IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 11, NOVEMBER 2003 1367 Image Analysis by Krawtchouk Moments Pew-Thian Yap, Raveendran Paramesran, Senior Member, IEEE, and Seng-Huat Ong Abstract—In this paper, a new set of orthogonal moments based on the discrete classical Krawtchouk polynomials is introduced. The Krawtchouk polynomials are scaled to ensure numerical sta- bility, thus creating a set of weighted Krawtchouk polynomials. The set of proposed Krawtchouk moments is then derived from the weighted Krawtchouk polynomials. The orthogonality of the pro- posed moments ensures minimal information redundancy. No nu- merical approximation is involved in deriving the moments, since the weighted Krawtchouk polynomials are discrete. These prop- erties make the Krawtchouk moments well suited as pattern fea- tures in the analysis of two-dimensional images. It is shown that the Krawtchouk moments can be employed to extract local fea- tures of an image, unlike other orthogonal moments, which gen- erally capture the global features. The computational aspects of the moments using the recursive and symmetry properties are dis- cussed. The theoretical framework is validated by an experiment on image reconstruction using Krawtchouk moments and the re- sults are compared to that of Zernike, Pseudo-Zernike, Legendre, and Tchebichef moments. Krawtchouk moment invariants is con- structed using a linear combination of geometric moment invari- ants and an object recognition experiment shows Krawtchouk mo- ment invariants perform significantly better than Hu’s moment in- variants in both noise-free and noisy conditions. Index Terms—Discrete orthogonal systems, Krawtchouk moments, Krawtchouk polynomials, local features, orthogonal moments, region-of-interest, weighted Krawtchouk polynomials. I. INTRODUCTION M OMENTS due to its ability to represent global features have found extensive applications in the field of image processing [1]–[9]. In 1961, Hu [1] introduced moment invari- ants. Based on the theory of algebraic invariants he derived a set of moment invariants, which are position, size and orienta- tion independent. Dudani et al. [3] used Hu’s moment invari- ants up to the third order in the recognition of images of air- craft. The same invariants were also used for recognition of ships [4]. Markandey et al. [9] developed techniques for robot sensing based on high dimensional moment invariants and ten- sors. However, regular moments are not orthogonal and as a consequence, reconstructing the image from the moments is deemed to be a difficult task. Teague [2], based on the theory of continuous orthogonal polynomials, has shown that the image can be reconstructed easily from a set of orthogonal moments, such as Zernike and Manuscript received June 11, 2002; revised May 12, 2003. The associate ed- itor coordinating the review of this manuscript and approving it for publication was Dr. Nasser Kehtarnavaz. P.-T. Yap and R. Paramesran are with the Department of Electrical En- gineering, University of Malaya, 50603 Kuala Lumpur, Malaysia (e-mail: ptyap@time.net.my; ravee@fk.um.edu.my). S.-H. Ong is with the Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia (e-mail: ongsh@um.edu.my). Digital Object Identifier 10.1109/TIP.2003.818019 Legendre moments. Zenike moments are able to store image in- formation with minimal information redundancy and have the property of being rotational invariant. Khotanzad et al. [5] used Zernike moment invariants in recognition and pose estimation of three-dimensional objects. Belkasim et al. [7], [6] did a com- parative study on Zernike moment invariants and used them in shape recognition. Ghosal et al. [8] utilize Zernike moments in composite-edge detection of three-dimensional objects. Legendre moments are constructed using the Legendre polynomials. The image representation capability, information redundancy, noise sensitivity and computation aspects of Legendre moments are studied in [2], [10]–[12]. Bailey et al. [13] used Legendre moments for the recognition of handwritten Arabic numerals. One common problem with the aforementioned moments is the discretization error, which accumulates as the order of the moments increases, and hence limits the accuracy of the computed moments [14]–[16]. Liao and Pawlak [15] did a discretization error analysis of moments and proposed a varia- tion of Simpson’s rule to keep the approximation error under certain level. Some studies concerning the discretization error in the case of geometric moments were also performed by Teh and Chin [17]. A general treatment on the quantization error can be found in [16]. Besides the discretization error, other problems associated with continuous orthogonal moments are large variations in the dynamic range of values and the need to transform coordinate spaces for Zernike and Legendre moments. Recently, to remedy this problem, a set of discrete orthog- onal moment functions based on the discrete Tchebichef poly- nomials was introduced [18]. Their study showed that the im- plementation of Tchebichef moments does not involve any nu- merical approximation since the basis set is orthogonal in the discrete domain of the image coordinate space. This property makes Tchebichef moments superior to the conventional contin- uous orthogonal moments in terms of preserving the analytical property needed to ensure information redundancy in a moment set. In this paper, another new set of orthogonal moments is proposed. It is based on the discrete classical Krawtchouk poly- nomials [19]–[24]. Similar to Tchebichef moments, there is no need for spatial normalization; hence, the error in the computed Krawtchouk moments due to discretization is nonexistent. Unlike the above-mentioned moments, Krawtchouk moments have the ability of being able to extract local features from any region-of-interest in an image. This can be accomplished by varying parameter of the binomial distribution associated with the Krawtchouk polynomials. For a list of discrete polynomials, the readers are referred to [20], [21]. 1057-7149/03$17.00 © 2003 IEEE