correspond to independent characteristics of the image. Zernike moments are however computationally more complex and lengthy compared to that of geometric moments and Legendre moments. This is because the ∗ Corresponding author. E-mail address: chong.chee.way@mmu.edu.my (C.-W. Chong). denition of Zernike radial polynomials is heavily depen- dent on factorial functions and only a single Zernike mo- ment can be obtained at one time. The computation has to be repeated to obtain the entire set of Zernike moments of order p with q = p - 2;p - 4;p - 6, etc. The computation time increases substantially as the order p increases. This limitation has prompted considerable study on algorithms for fast evaluation of Zernike moments [8–11]. Mukundan et al. have proposed two fast algorithms namely contour integration method and square-to-circular transformation method to compute Zernike moments [8]. However, these proposed algorithms have limitations. In the case of the contour integration method, it is applica- ble only for binary image and it requires o-line analysis to extract the image boundary points. The accuracy of the Zernike moments has to be compromised when the square-to-circular transformation method is used. Belkasim too has introduced an algorithm for fast compu- tation of Zernike moments based on the series angular and 0031-3203/02/$22.00 ? 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII:S0031-3203(02)00091-2