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.     

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.     

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.     

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.     

一种快速的具有旋转不变性的模板匹配方法

传统的基于相关的匹配方法计算量相当大,而且当模板相对于搜索图有角度旋转时,匹配的计算量更大。用圆警影和zemike矩的方法对快速的且具有旋转不变性的模板匹配方法进行了研究,圆投影将图像由二维变换成一维,这样就降低了计算复杂度,通过相似性度量可进行快速地粗匹配,然后在可能的匹配点中,再用Zemike矩实现精匹配。     

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.     

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.     

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.     

Efficient Legendre moment computation for grey level images

Legendre orthogonal moments have been widely used in the field of image analysis. Because their computation by a direct method is very time expensive, recent efforts have been devoted to the reduction of computational complexity. Nevertheless, the existing algorithms are mainly focused on binary images. We propose here a new fast method for computing the Legendre moments, which is not only suitable for binary images but also for grey level images. We first establish a recurrence formula of one-dimensional (1D) Legendre moments by using the recursive property of Legendre polynomials. As a result, the 1D Legendre moments of order p, L_p=L_p(0), can be expressed as a linear combination of L_p-1(1) and L_p-2(0). Based on this relationship, the 1D Legendre moments L_p(0) can thus be obtained from the arrays of L₁(a) and L₀(a), where a is an integer number less than p. To further decrease the computation complexity, an algorithm, in which no multiplication is required, is used to compute these quantities. The method is then extended to the calculation of the two-dimensional Legendre moments L_pq. We show that the proposed method is more efficient than the direct method.     

A high capacity image adaptive watermarking scheme with radial harmonic Fourier moments

In this paper, we introduce a novel image adaptive technique for high capacity watermarking scheme using accurate and fast radial harmonic Fourier moments (RHFMs). The high embedding capacity is achieved by improving the hiding ratio after reducing inaccuracies in the computation of RHFMs. The binary watermark is embedded by performing the conditional quantization of selected RHFMs magnitudes to minimize the spatial distortion added to the host image. In addition, fast algorithms based on 8-way symmetry/anti-symmetry properties and recursive relations for the computation of sinusoidal kernel functions are adopted to enhance the speed of RHFMs-based watermarking process. Experimental studies show that the proposed watermarking scheme provides higher embedding capacity, good visual imperceptibility, better robustness to geometric distortions and common signal processing transformations, and lower computational complexity compared to the existing Zernike and pseudo-Zernike moments (ZMs/PZMs)-based watermarking schemes.     

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.     

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。     

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.     

伪Zernike矩不变性分析及其改进研究

伪 Zernike矩是基于图象整个区域的形状描述算子 ,而基于轮廓的形状描述子 ,例如曲率描述子、傅立叶描述子和链码描述子等是不能正确描述由几个不连接区域组成的形状的 ,因为这些算子只能描述单个的轮廓形状 .同时 ,由于伪 Zernike矩的基是正交径向多项式 ,和 Hu矩相比 ,除了具有旋转不变性、高阶矩和低阶矩能表达不同信息等特征外 ,还具有冗余性小、可以任意构造高阶矩等特点 ,另外 ,伪 Zernike矩还可以用于目标重构 .目前 ,伪 Zernike矩没有得到广泛的应用 ,其中的一个主要原因是 ,它不具备真正意义上的比例不变性 .为了能使伪Zernike矩得到更广泛的应用 ,在详细分析伪 Zernike矩不变性的基础上 ,提出了伪 Zernike矩的改进方法 ,使改进后的伪 Zernike矩在保持旋转不变性的同时 ,还具有真正意义上的比例不变性 ,同时给出了部分的实验分析结果 .实验结果证明 ,该改进后的伪 Zernike矩较改进前的伪 Zernike矩 ,具有更好的旋转和比例不变性 .     

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 adaptive algorithms for minor component analysis using Householder transformation

In this paper, we propose new adaptive algorithms for the extraction and tracking of the least (minor) or eventually, principal eigenvectors of a positive Hermitian covariance matrix. The main advantage of our proposed algorithms is their low computational complexity and numerical stability even in the minor component analysis case. The proposed algorithms are considered fast in the sense that their computational cost is O(np) flops per iteration where n is the size of the observation vector and p<n is the number of eigenvectors to estimate.We consider OJA-type minor component algorithms based on the constraint and non-constraint stochastic gradient technique. Using appropriate fast orthogonalization procedures, we introduce new fast algorithms that extract the minor (or principal) eigenvectors and guarantee good numerical stability as well as the orthogonality of their weight matrix at each iteration. In order to have a faster convergence rate, we propose a normalized version of these algorithms by seeking the optimal step-size. Our algorithms behave similarly or even better than other existing algorithms of higher complexity as illustrated by our simulation results.     

Image reconstruction from continuous Gaussian–Hermite moments implemented by discrete algorithm

The problem of image reconstruction from its statistical moments is particularly interesting to researchers in the domain of image processing and pattern recognition. Compared to geometric moments, the orthogonal moments offer the ability to recover much more easily the image due to their orthogonality, which allows reducing greatly the complexity of computation in the phase of reconstruction. Since the 1980s, various orthogonal moments, such as Legendre moments, Zernike moments and discrete Tchebichef moments have been introduced early or late to image reconstruction. In this paper, another set of orthonormal moments, the Gaussian–Hermite moments, based on Hermite polynomials modulated by a Gaussian envelope, is proposed to be used for image reconstruction. Especially, the paper's focus is on the determination of the optimal scale parameter and the improvement of the reconstruction result by a post-processing which make Gaussian–Hermite moments be useful and comparable with other moments for image reconstruction. The algorithms for computing the values of the basis functions, moment computation and image reconstruction are also given in the paper, as well as a brief discussion on the computational complexity. The experimental results and error analysis by comparison with other moments show a good performance of this new approach.     

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.     

New algorithms for max restricted path consistency

Max Restricted Path Consistency (maxRPC) is a local consistency for binary constraints that enforces a higher order of consistency than arc consistency. Despite the strong pruning that can be achieved, maxRPC is rarely used because existing maxRPC algorithms suffer from overheads and redundancies as they can repeatedly perform many constraint checks without triggering any value deletions. In this paper we propose and evaluate techniques that can boost the performance of maxRPC algorithms by eliminating many of these overheads and redundancies. These include the combined use of two data structures to avoid many redundant constraint checks, and the exploitation of residues to quickly verify the existence of supports. Based on these, we propose a number of closely related maxRPC algorithms. The first one, maxRPC3, has optimal O(end ³) time complexity, displays good performance when used stand-alone, but is expensive to apply during search. The second one, maxRPC3 ^rm , has O(en ² d ⁴) time complexity, but a restricted version with O(end ⁴) complexity can be very efficient when used during search. The other algorithms are simple modifications of maxRPC3 ^rm . All algorithms have O(ed) space complexity when used stand-alone. However, maxRPC3 has O(end) space complexity when used during search, while the others retain the O(ed) complexity. Experimental results demonstrate that the resulting methods constantly outperform previous algorithms for maxRPC, often by large margins, and constitute a viable alternative to arc consistency on some problem classes.