Rotation,scaling and translation invariant texture recognition by Bessel-Fourier moments

The ideal of Bessel-Fourier moments (BFMs) for image analysis and only rotation invariant image cognition has been proposed recently. In this paper, we extend the previous work and propose a new method for rotation, scaling and translation (RST) invariant texture recognition using Bessel-Fourier moments. Compared with the others moments based methods, the radial polynomials of Bessel-Fourier moments have more zeros and these zeros are more evenly distributed. It makes Bessel-Fourier moments more suitable for invariant texture recognition as a generalization of orthogonal complex moments. In the experiment part, we got three testing sets of 16, 24 and 54 texture images by way of translating, rotating and scaling them separately. The correct classification percentages (CCPs) are compared with that of orthogonal Fourier-Mellin moments and Zernike moments based methods in both noise-free and noisy condition. Experimental results validate the conclusion of theoretical derivation: BFM performs better in recognition capability and noise robustness in terms of RST texture recognition under both noise-free and noisy condition when compared with orthogonal Fourier-Mellin moments and Zernike moments based methods.     

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.     

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。     

Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments

In this paper, we consider the use of orthogonal moments for invariant classification of alphanumeric characters of different size. In addition to the Zernike and pseudo-Zernike moments (ZMs and PZMs) which have been previously proposed for invariant character recognition, a new method of combining Orthogonal Fourier-Mellin moments (OFMMs) with centroid bounding circle scaling is introduced, which is shown to be useful in characterizing images with large variability. Through extensive experimentation using ZMs and OFMMs as features, different scaling methodologies and classifiers, it is shown that OFMMs give the best overall performance in terms of both image reconstruction and classification accuracy.     

Radial shifted Legendre moments for image analysis and invariant image recognition

The rotation, scaling and translation invariant property of image moments has a high significance in image recognition. Legendre moments as a classical orthogonal moment have been widely used in image analysis and recognition. Since Legendre moments are defined in Cartesian coordinate, the rotation invariance is difficult to achieve. In this paper, we first derive two types of transformed Legendre polynomial: substituted and weighted radial shifted Legendre polynomials. Based on these two types of polynomials, two radial orthogonal moments, named substituted radial shifted Legendre moments and weighted radial shifted Legendre moments (SRSLMs and WRSLMs) are proposed. The proposed moments are orthogonal in polar coordinate domain and can be thought as generalized and orthogonalized complex moments. They have better image reconstruction performance, lower information redundancy and higher noise robustness than the existing radial orthogonal moments. At last, a mathematical framework for obtaining the rotation, scaling and translation invariants of these two types of radial shifted Legendre moments is provided. Theoretical and experimental results show the superiority of the proposed methods in terms of image reconstruction capability and invariant recognition accuracy under both noisy and noise-free conditions.     

基于傅里叶-梅林矩的抗几何攻击图像水印算法

如何有效抵抗几何攻击是数字图像水印研究领域的热点问题之一,一个微弱不可觉察的几何攻击就可能使绝大多数水印算法失效。以不变矩理论为基础,提出了一种基于正交傅里叶-梅林矩的可有效抵抗几何攻击的图像水印新算法。结合傅里叶-梅林矩的几何不变特性,计算出原始图像的傅里叶-梅林矩;根据稳定矩的选取规则选取部分稳定的傅里叶-梅林矩,采用量化调制策略将水印信息嵌入到所选矩的幅值中;将傅里叶-梅林矩修改前后的重构差值图像叠加到原始载体图像中,得到含水印图像。仿真实验表明,该算法不仅具有较好的不可感知性,而且对常规信号处理和几何攻击均具有较好的鲁棒性。     

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.     

An efficient and robust multi-frame image super-resolution reconstruction using orthogonal Fourier-Mellin moments

Multi-frame image super-resolution (SR) has recently become an active area of research. The orthogonal rotation invariant moments (ORIMs) have several useful characteristics which make them very suitable for multi-frame image super-resolution application. Among the various ORIMs, Zernike moments (ZMs) and pseudo-Zernike moments (PZMs)-based SR approaches, i.e., NLM-ZMs and NLM-PZMs, have already shown improved SR performances for multi-frame image super-resolution. However, it is a well-known fact that among many ORIMs, orthogonal Fourier-Mellin moments (OFMMs) demonstrate better noise robustness and image representation capabilities for small images as compared to ZMs and PZMs. Therefore, in this paper, we propose a multi-frame image super-resolution approach using OFMMs. The proposed approach is based on the NLM framework because of its inherent capability of estimating motion implicitly. We have referred to this proposed approach as NLM-OFMMs-I. Also, a novel idea of using OFMMs-based interpolation in place of traditional Lanczos interpolation for obtaining an initial estimate of HR sequence has been presented in this paper. This variant of the proposed approach is referred to as NLM-OFMMs-II. Detailed experimental analysis demonstrates the effectiveness of the proposed OFMMs-based SR approaches to generate high-quality HR images in the presence of factors like image noise, global motion, local motion, and rotation in between the image frames.     

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.     

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。     

Hu新增不变矩在零水印中的应用

文章提出了一种使用Hu新增不变矩的零水印算法。该方法融合Hu不变矩及其新增的几个不变矩的特征,形成一组更为完备的特征矢量。文章利用这些特征矢量可以更好的构建零水印系统。在模式识别领域中使用这种方法可以实现对目标图像更为准确的识别;在图像检索领域中此方法比单一的Hu不变矩具有更好的检索性能。它不但保持原有Hu矩的平移、尺度、旋转不变性,而且比原有的Hu不变矩包含了更多的细节信息因此可以更全面地描述图像。所以将新增的几个不变矩和7个Hu不变矩应用到数字水印中,在一定程度上可以很好的提高水印系统的整体鲁棒性和可靠性。     

New set of adapted Gegenbauer–Chebyshev invariant moments for image recognition and classification

Orthogonal moments and their invariants to geometric transformations for gray-scale images are widely used in many pattern recognition and image processing applications. In this paper, we propose a new set of orthogonal polynomials called adapted Gegenbauer–Chebyshev polynomials (AGC). This new set is used as a basic function to define the orthogonal adapted Gegenbauer–Chebyshev moments (AGCMs). The rotation, scaling, and translation invariant property of (AGCMs) is derived and analyzed. We provide a novel series of feature vectors of images based on the adapted Gegenbauer–Chebyshev orthogonal moments invariants (AGCMIs). We practice a novel image classification system using the proposed feature vectors and the fuzzy k-means classifier. A series of experiments is performed to validate this new set of orthogonal moments and compare its performance with the existing orthogonal moments as Legendre invariants moments, the Gegenbauer and Tchebichef invariant moments using three different image databases: the MPEG7-CE Shape database, the Columbia Object Image Library (COIL-20) database and the ORL-faces database. The obtained results ensure the superiority of the proposed AGCMs over all existing moments in representation and recognition of the images.

     

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.     

Image representation using separable two-dimensional continuous and discrete orthogonal moments

This paper addresses bivariate orthogonal polynomials, which are a tensor product of two different orthogonal polynomials in one variable. These bivariate orthogonal polynomials are used to define several new types of continuous and discrete orthogonal moments. Some elementary properties of the proposed continuous Chebyshev–Gegenbauer moments (CGM), Gegenbauer–Legendre moments (GLM), and Chebyshev–Legendre moments (CLM), as well as the discrete Tchebichef–Krawtchouk moments (TKM), Tchebichef–Hahn moments (THM), Krawtchouk–Hahn moments (KHM) are presented. We also detail the application of the corresponding moments describing the noise-free and noisy images. Specifically, the local information of an image can be flexibly emphasized by adjusting parameters in bivariate orthogonal polynomials. The global extraction capability is also demonstrated by reconstructing an image using these bivariate polynomials as the kernels for a reversible image transform. Comparisons with the known moments are performed, and the results show that the proposed moments are useful in the field of image analysis. Furthermore, the study investigates invariant pattern recognition using the proposed three moment invariants that are independent of rotation, scale and translation, and an example is given of using the proposed moment invariants as pattern features for a texture classification application.     

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 .     

基于Radon和解析Fourier-Mellin变换的尺度与旋转不变目标识别算法

由于正交矩对噪声鲁棒性强、重建效果好,因此被广泛应用于目标识别与分类中,但是正交矩本质上缺乏尺度变换不变性,而且必要的图像二值化与规一化过程会引入重采样与重量化误差。为此,在研究现有正交矩的基础上,提出了一种基于Radon变换和解析Fourier-Mellin变换的尺度与旋转不变的目标识别算法。该算法首先直接对目标灰度图像进行Radon变换,然后对Radon变换结果进行进一步解析,通过Fourier-Mellin变换将原图像的旋转变化转化为相位变化,将原图像的尺度变化转化为幅度变化;最后,通过定义一旋转与尺度不变函数,同时利用不变函数的4种特征,再应用k-近邻法实现分类。理论与实验结果表明,由于避免了正交矩方法存在的重采样与重量化误差,该算法的分类精度高于基于正交矩的分类方法,而且对白噪声的鲁棒性也显著高于基于正交矩的识别与分类方法。