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.     

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.     

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.     

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。     

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.     

Fast computation of exact Zernike moments using cascaded digital filters

Zernike moments have been extensively used and have received much research attention in a number of fields: object recognition, image reconstruction, image segmentation, edge detection and biomedical imaging. However, computation of these moments is time consuming. Thus, we present a fast computation technique to calculate exact Zernike moments by using cascaded digital filters. The novelty of the method proposed in this paper lies in the computation of exact geometric moments directly from digital filter outputs, without the need to first compute geometric moments. The mathematical relationship between digital filter outputs and exact geometric moments is derived and then they are used in the formulation of exact Zernike moments. A comparison of the speed of performance of the proposed algorithm with other state-of-the-art alternatives shows that the proposed algorithm betters current computation time and uses less memory.     

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.     

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.     

特征点和不变矩结合的遥感图像飞机目标识别

传统的飞机目标识别算法一般是通过目标分割,然后提取不变特征进行训练来完成目标的识别。但是,对于实际情况比较复杂的遥感图像飞机目标,至今没有一种适合多种机型的分割和识别算法。针对现有识别算法的不足,本研究提出一种基于特征点空间分布、颜色不变矩和Zernike不变矩相结合的遥感图像飞机目标识别算法。方法：首先,对预处理后的遥感图像和模板图像进行小波变换,在低分辨率图像下采用圆投影特征进行粗匹配,确定候选目标;粗匹配结束后,提取高分辨率图像的多尺度Harris-laplace角点,并画出Delaunay三角网,同时提取出颜色不变矩和Zernike不变矩;然后使用欧氏距离作为这三种特征的相似性度量,并和样本图像进行加权匹配;最后选取欧式距离最小的图像作为最终的识别目标。结果：实验表明,本文算法飞机检测精度比现有算法高2.2%,飞机识别精度比现有算法高1.4%-10.4%。该算法能从遥感图像中精确识别出十大飞机目标,并对背景、噪声、视角变化等多种干扰具有良好的鲁棒性。结论：提出了一种基于特征点空间分布、颜色不变矩和Zernike不变矩相结合的飞机识别算法,该算法使用了图像的多种信息,包括特征点和不变矩,有效地克服了使用单一特征无法描述多种信息的不足。实验结果表明,本文采用基于特征点和不变矩的飞机识别算法比其他算法具有更强的抗干扰能力和识别精度。     

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。     

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.     

Zernike亚像素图像矩边缘检测算子方向角模型研究

基于Zernike图像矩的理想边缘模型,深入研究了方向角模型与亚像素判据间的关系。利用Zernike矩定义及其旋转不变特性,提出一种新的基于4阶方向角的Zernike矩亚像素边缘检测算子。为了提高边缘算子定位速度,首先基于9×9尺寸模板对Zernike图像矩0～4阶正交复数多项式进行了计算,推导出基于4阶方向角的边缘检测算子参数模型。最后将边缘算子应用在理想图像与实际图像上,检测结果表明:相比于传统的Zernike矩算子,基于4阶方向角的边缘检测算子具有更高的检测精度。     

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     

Higher order orthogonal moments for invariant facial expression recognition

Automatic facial expression recognition (FER) is a sub-area of face analysis research that is based heavily on methods of computer vision, machine learning, and image processing. This study proposes a rotation and noise invariant FER system using an orthogonal invariant moment, namely, Zernike moments (ZM) as a feature extractor and Naive Bayesian (NB) classifier. The system is fully automatic and can recognize seven different expressions. Illumination condition, pose, rotation, noise and others changing in the image are challenging task in pattern recognition system. Simulation results on different databases indicated that higher order ZM features are robust in images that are affected by noise and rotation, whereas the computational rate for feature extraction is lower than other methods.