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.     

基于DWT-DCT和Zernike矩的灰度级鲁棒性水印算法

鲁棒性是数字水印极为重要的指标.目前大多数文献中的鲁棒性水印方案对常规信号处理攻击均具有良好的鲁棒性,但抗几何攻击的鲁棒性相对较弱.提出了一种新的灰度级鲁棒性水印算法,对常规信号处理攻击和几何攻击均具有良好的鲁棒性.水印算法的鲁棒性由以下3方面获得:1)选择载体图像的内切圆盘作为Zernike矩的计算域,选择内切圆盘的内接正方形作为水印的嵌入区域.2)在载体图像嵌入区域的层叠DWT-DCT域嵌入水印,获得对常规信号处理攻击的强鲁棒性.3)保存原始含水印图像的2个Zernike矩作为校正几何攻击的参数,获得抗几何攻击能力.实验结果表明,该水印算法嵌入容量大,不仅对大角度旋转和大尺度缩放等几何攻击具有很好的鲁棒性,而且可以有效地抵抗有损压缩和常规信号处理攻击.     

一种新的基于伪Zernike矩的图像盲水印算法

抵抗几何攻击的鲁棒性是目前数字水印技术中的热点也是难点。提出一种新的基于伪Zernike矩的图像盲水印算法,首先计算图像归一化后的伪Zernike矩,然后选取部分合适的矩通过量化调制嵌入水印信息。水印提取时,利用伪Zernike矩的相位信息估计旋转角度进行几何校正,以提高矩的几何不变性。实验结果表明,本算法对于抗几何攻击尤其是旋转攻击具有很好的鲁棒性,同时也能抵抗常规的信号处理攻击。     

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 moment-based nonlocal-means algorithm for image denoising

Image denoising is a crucial step to increase image quality and to improve the performance of all the tasks needed for quantitative imaging analysis. The nonlocal (NL) means filter is a very successful technique for denoising textured images. However, this algorithm is only defined up to translation without considering the orientation and scale for each image patch. In this paper, we introduce the Zernike moments into NL-means filter, which are the magnitudes of a set of orthogonal complex moments of the image. The Zernike moments in small local windows of each pixel in the image are computed to obtain the local structure information for each patch, and then the similarities according to this information are computed instead of pixel intensity. For the rotation invariant of the Zernike moments, we can get much more pixels or patches with higher similarity measure and make the similarity of patches translation-invariant and rotation-invariant. The proposed algorithm is demonstrated on real images corrupted by white Gaussian noise (WGN). The comparative experimental results show that the improved NL-means filter achieves better denoising performance.     

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。     

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。     

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.     

基于Zernike矩和NSCT-SVD的数字水印算法研究

以非下采样Contourlet变换、奇异值分解以及Zernike 矩知识为基础,结合图像不变质心在几何攻击前后相对位置不变的特性和Zernike 矩对旋转、缩放攻击的不变性以及对噪声不敏感的特性,提出了一种利用 Zernike 矩对图像几何校正的 NSCT-SVD几何鲁棒盲水印算法。该方案中宿主图像通过非下采样 Contourlet变换分解提取出低频区域,通过量化每块奇异值矩阵的欧氏范数来嵌入水印。水印检测时,先利用Zernike 矩和不变质心等几何参数对被检测图像进行几何攻击校正,恢复水印的同步信息后再提取水印。实验结果表明,该算法对噪声、滤波、压缩以及各类几何攻击具有较好的鲁棒性。     

Improved quality of reconstructed images using floating point arithmetic for moment calculation

Zernike moments which are superior to geometric moments because of their special properties of image reconstruction and immunity to noise, suffer from several discretization errors. These errors lead to poor quality of reconstructed image and wide variations in the numerical values of the moments. The predominant factor, as observed in this paper, is due to the discrete integer implementation of the steps involved in moment calculation. It is shown in this paper that by modifying the algorithms to include discrete float implementation, the quality of the reconstructed image improves significantly and the first-order moment becomes zero. Low-order Zernike moments have been found to be stable under linear transformations while the high-order moments have large variations. The large variations in high-order moments, however, do not greatly affect the quality of the reconstructed image, implying that they should be ignored when numerical values of moments are used as features. The 11 functions based on geometric moments have also been found to be stable under linear transformations and thus these can be used as features. Pixel level analysis of the images has been carried out to strengthen the results.     

基于DWT-DCT和Zernike矩的鲁棒视频水印算法

为了提供数字视频版权的有效保护,提出一种基于DWT-DCT和Zernike矩的鲁棒视频水印算法。在水印嵌入阶段,先将视频帧从RGB空间转换到YCbCr颜色空间;从YCbCr图像帧的亮度分量中选择出嵌入区并进行分块,通过伪随机法选择出多个子块;执行DWT-DCT获得前n个交流分量,并将水印嵌入到这些分量中;执行DWT-DCT反变换获得水印图像,并计算Zernike矩作为校正几何攻击的参数。在水印提取过程中,通过水印嵌入阶段的伪随机密钥来确定嵌入位置,以此进一步提高水印安全性。实验结果表明,该算法具有较高的水印隐蔽性和鲁棒性。     

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.     

改进后的Zernike矩在文物图像检索中的应用

Zernike矩作为一种基于区域的形状描述子,具有良好的旋转不变性,能够很好地提取图像内部的形状信息,广泛应用于基于内容的图像检索(CBIR)、图像描述、物体定向及模式识别等各个领域.介绍了Zernike矩的定义和基本特性,并对Zernike矩算法提出了改进,使其具有较好的比例不变性,最后讨论了Zernike矩在基于内容的文物图像检索系统中的应用,同时给出部分实验数据和实验结果.     

一种起始点无关的小波系数形状匹配

小波变换的多分辨率特征使其在计算机视觉中得到广泛的应用,在形状匹配中,小波变换对起始点的依赖制约了小波变换的应用。为了克服小波变换对起始点的依赖,引入Zernike矩,提出一种起始点无关的小波系数形状匹配算法。对输入图像进行预处理后提取目标轮廓,生成具有平移、尺度不变的形状链状表达,并通过小波变换进行多尺度分析。最后计算各个尺度下的各阶Zernike矩,来解决小波变换的起始点问题,实现形状表达的旋转不变性。实验结果表明该算法适用于轮廓较明显的目标,同时具有速度快、精度高、鲁棒性强的优点。     

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.     

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。     

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.     

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.