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.     

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 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 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.     

基于Zernike矩的快速PPB相干斑抑制算法

应用非局部均值算法到SAR图像相干斑抑制中时存在计算量大、图像自相似性利用不足等缺点,严重制约该类算法的实际应用.针对该问题,提出一种SAR图像快速PPB滤波算法.给出基于积分图的PPB快速算法,引入Zemike矩构造一种图块相似性度量,利用Cosmo实测SAR数据进行实验.实验结果表明,该算法提高了图像自相似特征利用率,显著改善了滤波效果.     

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.     

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.     

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不变矩的零水印算法

为了实施图像的版权保护,提出了一种基于Zernike不变矩的零水印算法。利用Zernike矩幅度的旋转不变性提取图像特征点,将特征点通过设定的阈值量化为二值序列作为零水印保存到零水印信息库（IPR）。证明版权利用Zernike矩提取待检测图像特征点,以同样的阈值量化为二值水印序列与零水印作相关性检测。实验结果表明,该方法很好地解决了传统水印鲁棒性和不可感知性的矛盾问题,对常用的图像处理和几何攻击具有较好的鲁棒性。     

基于局部Zernike矩的RST不变零水印算法

旋转、缩放和平移(RST)等几何攻击能够破坏水印检测的同步性,使得水印检测失败。针对此问题,提出了一种基于图像局部Zernike矩的RST不变零水印算法。Zernike矩的幅度具有旋转不变性,再结合图像归一化,使其具有缩放和平移不变性。由于Zernike矩的图像重构效果不理想且重构过程中复杂度高,因此水印嵌入选择零水印方案。实验结果表明,该算法对旋转、缩放和平移(RST)的攻击具有很好的鲁棒性,同时对JPEG压缩、加噪、滤波等常见的图像处理操作也具有很好的鲁棒性。     

An approach for on-line signature authentication using Zernike moments

In this work, shape analysis of the acceleration plot, using lower order Zernike moments is performed for authentication of on-line signature. The on-line signature uses time functions of the signing process. The lower order Zernike moments represent the global shape of a pattern. The derived feature, acceleration vector is computed for the sample signature which comprises on-line pixels. The Zernike moment represent the shape of the acceleration plot. The summation value of a Zernike moment for a signature sample is obtained on normalized acceleration values. This type of substantiation decreases the influence of primary features with respect to translation, scaling and rotation at preprocessing stage. Zernike moments provide rotation invariance. In this investigation it was evident that the summation of magnitude of a Zernike moment for a genuine sample was less as compared to the summation of magnitude of a imposter sample. The number of derivatives of acceleration feature depends on the structural complexity of the signature sample. The computation of best order by polynomial fitting and reference template of a subject is discussed. The higher order derivatives of acceleration feature are considered. Signatures with higher order polynomial fitting and complex structure require higher order derivatives of acceleration. Each derivative better represents a portion of signature. The best result obtained is 4% of False Rejection Rate [FRR] and 2% of False Acceptance Rate [FAR].     

On the accuracy of image normalization by Zernike moments

Zernike Moment (ZM) is an effective region-based shape representation technique. The extracted ZM features should be independent of scale, position and orientation, which can be achieved by ZM-based image normalization. Nevertheless, due to the discretization of digital image and the presence of noise, the normalization is imperfect. Thus, in practice Zernike Moment Invariants (ZMI) cannot perfectly preserve the invariant properties. In this paper, firstly the ZM-based image normalization criteria are derived, and then I theoretically and experimentally evaluate the accuracy of the ZM-based image normalization. Our theoretical and experimental results not only disclose some essential facts, but also have some new findings. The relations between the accuracy of ZM-based image normalization and its influencing factors are established. A creative pseudo-polar coordinate is proposed to cut down the geometrical errors to the greatest extent. Furthermore, I suggest several techniques to improve the accuracy of image normalization. By combining moment-based image normalization with the image regularization theory and the scale-space theory, and several new conclusions are drawn. Our experimental results show that the proposed techniques can preserve the invariance of image normalization and restrain the influence of noise quite effectively.     

A comment on: “Fast and numerically stable methods for the computation of Zernike moments” by Singh et al. [Pattern Recognition, 43(2010), Pages 2497-2506]

The aim of the present comment is to point out that the q-Recursive relation introduced by Singh et al. is wrong. These errors have been corrected in the comment. The numerical experiments show that the correct recurrence relation is consistent with the original q-Recursive method.     

基于Zernike形状矩的地图匹配算法

地图匹配算法的有效性和可靠性对于智能交通系统而言是非常重要的,而目前存在的地图匹配算法在一些复杂环境下（如道路交叉口）仍然不能提供合理的输出。采用D-S证据理论融合当前车辆位置信息和方向信息可以有效地扩大待匹配道路之间的差异,但在复杂路网下信息量的不足会降低其匹配精度。因此,为了提高道路网络中的地图匹配精度,提出了基于Zernike形状矩的地图匹配算法。新算法引入Zernike矩描述轨迹曲线的形状,进一步修正了错误结果。通过仿真和实验表明,新算法在复杂环境下具有较强的有效性和可靠性。     

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。     

Shot boundary detection using Zernike moments in multi-GPU multi-CPU architectures

This paper presents an analysis of a Multi-GPU Multi-CPU environment, along with the different possible hybrid combinations. The analysis has been performed for a shot boundary detection application, based on Zernike moments, although it is general enough to be applied to many different application areas. A deep study of the performance, bottlenecks and design challenges is carried out showing the validity of this approach and achieving very high frame per second rates. In this paper, Zernike calculations are carried out on GPUs, taking advantage of a packing strategy proposed to minimize host-device communication time.     

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 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.     

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.     

基于Zernike矩亚像素边缘检测的快速算法

为了克服传统的Zernike法在边缘检测过程中,由于人工手动选取阈值而带来的低效率、高误判等不足,将原算法与Otsu法相结合,提出了一种边缘检测的快速算法。利用传统的Zernike法计算出图像的阶跃灰度矩阵,再将该矩阵作为计算对象,用Otsu法直接得到最优的阶跃灰度阈值进行边缘判别,并考虑了由于边缘模型带来的误差,在保证检测效果的同时缩短了检测时间。实验结果表明,改进的算法能够更有效地完成边缘检测,补偿后的亚像素定位更准确。