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 novel approach to the fast computation of Zernike moments

This paper presents a novel approach to the fast computation of Zernike moments from a digital image. Most existing fast methods for computing Zernike moments have focused on the reduction of the computational complexity of the Zernike 1-D radial polynomials by introducing their recurrence relations. Instead, in our proposed method, we focus on the reduction of the complexity of the computation of the 2-D Zernike basis functions. As Zernike basis functions have specific symmetry or anti-symmetry about the x-axis, the y-axis, the origin, and the straight line y=x, we can generate the Zernike basis functions by only computing one of their octants. As a result, the proposed method makes the computation time eight times faster than existing methods. The proposed method is applicable to the computation of an individual Zernike moment as well as a set of Zernike moments. In addition, when computing a series of Zernike moments, the proposed method can be used with one of the existing fast methods for computing Zernike radial polynomials. This paper also presents an accurate form of Zernike moments for a discrete image function. In the experiments, results show the accuracy of the form for computing discrete Zernike moments and confirm that the proposed method for the fast computation of Zernike moments is much more efficient than existing fast methods in most cases.     

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.     

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。     

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.     

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.     

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.     

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.     

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.     

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.     

一种Pseudo-Zernike矩的改进算法

Pseudo—Zernike矩在模式识别中被广泛采用。但由于Pseudo-Zernike矩的复杂性，相关算法研究尚未得到良好解决。文中对常用算法进行改进，提出了一种直接将矩形图像映射到单位圆内进而求取Pseudo—Zernike矩的算法，并将其应用于识别阿拉伯数字。实验结果表明该算法在计算速度和识别精度上都有明显提高。     

正交傅里叶—梅林矩的一种高效算法*

提出了一种高效计算图像正交傅里叶—梅林矩的算法。该算法通过消除正交多项式中的阶乘项和提取该图像矩的公共项以提高图像矩值的计算性能。实验分析表明,与传统的直接计算方法相比,该算法可有效节省计算时间,尤其是在计算高阶连续矩情况下性能更好。     

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.     

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.     

基于Zernike矩的视频对象零水印算法

针对几何攻击所导致的水印不同步问题,提出了一种结合Zernike矩和图像归一化的有效视频对象水印算法。Zernike矩的幅度具有旋转不变性,缩放和平移不变性通过图像归一化取得。水印嵌入采用零水印方案,解决了基于Zernike矩的图像重构效果不理想和重构过程中复杂度高的问题。实验结果表明,该水印算法对旋转、缩放等几何操作具有鲁棒性,同时对压缩、滤波、高斯噪声等常见的图像处理操作也具有鲁棒性。     

On the fast computation of orthogonal Fourier–Mellin moments with improved numerical stability

A fast and numerically stable recursive method for the computation of orthogonal Fourier?CMellin moments (OFMMs) is proposed. Fast recursive method is developed for the radial polynomials which occur in the kernel function of the OFMMs, thus enhancing the overall computation speed. The proposed method is free from any overflow situations as it does not consist of any factorial term. It is also free from underflow situations as no power terms are involved. The proposed recursive method is claimed to be fastest in comparison with the direct and other methods to compute OFMMs till date. The elimination of the computation of factorial terms makes the moments very stable even up to an order of 200, which become instable in conventional or in any other recursive methods proposed earlier wherein instability occurs at moment order ??25. Experiments are performed on standard test images to prove the superiority of the proposed method on existing methods in terms of speed and numerical stability.     

Fast and numerically stable methods for the computation of Zernike moments

Zernike moments (ZMs) are used in many image processing applications due to their superior performance over other moments. However, they suffer from high computation cost and numerical instability at high order of moments. In the past many recursive methods have been developed to improve their speed performance and considerable success has been achieved. The analysis of numerical stability has also gained momentum as it affects the accuracy of moments and their invariance property. There are three recursive methods which are normally used in ZMs calculation—Prata’s, Kintner’s and q-recursive methods. The earlier studies have found the q-recursive method outperforming the two other methods. In this paper, we modify Prata’s method and present a recursive relation which is proved to be faster than the q-recursive method. Numerical instability is observed at high orders of moments with the q-recursive method suffering from the underflow problem while the modified Prata’s method suffering from finite precision error. The modified Kintner’s method is the least susceptible to these errors. Keeping in view the better numerical stability, we further make the modified Kintner’s method marginally faster than the q-recursive method. We recommend the modified Prata’s method for low orders (≤90) and Kintner’s fast method for high orders (>90) of ZMs.