Ramakrishnan Mukundan

Image analysis by Tchebichef moments

This paper introduces a new set of orthogonal moment functions based on the discrete Tchebichef polynomials. The Tchebichef moments can be effectively used as pattern features in the analysis of two-dimensional images. The implementation of moments proposed in this paper does not involve any numerical 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 orthogonal moments such as Legendre moments and Zernike moments, in terms of preserving the analytical properties needed to ensure information redundancy in a moment set. The paper also details the various computational aspects of Tchebichef moments and demonstrates their feature representation capability using the method of image reconstruction.

Moment functions in image analysis : theory and applications

A comprehensive treaty on the theory and applications of moment functions in image analysis. Moment functions are widely-used in various realms of computer vision and image processing. Numerous algorithms and techniques have been developed using image moments, in the areas of pattern recognition, object identification, three-dimensional object pose estimation, robot sensing, image coding and reconstruction. This is a compilation of the theoretical aspects related to different types of moment functions, and their applications in the above areas. The book is organized in two parts. The first part discusses the fundamental concepts behind important moments such as geometric moments, complex moments, Legendre moments, Zernike moments, and moment tensors. Most of the commonly-used properties of moment functions and the mathematical framework for the derivation of basic theorems and results are discussed. This includes the derivation of moment invariants, implementation aspects of moments, transport properties, and fast methods for computing the moment functions for both binary and gray-level images. The second part presents the key application areas of moments, such as pattern recognition, object identification, image-based pose estimation, edge detection, clustering, segmentation, coding and reconstruction. Important algorithms in each of these areas are discussed. A list of bibliographic references on image moments is also included.

A comparative analysis of algorithms for fast computation of Zernike moments

Abstract This paper details a comparative analysis on time taken by the present and proposed methods to compute the Zernike moments, Z pq . The present method comprises of Direct, Belkasims, Pratas, Kintners and Coefficient methods. We propose a new technique, denoted as q -recursive method, specifically for fast computation of Zernike moments. It uses radial polynomials of fixed 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 specific order p using binary and grayscale images. In both the cases, the q -recursive method takes the shortest time to compute Zernike moments.

Fast computation of Legendre and Zernike moments

This paper presents recursive algorithms for fast computation of Legendre and Zernike moments of a grey-level image intensity distribution. For a binary image, a contour integration method is developed for the evaluation of Legendre moments using only the boundary information. A method for recursive calculation of Zernike polynomial coefficients is also given. A square-to-circular image transformation scheme is introduced to minimize the computation involved in Zernike moment functions. The recursive formulae can also be used in inverse moment transforms to reconstruct the original image from moments. The mathematical framework of the algorithms is given in detail, and illustrated with binary and grey-level images.

Some computational aspects of discrete orthonormal moments

Discrete orthogonal moments have several computational advantages over continuous moments. However, when the moment order becomes large, discrete orthogonal moments (such as the Tchebichef moments) tend to exhibit numerical instabilities. This paper introduces the orthonormal version of Tchebichef moments, and analyzes some of their computational aspects. The recursive procedure used for polynomial evaluation can be suitably modified to reduce the accumulation of numerical errors. The proposed set of moments can be used for representing image shape features and for reconstructing an image from its moments with a high degree of accuracy.

Translation Invariants of Zernike Moments

Moment functions defined using a polar coordinate representation of the image space, such as radial moments and Zernike moments, are used in several recognition tasks requiring rotation invariance. However, this coordinate representation does not easily yield translation invariant functions, which are also widely sought after in pattern recognition applications. This paper presents a mathematical framework for the derivation of translation invariants of radial moments defined in polar form. Using a direct application of this framework, translation invariant functions of Zernike moments are derived algebraically from the corresponding central moments. Both derived functions are developed for non-symmetrical as well as symmetrical images. They mitigate the zero-value obtained for odd-order moments of the symmetrical images. Vision applications generally resort to image normalization to achieve translation invariance. The proposed method eliminates this requirement by providing a translation invariance property in a Zernike feature set. The performance of the derived invariant sets is experimentally confirmed using a set of binary Latin and English characters.

Translation and scale invariants of Legendre moments

Abstract By convention, the translation and scale invariant functions of Legendre moments are achieved by using a combination of the corresponding invariants of geometric moments. They can also be accomplished by normalizing the translated and/or scaled images using complex or geometric moments. However, the derivation of these functions is not based on Legendre polynomials. This is mainly due to the fact that it is difficult to extract a common displacement or scale factor from Legendre polynomials. In this paper, we introduce a new set of translation and scale invariants of Legendre moments based on Legendre polynomials. The descriptors remain unchanged for translated, elongated, contracted and reflected non-symmetrical as well as symmetrical images. The problems associated with the vanishing of odd-order Legendre moments of symmetrical images are resolved. The performance of the proposed descriptors is experimentally confirmed using a set of binary English, Chinese and Latin characters. In addition to this, a comparison of computational speed between the proposed descriptors and the aforesaid geometric moments-based method is also presented.

An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments

Pseudo-Zernike moments have better feature representation capability, and are more robust to image noise than those of the conventional Zernike moments. However, due to the computation complexity of pseudo-Zernike polynomials, pseudo-Zernike moments are yet to be extensively used as feature descriptors as compared to Zernike moments. In this paper, we propose two new algorithms, namely coefficient method and p-recursive method, to accelerate the computation of pseudo-Zernike moments. Coefficient method calculates polynomial coefficients recursively. It eliminates the need of using factorial functions. Individual order or index of pseudo-Zernike moments can be derived independently, which is useful if selected orders or indices of moments are needed as pattern features. p-recursive method uses a combination of lower order polynomials to derive higher order polynomials with the same index q. Fast computation is achieved because it eliminates the requirements of calculating polynomial coefficients, Bpqk, and power of radius, rk, in each polynomial. The performance of the proposed algorithms on moment computation and image reconstruction, as compared to those of the present methods, are experimentally verified using a set of binary and grayscale images.

The scale invariants of pseudo-Zernike moments

The definition of pseudo-Zernike moments has a form of projection of the image intensity function onto the pseudo-Zernike polynomials, and they are defined using a polar coordinate representation of the image space. Hence, they are commonly used in recognition tasks requiring rotation invariance. However, this coordinate representation does not easily yield a scale invariant function because it is difficult to extract a common scale factor from the radial polynomials. As a result, vision applications generally resort to image normalisation method or using a combination of scale invariants of geometric orradial moments to achieve the corresponding invariants of pseudo-Zernike moments. In this paper, we present a mathematical framework to derive a new set of scale invariants of pseudo-Zernike moments based on pseudo-Zernike polynomials. They are algebraically obtained by eliminating the scale factor contained in the scaled pseudo-Zernike moments. They remain unchanged under equal-shape expansion, contraction and reflection of theoriginal image. They can be directly computed from any scaled image without prior knowledge of the normalisation parameters, or assistance of geometric or radial moments. Their performance is experimentally verified using a set of Chinese and Latin characters. In addition, a comparison of computational speed between the proposed descriptors and the present methods is also presented.

A Fast 4

The discrete Tchebichef transform (DTT) is a transform method based on discrete orthogonal Tchebichef polynomials, which have applications recently found in image analysis and compression. This letter introduces a new fast 4 x 4 forward DTT algorithm. The new algorithm requires only 32 multiplications and 66 additions, while the best-known method using two properties of the DTT requires 64 multiplications and 96 additions. The proposed method could be used as the base case for recursive computation of transform coefficients. Experimental results showing performance improvement over existing techniques are presented.