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.     

Determination of blur and affine combined invariants by normalization

The determination of invariant characteristics is an important problem in pattern recognition. In many situations, images to be processed are usually subjected to geometric distortion and/or blur degradation. In this paper, we introduce an approach to derive blur and affine combined invariants (BAI). Firstly, we normalize the image to a standard form by using blur invariant moments as normalization constraints. Then, we construct the blur and affine combined invariants at the standard form. Using the method proposed in this paper, a set of blur and affine combined invariant features can be obtained easily and effectively. Several experimental results are presented to illustrate the performance of the invariants for simultaneously affine deformed and blur degraded images.     

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.

     

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.     

Combined blur,translation, scale and rotation invariant image recognition by Radon and pseudo-Fourier–Mellin transforms

The idea of scale and rotation invariant image recognition based on Radon and Fourier–Mellin transforms has been presented recently. In this paper, we extend the previous work and propose a new method to construct a set of combined blur, translation, scale and rotation invariants using Radon and pseudo-Fourier–Mellin transforms, named Radon and pseudo-Fourier–Mellin invariants (RPFMI). The proposed method is robust to additive white noise as a result of summing pixel values to generate projections in the Radon transform step. We also present a mathematical framework of obtaining the Radon and pseudo-Fourier–Mellin transforms of blurred images, and a framework of deriving the combined blur, scale and rotation invariants. Theoretical and experimental results show the superiority of the proposed method and its robustness to additive white noise in comparison with some recent methods.     

Image analysis by Bessel-Fourier moments

In this paper, we proposed a new set of moments based on the Bessel function of the first kind, named Bessel-Fourier moments (BFMs), which are more suitable than orthogonal Fourier-Mellin and Zernike moments for image analysis and rotation invariant pattern recognition. Compared with orthogonal Fourier-Mellin and Zernike polynomials of the same degree, the new orthogonal radial polynomials have more zeros, and these zeros are more evenly distributed. The Bessel-Fourier moments can be thought of as generalized orthogonalized complex moments. Theoretical and experimental results show that the Bessel-Fourier moments perform better than the orthogonal Fourier-Mellin and Zernike moments (OFMMs and ZMs) in terms of image reconstruction capability and invariant recognition accuracy in noise-free, noisy and smooth distortion conditions.     

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.     

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.     

纹理图像识别中的旋转不变性分析

在对纹理图像进行分类识别过程中,许多具有相同纹理特性的不同图像经常在方向上呈现多样性。这些图像应该被归为一类。针对这一问题,有许多方法可以得到旋转不变性特征,例如：几何矩,正交矩,灰度共生矩阵等,然而,前两种方法计算量很大,第三种方法效果也不令人满意。提出了一种基于灰度-梯度共生矩阵的方法来得到旋转不变特征量,并且提出了一种快速计算灰度-梯度共生矩阵的算法。实验表明利用灰度-梯度共生矩阵的方法得到旋转不变量的方法非常有效,快速计算灰度-梯度共生矩阵的算法也大大减小了计算量。     

基于Gaussian-Hermite矩的旋转运动模糊不变量

目的 模糊图像的分析与识别是图像分析与识别领域的重要方向。有些图像形成过程中成像系统与物体之间存在相对旋转运动,如因导弹高速自旋转造成的制导图像的旋转运动模糊。大多数对于这类图像的识别都需要先对模糊图像进行“去模糊”的预处理,且该类方法存在计算时间复杂度较高及不适定的问题。对此,提出一种直接提取旋转运动模糊图像中的不变特征,用于旋转运动模糊图像目标检索和识别。方法 本文以旋转运动模糊的退化模型为出发点,提出了旋转运动模糊Gaussian-Hermite (GH)矩,构造了一组由5个对旋转变换和旋转运动模糊保持不变性的GH矩不变量组成的特征向量(rotational motion blur Gaussian-Hermite moment invariants,RMB_GHMI-5),可从旋转变换和旋转运动模糊的图像中直接进行目标检索和识别,无需前置复杂的“去模糊”预处理过程。结果 在USC-SIPI (University of Southern California — Signal and Image Processing Institute)数据集上进行不变性实验,对原图进行不同程度的旋转变换叠加旋转运动模糊处理,证明RMB_GHMI-5对于旋转变换和旋转运动模糊具有良好的稳定性和不变性。在两个数据集上与同类4种方法进行图像检索实验比较,在80%召回率下,本文方法维数更少,相比性能第2的特征向量,在Flavia数据集中,高斯噪声、椒盐噪声、泊松噪声和乘性噪声干扰下的准确率分别提高25.89%、39.95%、22.79%和35.80%;在Butterfly Image数据集中,高斯噪声、椒盐噪声、泊松噪声和乘性噪声干扰下的准确率分别提高4.79、7.63%、5.65%和18.31%。同时,在上述8个测试数据集中进行对比实验以验证融合算法的有效性,结果表明本文提出的GH矩和几何矩相融合算法显著改善了图像检索效果。结论 本文提出的RMB_GHMI-5特征向量在旋转变换和旋转运动模糊下具有良好的不变性与稳定性,在图像检索抗噪性能方面表现优异。相比同类方法,本文方法更具实际应用价值。     

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.     

Zernike Moments-Based Gurumukhi Character Recognition

In this article, use of Zernike moments is presented for invariant recognition of Gurumukhi characters. Zernike moments belong to a class of continuous orthogonal moments defined over a unit circle. So, for a square image, computations of Zernike moments involve a certain square-to-circle mapping to map the pixel coordinates within the range of the unit circle. There exist two forms of mapping, which are used by various authors in their research works. Here, a computational framework has been proposed for calculation of Zernike moments using both mapping techniques, and a comparison between the performances of both of these mapping techniques is shown through a series of extensive experiments.     

Invariant image recognition by Zernike moments

The problem of rotation-, scale-, and translation-invariant recognition of images is discussed. A set of rotation-invariant features are introduced. They are the magnitudes of a set of orthogonal complex moments of the image known as Zernike moments. Scale and translation invariance are obtained by first normalizing the image with respect to these parameters using its regular geometrical moments. A systematic reconstruction-based method for deciding the highest-order Zernike moments required in a classification problem is developed. The quality of the reconstructed image is examined through its comparison to the original one. The orthogonality property of the Zernike moments, which simplifies the process of image reconstruction, make the suggest feature selection approach practical. Features of each order can also be weighted according to their contribution to the reconstruction process. The superiority of Zernike moment features over regular moments and moment invariants was experimentally verified     

伪Zernike矩的不变性分析及其应用

本文从几何矩的概念出发,详细介绍了Zernike和伪Zernike矩。然后重点分析了伪Zernike矩的旋转、位移及尺度不变性。本文利用脱机手写签名图像对此进行了验证,实验结果证明了其不变性特征。故本文提出利用伪Zernike矩来进行脱机手写签名图像的特征提取。     

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.