一种快速计算Zernike矩的改进q-递归算法

提出了一种快速计算Zernike矩的改进q-递归算法,该方法通过同时降低核函数中Zernike多项式和Fourier函数的计算复杂度以提高Zernike矩的计算效率。采用 q-递归法快速计算Zernike多项式以避免复杂的阶乘运算,再利用x轴、y轴、x=y和x=-y 4条直线将图像域分成8等分。计算Zernike矩时,仅计算其中1个区域的核函数的值,其他区域的值可以通过核函数关于4条直线的对称性得到。该方法不仅减少了核函数的存储空间,而且大大降低了Zernike矩的计算时间。试验结果表明,与现有方法相比,改进q-递归算法具有更好的性能。     

Efficient computation of radial moment functions using symmetrical property

The applications of radial moment functions such as orthogonal Zernike and pseudo-Zernike moments in real-world have been limited by the computational complexity of their radial polynomials. The common approaches used in reducing the computational complexity include the application of recurrence relations between successive radial polynomials and coefficients. In this paper, a novel approach is proposed to further reduce the computation complexity of Zernike and pseudo-Zernike polynomials based on the symmetrical property of radial polynomials. By using this symmetrical property, the real-valued radial polynomials computation is reduced to about one-eighth of the full set polynomials while the computation of the exponential angle values is reduced by half. This technique can be integrated with existing fast computation methods to further improve the computation speed. Besides significant reduction in computation complexity, it also provides vast reduction in memory storage.     

A comparative analysis of algorithms for fast computation of Zernike moments

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, Belkasim's, Prata's, Kintner's 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.     

A systematic method for efficient computation of full and subsets Zernike moments

A new method is proposed for fast and accurate computation of Zernike moments. This method presents a novel formula for computing exact Zernike moments by using exact complex moments where the exact values of complex moments are computed by mathematical integration of the monomials over digital image pixels. The proposed method is applicable to compute the full set of Zernike moments as well as the subsets of individual order, repetition and an individual moment. A comparison with other conventional methods is performed. The results show the superiority of the proposed method.     

A novel algorithm for fast computation of Zernike moments

Zernike moments (ZMs) have been successfully used in pattern recognition and image analysis due to their good properties of orthogonality and rotation invariance. However, their computation by a direct method is too expensive, which limits the application of ZMs. In this paper, we present a novel algorithm for fast computation of Zernike moments. By using the recursive property of Zernike polynomials, the inter-relationship of the Zernike moments can be established. As a result, the Zernike moment of order n with repetition m, Z_nm, can be expressed as a combination of Z_n−2,m and Z_n−4,m. Based on this relationship, the Zernike moment Z_nm, for n>m, can be deduced from Z_mm. To reduce the computational complexity, we adopt an algorithm known as systolic array for computing these latter moments. Using such a strategy, the multiplication number required in the moment calculation of Z_mm can be decreased significantly. Comparison with known methods shows that our algorithm is as accurate as the existing methods, but is more efficient.     

Fast computation of pseudo Zernike moments

A fast and numerically stable method to compute pseudo Zernike moments is proposed in this paper. Several pseudo Zernike moment computation architectures are also implemented and some have overflow problems when high orders are computed. In addition, a correction to a previous two stage p-recursive pseudo Zernike radial polynomial algorithm is introduced. The newly proposed method that is based on computing pseudo Zernike radial polynomials through their relation to Zernike radial polynomials is found to be one and half times faster than the best algorithm reported up to date.     

Fast computation of exact Zernike moments using cascaded digital filters

Zernike moments have been extensively used and have received much research attention in a number of fields: object recognition, image reconstruction, image segmentation, edge detection and biomedical imaging. However, computation of these moments is time consuming. Thus, we present a fast computation technique to calculate exact Zernike moments by using cascaded digital filters. The novelty of the method proposed in this paper lies in the computation of exact geometric moments directly from digital filter outputs, without the need to first compute geometric moments. The mathematical relationship between digital filter outputs and exact geometric moments is derived and then they are used in the formulation of exact Zernike moments. A comparison of the speed of performance of the proposed algorithm with other state-of-the-art alternatives shows that the proposed algorithm betters current computation time and uses less memory.     

Orthogonal moments based on exponent functions: Exponent-Fourier moments

In this paper, we propose a new set of orthogonal moments based on Exponent functions, named Exponent-Fourier moments (EFMs), which are suitable for image analysis and rotation invariant pattern recognition. Compared with Zernike polynomials of the same degree, the new radial functions have more zeros, and these zeros are evenly distributed, this property make EFMs have strong ability in describing image. Unlike Zernike moments, the kernel of computation of EFMs is extremely simple. Theoretical and experimental results show that Exponent-Fourier moments perform very well in terms of image reconstruction capability and invariant recognition accuracy in noise-free, noisy and smooth distortion conditions. The Exponent-Fourier moments can be thought of as generalized orthogonal complex moments.     

Fast computation of accurate Zernike moments

Zernike polynomials are continuous orthogonal polynomials defined in polar coordinates over a unit disk. Zernike moment’s computation using conventional methods produced two types of errors namely approximation and geometrical. Approximation errors are removed by using exact Zernike moments. Geometrical errors are minimized through a proper mapping of the image. Exact Zernike moments are expressed as a combination of exact radial moments, where exact values of radial moments are computed by mathematical integration of the monomial polynomials over digital image pixels. A fast algorithm is proposed to accelerate the moment’s computations. A comparison with other conventional methods is performed. The obtained results explain the superiority of the proposed method.     

Algorithms for fast computation of Zernike moments and their numerical stability

Accuracy, speed and numerical stability are among the major factors restricting the use of Zernike moments (ZMs) in numerous commercial applications where they are a tool of significant utility. Often these factors are conflicting in nature. The direct formulation of ZMs is prone to numerical integration error while in the recent past many fast algorithms are developed for its computation. On the other hand, the relationship between geometric moments (GMs) and ZMs reduces numerical integration error but it is observed to be computation intensive. We propose fast algorithms for both the formulations. In the proposed method, the order of time complexity for GMs-to-ZMs formulation is reduced and further enhancement in speed is achieved by using quasi-symmetry property of GMs. The existing q-recursive method for direct formulation is further modified by incorporating the recursive steps for the computation of trigonometric functions. We also observe that q-recursive method provides numerical stability caused by finite precision arithmetic at high orders of moment which is hitherto not reported in the literature. Experimental results on images of different sizes support our claim.     

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.     

Efficient computation of Zernike and Pseudo-Zernike moments for pattern classification applications

Two novel algorithms for the fast computation of the Zernike and Pseudo-Zernike moments are presented in this paper. The proposed algorithms are very useful, particularly in the case of using the computed moments, as discriminative features in pattern classification applications, where the computation of single moments of several orders is required. The derivation of the algorithms is based on the elimination of the factorial computations, by computing recursively the fractional terms of the orthogonal polynomials being used. The newly introduced algorithms are compared to the direct methods, which are the only methods that permit the computation of single moments of any order. The computational complexity of the proposed method is O(p ²) in multiplications, with p being the moment order, while the corresponding complexity of the direct method is O(p ³). Appropriate experiments justify the superiority of the proposed recursive algorithms over the direct ones, establishing them as alternative to the original algorithms, for the fast computation of the Zernike and Pseudo-Zernike moments.     

A new class of Zernike moments for computer vision applications

A Modified Direct Method for the computation of the Zernike moments is presented in this paper. The presence of many factorial terms, in the direct method for computing the Zernike moments, makes their computation process a very time consuming task. Although the computational power of the modern computers is impressively increasing, the calculation of the factorial of a big number is still an inaccurate numerical procedure. The main concept of the present paper is that, by using Stirling’s Approximation formula for the factorial and by applying some suitable mathematical properties, a novel, factorial-free direct method can be developed. The resulted moments are not equal to those computed by the original direct method, but they are a sufficiently accurate approximation of them. Besides, their variability does not affect their ability to describe uniquely and distinguish the objects they represent. This is verified by pattern recognition simulation examples.     

几何误差和数值误差最小化的Zernike矩

形状特征提取和表示是基于内容图像检索的重要研究内容之一.提出一种几何误差和数值误差最小化的Zernike矩方法,并且把这种方法应用于形状特征提取和表示.该方法把图像中的兴趣区域映射到单位圆里,通过计算变换后图像在Zernike多项式上的投影来取得Zernike矩,并且通过把心理生理学的研究成果引入Zernike矩的计算过程来提高系统的检索性能.通过实验对传统Zernike矩方法、几何误差和数值误差最小化的Zernike矩方法进行了比较,发现从重构角度采用几何误差和数值误差最小化的Zernike矩方法优于采用传统Zernike矩方法.而从检索角度采用几何误差和数值误差最小化的Zernike矩方法的系统比采用传统Zernike矩方法的系统具有更好的检索性能.     

Practical fast computation of Zernike moments

The fast computation of Zernike moments from normalized gometric moments has been developed in this paper,The computation is multiplication free and only additions are needed to generate Zernike moments .Geometric moments are generated using Hataming‘s filter up to high orders by a very simple and straightforward computaion scheme.Other kings of monents(e.g.,Legendre,pseudo Zernike)can be computed using the same algorithm after giving the proper transformaitons that state their relations to geometric moments.Proper normaliztions of geometric moments are necessary so that the method can be used in the efficient computation of Zernike moments.To ensure fair comparisons,recursive algorithms are used to generate Zernike polynoials and other coefficients.The computaional complexity model and test programs show that the speed-up factor of the proposed algorithm is superior with respect ot other fast and /or direct computations It perhaps is the first time that Zernike moments can be computed in real time rates,which encourages the use of Zernike moment features in different image retrieval systems that support huge databases such as the XM experimental model stated for the MPEG-7 experimental core.It is concluded that choosing direct copmutation would be impractical.     

Computing Bessel functions of the second kind in extreme parameter regimes

A useful method of computing the integral order Bessel functions of the second kind Y_n(x+iy) when either, the absolute value of the real part, or the imaginary part of the argument z=x+iy is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when x∼0 (or y∼0) and is useful because a number existing algorithms and methods fail to give correct results for small x or small y. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.     

基于两变量Krawtchouk矩的图像重建

传统的离散正交Krawtchouk矩的基函数由两个单变量的Krawtchouk多项式乘积构成,它割裂平面两个方向之间的联系。提出了一种新的、以两变量Krawtchouk正交多项式为基函数的图像矩,并推导了正则化后两变量多项式的简单的计算方法。重建实验结果表明,相对于同系数的单变量的离散正交矩,两变量离散正交矩的重建误差更小。     

Image analysis by Bessel-Fourier moments

In this paper, we proposed a new set of moments based on the Bessel function of the first kind, named Bessel-Fourier moments (BFMs), which are more suitable than orthogonal Fourier-Mellin and Zernike moments for image analysis and rotation invariant pattern recognition. Compared with orthogonal Fourier-Mellin and Zernike polynomials of the same degree, the new orthogonal radial polynomials have more zeros, and these zeros are more evenly distributed. The Bessel-Fourier moments can be thought of as generalized orthogonalized complex moments. Theoretical and experimental results show that the Bessel-Fourier moments perform better than the orthogonal Fourier-Mellin and Zernike moments (OFMMs and ZMs) in terms of image reconstruction capability and invariant recognition accuracy in noise-free, noisy and smooth distortion conditions.     

Image representation using accurate orthogonal Gegenbauer moments

Image representation by using polynomial moments is an interesting theme. In this paper, image representation by using orthogonal Gegenbauer function is presented. A novel method for accurate and fast computation of orthogonal Gegenbauer moments is proposed. The accurate values of Gegenbauer moments are obtained by mathematically integrating Gegenbauer polynomials multiplied by their weight functions over the digital image pixels. A novel recurrence formula is derived for the kernel generation. The proposed method removes the numerical approximation errors involved in conventional method. A fast algorithm is proposed to accelerate the moment’s computations. A comparison with the conventional method is performed. The obtained results explain the efficiency and the superiority of the proposed method.