Zernike矩在图像旋转与抗噪声中的应用

Zernike矩是基于称作Zernike多项式的正交化函数。尽管同几何矩和Legendre矩相比其计算更加复杂,但Zernike矩在图像的旋转和低的噪声敏感度方面是有较大的优越性。特别是本文采取了Zernike矩的一种快速算法,使计算量大大减少,并且采取了方-圆变换的方法,对矩形图像仍能起到同样的作用。     

基于差分法的灰度图像矩的快速算法

本文提出了一种新的对于灰度图像的几何矩的快速算法。首先运用图像差分法,将图像函数f（x,y）变换为图像函数d（x,y）。其次,从x^n（n=1,2,3）的递推求和得到一组数组。灰度图像的几何矩可以由该数组和函数d（x,y）计算获得。这种方法的优点在于：图像行（列）中具有相同像素值的连续部分,经差分后,除端点外的其它部分都为0,求矩无需考虑值为0的像素。所以,求矩计算量大大地降低了。文中给出了实验结果,和其它灰度图像求矩算法相比,文中算法在大多数情形下都极大地降低了计算复杂度。该算法乘法和加法的运算次数大约是Belkasim’s算法的47．4％和59．8％,大约是Yang’s算法的35％和51．8％。     

伪Zernike矩归一化的抗几何攻击图像水印

以改进的伪Zernike矩相关知识为基础,提出了一种改进的抗几何攻击的数字水印算法。该算法首先计算载体图像的伪Zernike矩,然后对其进行归一化处理。最后选择部分低阶矩幅值量化嵌入水印。实验结果表明该数字水印算法不仅可以抵抗常规的数字信号处理,而且较基于伪Zernike矩的抗几何攻击的数字水印算法有对旋转和缩放的联合攻击具有较好的抵抗能力。     

一类新的正交矩-Franklin矩及其图像表达

该文定义了一类以Franklin函数为核的正交矩,称之为Franklin矩.Franklin函数是一类完备正交一次样条函数系.传统的Legendre矩、Zernike矩等多项式矩,由于涉及高次多项式的计算,往往会导致计算不稳定,特征空间维数扩展受到制约.Franklin函数是正交的,相应的矩函数可以使得图像分解后的信息具有独立性,没有信息的冗余.而且,Franklin函数仅由一次分段多项式组成,在计算过程中,避免了高次多项式的计算,兼具复杂度低、数值稳定的优点.通过对图像的重构实验表明,Franklin矩比传统正交多项式矩具有更好的特征表达能力.     

Ackermann函数值的计算

１Ackermann函数Ackermann函数最初是由德国数学家W．Ac－kermann在１９２８年提出，并以其名字命名的。该函数通常由以下形式给出：ack（０，y）＝y＋１ack（x，０）＝ack（x－１，１）ack （x，y）＝ack（x－１，ack （x，y－１））函数看上去十分简单，但想计算出它的值可不是一件容易的事。因为它的函数增长率超过了所有原始递归函数（primitiverecursivefunction），是一个可计算的非原始递归函数犤２犦。２对Ackermann函数递归公式的改进计算Ackermann函数值的困难主要有二：精度和递归次数问题。前者可以通过高精度运算例程来解决…     

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矩快速算法的步态识别

在图像识别问题的研究中,步态识别是生物识别领域较活跃的研究课题,在视频监控等方面有广阔的应用前景.为提高轮廓识别精度和准确性,提出一种提取步态序列图像关键帧的特征并进行身份识别的方法,用来提高图像的精确性.首先把序列图像的人体部分模板化,然后利用q-递归算法计算关键帧的Zernike矩值、对序列图像进行矩特征描述,利用PCA变换进行特征数据的降维,利用支持向量机(SVM)等方法对数据进行分类.对不同的Zernike矩阶数、不同的训练方法、识别方法进行实验,并对识别结果进行分析比较.仿真实验结果证明了特征提取方法的有效性.     

一种新的灰度图像Legendre矩的快速算法

Legendre正交矩在模式识别和图像分析等领域有着广泛的应用，但由于计算的复杂性，相关的快速算法尚未得到很好的解决，已有方法均局限于二值图像．文章提出了一种灰度图像的Legendre正交矩的快速算法，借助于Legendre多项式的递推公式推导出计算一维Legendre矩的递归公式．利用该关系式，一维Legendre矩Lp可以用一系列初始值L1(a)，a＜p，Lo(a)，a＜p-1来得到．而二维Legendre矩pq可以利用一维算法进行计算,为了降低算法复杂度，文中采用基于Systolic阵列的快速算法进行计算L1(a)，Lo(a),与直接方法相比，快速算法可以大幅度减少乘法的次数，从而达到了降低算法复杂度的目的。     

改进的电缆护套厚度检测方法

图像精密检测精度很大程度上依赖于亚像素边缘检测精度,目前应用较广的是基于Zernike旋转不变矩的边缘检测技术。分析了传统Zernike矩边缘检测不足之处,分别在边缘灰度值函数建模、曲率较大处亚像素定位和实时性能等方面给出改进与创新,进而提出一种改进的基于旋转不变矩亚像素定位算法思想。并用此算法做了电缆护套厚度测试实验。从实验结果看,该算法定位精度有了改进,算法思想可行。     

一种支持多线宽直线反走样算法

直线绘制中出现的锯齿现象称为走样,消除走样的方法称为反走样。文中通过对直线走样产生的原因进行理论上的分析,总结了现有的反走样技术。通过对经典的DDA直线绘制算法和Wu直线反走样绘制算法的研究,在二者结合的基础上,给出了一种任意宽度和复杂背景色下的直线反走样快速绘制算法：对于直线f（x）=mx＋b,0≤m≤1,x轴上每移动一个像素单位,根据直线所需绘制的宽度,在y轴上进行跨度像素着色,填充的色深值取决于该像素到对应直线边缘线的距离、原有背景色深和当前直线绘制色深。对算法进行了去浮点优化,给出了复杂度分析和实验结果,实践证明,该算法有很好的执行效率和反走样效果。     

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 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 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.     

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 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.     

参数可调的通用半正交图像矩模型

目的 为了提高以正交多项式为核函数构造的高阶矩数值的稳定性,增强低阶矩抗噪和滤波的能力,将仅具有全局描述能力的常规正交矩推广到可以局部化提取图像特征的矩模型,从频率特性分析的角度定义一种参数可调的通用半正交矩模型。方法 首先,对传统正交矩的核函数进行合理的修正,以修正后的核函数（也称基函数）替代传统正交矩中的原核函数,使其成为修改后的特例之一。经过修正后的基函数可以有效消除图像矩数值不稳定现象。其次,采用时域的分析方法能够对图像的低阶矩作定量的分析,但无法对图像的高频部分（对应的高阶矩）作更合理的表述。因此提出一种时—频对应的方法来分析和增强不同阶矩的稳定性,通过对修正后核函数的频带宽度微调可以建立性能更优的不同阶矩。最后,利用构建的半正交—三角函数矩研究和分析了通用半正交矩模型的特点及性质。结果 将三角函数为核函数的图像矩与现有的Zernike、伪Zernike、正交傅里叶—梅林矩及贝塞尔—傅里叶矩相比,由于核函数组成简单,且其值域恒定在[-1,1]区间,因此在图像识别领域具有更快的计算速度和更高的稳定性。结论 理论分析和一系列相关图像的仿真实验表明,与传统的正交矩相比,在数值稳定性、图像重构、图像感兴趣区域（ROI）特征检测、噪声鲁棒性测试及不变性识别方面,通用的半正交矩性能及效果更优。     

基于两变量Hahn矩的图像分析

提出了一种新的、以两变量离散正交Hahn多项式为核函数的图像矩,推导了正则化后,两变量离散正交Hahn多项式的简单的计算方法。对二值图像、灰度图像以及噪声图像的重建实验表明：相对于同系数的单变量的Hahn矩,两变量Hahn矩的重建误差更小。因此,它们能够更好地提取图像的特征。     

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.