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.     

Three dimensional radial Tchebichef moment invariants for volumetric image recognition

The property of rotation, scaling and translation invariant has a great important in 3D image classification and recognition. Tchebichef moments as a classical orthogonal moment have been widely used in image analysis and recognition. Since Tchebichef moments are represented in Cartesian coordinate, the rotation invariance can’t easy to realize. In this paper, we propose a new set of 3D rotation scaling and translation invariance of radial Tchebichef moments. We also present a theoretical mathematics to derive them. Hence, this paper we present a new 3D radial Tchebichef moments using a spherical representation of volumetric image by a one-dimensional orthogonal discrete Tchebichef polynomials and a spherical function. 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 Tchebichef 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. The result of experiments prove that the Tchebichef moments have done better than the Krawtchouk moments with and without noise. Simultaneously, the reconstructed 3D image converges quickly to the original image using 3D radial Tchebichef moments and the test images are clearly recognized from a set of images that are available in a PSB database.     

基于Tchebichef不变距的图形识别

论文提出一种新的基于Tchebichef不变距的图形识别方法,Tchebichef不变距通过对图像进行大小归一化、平移处理以及旋转处理并结合Tchebichef正交距得到;利用提出的Tchebichef不变距作为特征进行图形的识别,和采用Hu不变距以及Zernike不变矩的识别方法相比,无论是在无噪声还是在有噪声的情况下都有更高的识别精度;仿真试验证实了本方法的可行性。     

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.

     

基于递降阶乘计算的Tchebichef 矩的旋转不变量

传统Tchebichef矩的旋转不变量是用几何矩的旋转不变量来表示的,这就不能避免几何矩冗余信息多、对噪声敏感等缺点。提出了一种新的Tchebichef矩的旋转不变量,用降阶阶乘的性质,将它转化为可以利用Tchebichef矩直接计算的Tchebichef中心矩的线性组合。实验表明,提出的描述子具有更好的旋转不变性。     

3D radial invariant of dual Hahn moments

In this work, we propose new sets of 2D and 3D rotation invariants based on orthogonal radial dual Hahn moments, which are orthogonal on a non-uniform lattice. We also present theoretical mathematics to derive them. Thus, this paper presents in the first case new 2D radial dual Hahn moments based on polar representation of an image by one-dimensional orthogonal discrete dual Hahn polynomials and a circular function. The dual Hahn polynomials are general case of Tchebichef and Krawtchouk polynomials. In the second case, we introduce new 3D radial dual Hahn moments employing a spherical representation of volumetric image by one-dimensional orthogonal discrete dual Hahn polynomials and a spherical function, which are orthogonal on a non-uniform lattice. The 2D and 3D rotational invariants are extracts from the proposed 2D and 3D radial dual Hahn moments respectively. In order to test the proposed approach, three problems namely image reconstruction, rotational invariance and pattern recognition are attempted using the proposed moments. The result of experiments shows that the radial dual Hahn moments have performed better than the radial Tchebichef and Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial dual Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image and PSB database for 3D image.

     

基于矩特征的三维飞机目标识别

利用矩特征可以从二维图像识别三维目标,４阶矩的图像峰值反映了目标的形状,为了使它具有平移、比例和旋转不变性,作者对其进行了归一化处理。文中将未修正的不变矩、修正的不变矩及归一化的４阶矩用于三维飞机目标识别,仿真结果表明归一化的４阶矩和修正的正的不变矩可有效地用于三维目标的识别。     

基于Tchebichef矩的几何攻击不变性第二代水印算法

提出了一种基于原始图像的Tchebichef矩实现的几何攻击不变性第二代盲水印算法,利用原始图像的Tchebichef矩估计图像可能经历的几何攻击的参数来还原图像,其中,原始图像的Tchebichef矩可作为水印检测器的密钥.水印嵌入过程结合人类视觉系统的特性,且可在任何图像变换域中实现,给出了小波域的一种实现方法.水印检测过程采用独立分量分析技术不仅可以检测到水印而且可以提取水印,实现了真正意义上的盲检测.实验结果表明,该水印算法对于通用水印测试软件Stirmark提供的各种攻击具有很好的鲁棒性.     

Radial Hahn Moment Invariants for 2D and 3D Image Recognition

Recently, orthogonal moments have become efficient tools for two-dimensional and three-dimensional (2D and 3D) image not only in pattern recognition, image vision, but also in image processing and applications engineering. Yet, there is still a major difficulty in 3D rotation invariants. In this paper, we propose new sets of invariants for 2D and 3D rotation, scaling and translation based on orthogonal radial Hahn moments. We also present theoretical mathematics to derive them. Thus, this paper introduces in the first case new 2D radial Hahn moments based on polar representation of an object by one-dimensional orthogonal discrete Hahn polynomials, and a circular function. In the second case, we present new 3D radial Hahn moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Hahn polynomials and a spherical function. Further 2D and 3D invariants are derived from the proposed 2D and 3D radial Hahn moments respectively, which appear as the third case. In order to test the proposed approach, we have resolved three issues: the image reconstruction, the invariance of rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Hahn moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and Princeton shape benchmark (PSB) database for 3D image.     

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.     

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.     

基于Tchebichef矩的区域复制窜改检测算法*

提出了一种新的数字图像被动认证算法,用于检测同幅图像的区域复制窜改问题。算法首先利用离散小波变换提取图像的低频分量,再对低频分量进行分块并提取每一块的离散正交Tchebichef矩特征;然后将特征矢量进行字典排序,比较相邻两组特征矢量的相似性;最后利用阈值判别实现窜改伪造区域的检测和定位。实验结果证明,算法能较好地检测及定位出图像中复制与窜改区域,且具有运算量小、检测效率高、鲁棒性好等特点。     

Radial invariant of 2D and 3D Racah moments

In this paper, we introduce new sets of 2D and 3D rotation, scaling and translation invariants based on orthogonal radial Racah moments. We also provide theoretical mathematics to derive them. Thus, this work proposes in the first case a new 2D radial Racah moments based on polar representation of an object by one-dimensional orthogonal discrete Racah polynomials on non-uniform lattice, and a circular function. In the second case, we present new 3D radial Racah moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Racah polynomials and a spherical function. Further 2D and 3D invariants are extracted from the proposed 2D and 3D radial Racah moments respectively will appear in the third case. To validate the proposed approach, we have resolved three problems. The 2D/ 3D image reconstruction, the invariance of 2D/3D rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Racah moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges rapidly to the original image using 2D and 3D radial Racah moments, and the test 2D/3D images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and PSB database for 3D image.     

一种基于Krawtchouk矩的水印算法

提出一种基于Krawtchouk矩的水印算法,通过修改一些原始Krawtchouk矩并重构图像以获得水印图像.基于Krawtchouk矩与几何矩的关系,提出采用具有平移、比例缩放和旋转不变性的几何不变矩来检测水印.实验表明,与用Krawtchouk不变矩检测相比,该算法对于大角度旋转和图像平移的几何攻击具有更好的鲁棒性.     

The fast recursive computation of Tchebichef moment and its inverse transform based on Z-transform

The outputs of cascaded digital filters operating as accumulators are combined with a simplified Tchebichef polynomials to form Tchebichef moments (TMs). In this paper, we derive a simplified recurrence relationship to compute Tchebichef polynomials based on Z-transform properties. This paves the way for the implementation of second order digital filter to accelerate the computation of the Tchebichef polynomials. Then, some aspects of digital filter design for image reconstruction from TMs are addressed. The new proposed digital filter structure for reconstruction is based on the 2D convolution between the digital filter outputs used in the computation of the TMs and the impulse response of the proposed digital filter. They operate as difference operators and accordingly act on the transformed image moment sets to reconstruct the original image. Experimental results show that both the proposed algorithms to compute TMs and inverse Tchebichef moments (ITMs) perform better than existing methods in term of computation speed.