Accurate calculation of high order pseudo-Zernike moments and their numerical stability

The accuracy of pseudo-Zernike moments (PZMs) suffers from various errors, such as the geometric error, numerical integration error, and discretization error. Moreover, the high order moments are vulnerable to numerical instability. In this paper, we present a method for the accurate calculation of PZMs which not only removes the geometric error and numerical integration error, but also provides numerical stability to PZMs of high orders. The geometric error is removed by taking the square-grids and arc-grids, the ensembles of which maps exactly the circular domain of PZMs calculation. The Gaussian numerical integration is used to eliminate the numerical integration error. The recursive methods for the calculation of pseudo-Zernike polynomials not only reduce the computation complexity, but also provide numerical stability to high order moments. A simple computational framework to implement the proposed approach is also discussed. Detailed experimental results are presented which prove the accuracy and numerical stability of PZMs.     

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.     

一种可有效抵抗几何攻击的鲁棒图像水印算法

伪Zernike矩(Pesudo-Zernike Moments)是一组正交复数矩,其不仅具有旋转不变性、更低的噪声敏感性,而且还具有表达有效性、计算快速性以及多级表达性等特点.以伪Zernike矩理论为基础,提出了一种可有效抵抗几何攻击的数字图像水印新方案.该方案首先结合伪Zernike矩的旋转不变特性,计算出原始载体图像的伪Zernike矩;然后选取部分低阶伪Zernike矩;最后采纳量化调制策略将水印信息嵌入到伪Zernike矩幅值中.实验结果表明,该数字图像水印方案不仅具有良好的透明性,而且具有较强的抵抗常规信号处理、几何攻击等能力,整体性能优于Zernike矩图像水印方案.     

伪Zernike矩特征在图像重建中的应用

简单介绍了描述图像形状特征的伪Zernike矩,给出了伪Zernike矩的定义;在讨论伪Zernike矩性质的基础上,指出可以使用基于伪Zernike矩的形状特征数据来重建图像,阐述了基于伪Zernike矩的形状特征数据进行图像重建的理论基础;实验结果证明了基于伪Zernike矩进行图像重建的可行性。     

On the accuracy of image normalization by Zernike moments

Zernike Moment (ZM) is an effective region-based shape representation technique. The extracted ZM features should be independent of scale, position and orientation, which can be achieved by ZM-based image normalization. Nevertheless, due to the discretization of digital image and the presence of noise, the normalization is imperfect. Thus, in practice Zernike Moment Invariants (ZMI) cannot perfectly preserve the invariant properties. In this paper, firstly the ZM-based image normalization criteria are derived, and then I theoretically and experimentally evaluate the accuracy of the ZM-based image normalization. Our theoretical and experimental results not only disclose some essential facts, but also have some new findings. The relations between the accuracy of ZM-based image normalization and its influencing factors are established. A creative pseudo-polar coordinate is proposed to cut down the geometrical errors to the greatest extent. Furthermore, I suggest several techniques to improve the accuracy of image normalization. By combining moment-based image normalization with the image regularization theory and the scale-space theory, and several new conclusions are drawn. Our experimental results show that the proposed techniques can preserve the invariance of image normalization and restrain the influence of noise quite effectively.     

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.     

基于伪Zernike矩和Contourlet变换的抗几何攻击图像水印算法

在研究Contourlet变换和伪Zernike矩理论的基础上,提出了一种新颖的抗几何攻击数字图像水印算法。该算法首先 用Contourlet变换处理 原始图像,提取出图像的低频区域;然后依据人眼的视觉特性以及嵌入水印前、后系数的相关性,采取量化调制伪Zernike矩的幅值将数字水印信息嵌入图像的低频区域中。其无须借助于原始图像,就能进行水印提取,实现真正的盲检测。实验结果表明,该算法对平移变换的抵抗力最强,对旋转和缩放也有很好的抵抗力,而且经过JPEG压缩之后,数字水印也不会失真。     

基于伪Zenike矩的离线地图匹配算法

地图匹配算法分为在线和离线匹配,针对离线地图匹配中Marchal算法精度较低以及存在模糊多解的问题,利用伪Zenike矩对其进行改进,将行驶轨迹与道路曲线利用伪Zenike矩进行形状描述,然后对曲线进行特征匹配,获取道路点。实验结果表明,新算法可以较好地纠正矢量数据不完整时Marchal算法产生的错误结果,很大程度上提高了匹配的准确性,而且匹配的效率优于现有算法。     

采用圆谐-傅里叶矩的图像区域复制粘贴篡改检测

现有检测方法大多对图像区域复制粘贴篡改的后处理操作鲁棒性不高.针对这种篡改技术,提出一种新的基于圆谐-傅里叶矩的区域篡改检测算法.首先将图像分为重叠的小块;然后提取每个图像块的圆谐-傅里叶矩作为特征向量并对其进行排序;最后根据阈值确定相似块,利用位移矢量阈值去除错误相似块以定位篡改区域.实验结果表明,该算法能有效抵抗噪声、高斯模糊、旋转等图像后处理操作,且与基于HU矩的方法相比有更好的检测结果.     

Circularly orthogonal moments for geometrically robust image watermarking

Circularly orthogonal moments, such as Zernike moments (ZMs) and pseudo-Zernike moments (PZMs), have attracted attention due to their invariance properties. However, we find that for digital images, the invariance properties of some ZMs/PZMs are not perfectly valid. This is significant for applications of ZMs/PZMs. By distinguishing between the ‘good’ and ‘bad’ ZMs/PZMs in terms of their invariance properties, we design image watermarks with ‘good’ ZMs/PZMs to achieve watermark's robustness to geometric distortions, which has been considered a crucial and difficult issue in the research of digital watermarking. Simulation results show that the embedded information can be decoded at low error rates, robust against image rotation, scaling, flipping, as well as a variety of other common manipulations such as lossy compression, additive noise and lowpass filtering.     

A new SVM-based image watermarking using Gaussian-Hermite moments

Geometric attack is known as one of the most difficult attacks to resist, for it can desynchronize the location of the watermark and hence causes incorrect watermark detection. It is a challenging work to design a robust image watermarking scheme against geometric attacks. Based on the support vector machine (SVM) and Gaussian-Hermite moments (GHMs), we propose a robust image watermarking algorithm in nonsubsampled contourlet transform (NSCT) domain with good visual quality and reasonable resistance toward geometric attacks in this paper. Firstly, the NSCT is performed on original host image, and corresponding low-pass subband is selected for embedding watermark. Then, the selected low-pass subband is divided into small blocks. Finally, the digital watermark is embedded into host image by modulating adaptively the NSCT coefficients in small block. The main steps of digital watermark detecting procedure include: (1) some low-order Gaussian-Hermite moments of training image are computed, which are regarded as the effective feature vectors; (2) the appropriate kernel function is selected for training, and a SVM training model can be obtained; (3) the watermarked image is corrected with the well trained SVM model; (4) the digital watermark is extracted from the corrected watermarked image. Experimental results show that the proposed image watermarking is not only invisible and robust against common image processing operations such as filtering, noise adding, JPEG compression, etc., but also robust against the geometric attacks.     

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.     

Image reconstruction from limited range projections using orthogonal moments

A set of orthonormal polynomials is proposed for image reconstruction from projection data. The relationship between the projection moments and image moments is discussed in detail, and some interesting properties are demonstrated. Simulation results are provided to validate the method and to compare its performance with previous works.     

Algorithms for fast computation of Zernike moments and their numerical stability

Accuracy, speed and numerical stability are among the major factors restricting the use of Zernike moments (ZMs) in numerous commercial applications where they are a tool of significant utility. Often these factors are conflicting in nature. The direct formulation of ZMs is prone to numerical integration error while in the recent past many fast algorithms are developed for its computation. On the other hand, the relationship between geometric moments (GMs) and ZMs reduces numerical integration error but it is observed to be computation intensive. We propose fast algorithms for both the formulations. In the proposed method, the order of time complexity for GMs-to-ZMs formulation is reduced and further enhancement in speed is achieved by using quasi-symmetry property of GMs. The existing q-recursive method for direct formulation is further modified by incorporating the recursive steps for the computation of trigonometric functions. We also observe that q-recursive method provides numerical stability caused by finite precision arithmetic at high orders of moment which is hitherto not reported in the literature. Experimental results on images of different sizes support our claim.     

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.     

融合HU不变矩和SIFT特征的商标检索

利用商标图像的形状特征,提出了一种融合图像全局特征和局部特征的商标检索算法。其中全局特征反应了图像的整体信息,这些信息可用来较快地建立候选图像库,而局部特征则可以更准确地与候选图像进行匹配。提取图像的HU不变矩进行初步检索,按相似度排序,在此结果集的基础上对候选图像通过提取SIFT特征进行精确匹配。实验结果表明,该方法既保持了SIFT特征的良好描述能力,又减少了精确匹配需要的计算次数,降低了复杂度。