Circularity — a new principle underlying the design of accurate edge orientation operators

Currently available fast edge detection operators are reviewed. Their rationale is studied with particular reference to the estimation of angular orientation of edges. The idea of ‘circular’ edge detection operators is developed, and used to explain the surprising accuracy of the Sobel operator. A family of circular operators is designed for use in neighbourhoods of various sizes, and general ideas on edge operator design are developed. The concept of ‘closed bands’ of pixels for making neighbourhoods close to circular should be widely applicable in digital image analysis, not just in the ambit of fast edge detection.     

Laplacian operator-based edge detectors

Laplacian operator is a second derivative operator often used in edge detection. Compared with the first derivative-based edge detectors such as Sobel operator, the Laplacian operator may yield better results in edge localization. Unfortunately, the Laplacian operator is very sensitive to noise. In this paper, based on the Laplacian operator, a model is introduced for making some edge detectors. Also, the optimal threshold is introduced for obtaining a maximum a posteriori (MAP) estimate of edges     

Rotation Invariant Kernels and Their Application to Shape Analysis

Shape analysis requires invariance under translation, scale, and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modeling the resulting data using complex scalar-rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters and the nonlinear nature of their formulation. In the present paper, we show how a set of kernel functions that we refer to as rotation invariant kernels can be used to convert the original nonlinear problem into a linear one. As their name implies, these kernels are defined to provide the much needed rotation invariance property allowing one to bypass the difficulty of working with complex spherical distributions. The resulting approach provides an easy, fast mechanism for 2D & 3D shape analysis. Extensive validation using a variety of shape modeling and classification problems demonstrates the accuracy of this proposed approach.     

梯度算子对反向合成梯度算法的影响

反向合成梯度算法是一种基于局部指向性的反向合成图像对齐算法。与传统的反向合成图像对齐算法相比,该算法可有效地克服光照变化对匹配结果的影响[1]。由于局部指向性的计算在本质上是梯度的计算,而图像梯度的计算可以采用不同的梯度算子,因此采用4种不同的梯度算子（一阶差分算子,Roberts算子,Sobel算子和Prewitt算子）来计算局部指向性,并通过实验比较分析了4种梯度算子对反向合成梯度算法的影响。     

Recognition by symmetry derivatives and the generalized structure tensor

We suggest a set of complex differential operators that can be used to produce and filter dense orientation (tensor) fields for feature extraction, matching, and pattern recognition. We present results on the invariance properties of these operators, that we call symmetry derivatives. These show that, in contrast to ordinary derivatives, all orders of symmetry derivatives of Gaussians yield a remarkable invariance: they are obtained by replacing the original differential polynomial with the same polynomial, but using ordinary coordinates x and y corresponding to partial derivatives. Moreover, the symmetry derivatives of Gaussians are closed under the convolution operator and they are invariant to the Fourier transform. The equivalent of the structure tensor, representing and extracting orientations of curve patterns, had previously been shown to hold in harmonic coordinates in a nearly identical manner. As a result, positions, orientations, and certainties of intricate patterns, e.g., spirals, crosses, parabolic shapes, can be modeled by use of symmetry derivatives of Gaussians with greater analytical precision as well as computational efficiency. Since Gaussians and their derivatives are utilized extensively in image processing, the revealed properties have practical consequences for local orientation based feature extraction. The usefulness of these results is demonstrated by two applications: 1) tracking cross markers in long image sequences from vehicle crash tests and 2) alignment of noisy fingerprints.     

Observability analysis of conewise linear systems via directional derivative and positive invariance techniques

Belonging to the broad framework of hybrid systems, conewise linear systems (CLSs) form a class of Lipschitz piecewise linear systems subject to state triggered mode switchings. Motivated by state estimation of nonsmooth switched systems, this paper exploits directional derivative and positive invariance techniques to characterize finite-time and long-time local observability of a general CLS. For the former observability notion, directional derivative results are developed from the simple switching property, and they yield improved observability conditions. For the latter notion, we focus on the case where a nominal trajectory has finitely many switchings. In order to characterize long-time behaviors of the CLS, necessary and sufficient conditions are obtained for the interior of a positively invariant cone. By employing these conditions, we establish connections between finite-time and long-time local observability; underlying positive invariance properties are unveiled.     

Robust Visual Tracking Based on Convolutional Features with Illumination and Occlusion Handing

视觉追踪是在计算机视觉的一个重要区域。怎么处理照明和吸藏问题是一个挑战性的问题。这份报纸论述一篇小说和有效追踪算法处理如此的问题。一方面,一起始的外观总是有的目标清除轮廓,它对照明变化光不变、柔韧。在另一方面,特征在追踪起一个重要作用,在哪个之中 convolutional 特征显示出有利性能。因此,我们采用卷的轮廓特征代表目标外观。一般来说,一阶的衍生物边坡度操作员在由卷检测轮廓是有效的他们与图象。特别, Prewitt 操作员对水平、垂直的边更敏感,当 Sobel 操作员对斜边更敏感时。内在地, Prewitt 和 Sobel 与对方一起是补足的。技术上说,这份报纸设计二组 Prewitt 和 Sobel 边察觉者提取一套完全的 convolutional 特征,它包括水平、垂直、斜的边特征。在第一个框架,轮廓特征从目标被提取构造起始的外观模型。在有这些轮廓特征的试验性的图象的分析以后,明亮的部分经常提供更有用的信息描述目标特征,这能被发现。因此,我们建议一个方法比较候选人样品和我们仅仅使用明亮的象素的训练模型的类似,它使我们的追踪者有能力处理部分吸藏问题。在得到新目标以后,变化以便改编外观,我们建议相应联机策略逐渐地更新我们的模型。convolutional 特征由井综合的 Prewitt 和 Sobel 边察觉者提取了的实验表演能是足够有效的学习柔韧的外观模型。九个挑战性的序列上的众多的试验性的结果证明我们的建议途径与最先进的追踪者比较很有效、柔韧。     

Precision edge contrast and orientation estimation

The contrast and orientation estimation accuracy of several edge operators that have been proposed in the literature is examined both for the noiseless case and in the presence of additive Gaussian noise. The test image is an ideal step edge that has been sampled with a square-aperture grid. The effects of subpixel translations and rotations of the edge on the performance of the operators are studied. It is shown that the effect of subpixel translations of an edge can generate more error than moderate noise levels. Methods with improved results are presented for Sobel angle estimates and the Nevatia-Babu operator, and theoretical noise performance evaluations are also provided. An edge operator based on two-dimensional spatial moments is presented. All methods are compared according to worst-case and RMS error in an ideal noiseless situation and RMS error under various noise levels     

图像关于边缘提取算子的微分不变性

图像边缘是一类重要特征,该文研究图像关于边缘提取算子的微分不变性,其目的是考察图像在边缘提取的过程失去了什么信息,利用Ｌｉｅ变换群在节丛上的延拓公式以及由此导出的相关微分方程的不变性条件,文中给出了构造Ｌａｐｌａｃｅ边缘算子微分不变量的详细推导过程,为使该文自封闭,作者发展了所需的数学工具,包括一系列延拓Ｌｉｅ悍数的计算公式和有关微分方程解的不变性条件,最后给出了一个包含尺度、灰度、放置变换的数值     

Edge location to subpixel values in digital imagery

A new method for locating edges in digital data to subpixel values and which is invariant to additive and multiplicative changes in the data is presented. For one-dimensional edge patterns an ideal edge is fit to the data by matching moments. It is shown that the edge location is related to the so-called ``Christoffel numbers.' Also presented is the study of the effect of additive noise on edge location. The method is extended to include two-dimensional edge patterns where a line equation is derived to locate an edge. This in turn is compared with the standard Hueckel edge operator. An application of the new edge operator as an edge detector is also provided and is compared with Sobel and Hueckel edge detectors in presence and absence of noise.     

整数阶滤波的分数阶Sobel算子的边缘检测算法

针对传统的整数阶微分图像边缘检测算子存在的边缘模糊不清、受噪声影响大等问题,该算法从改进传统的整数阶微分Sobel算子入手,以分数阶微分理论为基础推导出了分数阶微分Sobel算子,结合Sobel算子边缘检测方法,将整数阶微分Sobel算子作为滤波器与分数阶微分Sobel算子作卷积运算,改进了整数阶微分Sobel算子。整数阶微分滤波后的分数阶微分Sobel算子成功地解决了传统的边缘检测算子存在的准确性低、抗噪性差等问题。理论研究与实验结果表明,该边缘检测算子对图像的边缘细节特征刻画得更精细,抗噪性更强,优于常用的整数阶微分边缘检测算子,边缘检测效果很好。     

Robust digital image watermarking method against geometrical attacks

This paper proposes a new image watermarking scheme which is robust to RST attacks with cropping by improving Fourier–Mellin transform based watermarking (FMW). The proposed scheme reorders and modifies function blocks of FMW for improvement of realization and performance. Unlike FMW, our method uses log-polar map (LPM) in the spatial domain for scaling invariance, while translation invariance is provided by the use of an invariant centroid (IC) as the origin of LPM. IC is a gravity center of a central area on gray scale image that is invariant although an image is attacked by RST. For this, its calculation method is proposed. Also since LPM includes the property which transforms rotation of Cartesian coordinates system into a cyclic shift, 2-D DFT is performed on the LPM image and the magnitude spectrum extracted to provide a domain that is rotation invariant. The resulting domain, which is invariant to RST, is then used as the watermark-embedding domain. Furthermore, to prevent the watermarked image from degrading due to the coordinate system conversion, only LPM image of watermark signal is inverse mapped to Cartesian coordinates and add to the original image. Experimental results demonstrate that the proposed scheme is robust to RST attacks.     

基于局部差分模式的不变性纹理分类

提出了局部差分变换和局部差分模式。局部差分变换具有灰度线性不变性,可消除光照变化对纹理分析的影响。基于局部差分变换的局部差分模式具有光照、旋转不变性和良好的多尺度分析能力。局部差分模式直方图可作为光照、平移、旋转不变性特征用于不变性纹理分类。实验表明,该方法的不变性纹理分类效果优于目前国际公认的基于LBP的方法。     

基于改进LPB特征算子的行人检测算法研究

LBP特征算子具有光照不变性、旋转不变性且计算简单等优点,是行人检测中描述人体常用的特征算子,但在场景光照与行人对比度较大时LBP算子对行人特征描述不太准确,容易造成跟踪失败。针对以上情况提出了改进的LBP算子,在计算LBP二进制码时不仅考虑中心点与邻域点像素值的大小,还将二者之间的对比度信息加入到LBP算子中。通过实验证明,改进的LBP特征算子对图像噪声不太敏感,且在光照不足的环境中对行人具有更强的表示能力和判别能力。     

Hierarchical kernel-based rotation and scale invariant similarity

Image similarity measure has been widely used in pattern recognition and computer vision. We usually face challenges in terms of rotation and scale changes. In order to overcome these problems, an effective similarity measure which is invariant to rotation and scale is proposed in this paper. Firstly, the proposed method applies the log-polar transform to eliminate the rotation and scale effect and produces a row and column translated log-polar image. Then the obtained log-polar image is passed to hierarchical kernels to eliminate the row and column translation effects. In this way, the output of the proposed method is invariant to rotation and scale. The theoretical analysis of invariance has also been given. In addition, an effective template sets construction method is presented to reduce computational complexity and to improve the accuracy of the proposed similarity measure. Through the experiments with several image data sets we demonstrate the advantages of the proposed approach: high classification accuracy and fast.     

Comparative evaluation of linear edge detection methods

A comparative evaluation of the most commonly used linear methods for edge detection in grayscale images are presented. Detectors based on the first and second derivatives of image brightness are considered. The method for automatic edge tracking in grayscale images is proposed. The model for assessing errors and artifacts caused by sampling during digitization of real input images is proposed. Investigation of edge detectors isotropy and errors caused by input images sampling is conducted. The advantage of the Isotropic operator for edge tracking is shown. The noise immunity of linear edge detection methods is assessed and the superiority of 3 × 3 gradient operators for noisy images is shown. Isotropic and Sobel operators are identified to be optimal on a basis of sampling errors, output noise level, and computational complexity.     

基于MATLAB 的遥感图像边缘检测算子的实验比较分析

MATLAB软件工具箱中,提供了edge函数,它能利用多种算子进行图像的边缘检测,语言结构简单。本文就数字图像处理中几种常用的用于检测的算子（Roberts算子、Prewitt算子、Sobel算子、log算子、Canny算子）,通过MATLAB软件来实现遥感图像的边缘检测,对抗噪声能力进行分析,得出各自的优缺点和适用范围。     

Computer modeling of hydrogen-like atoms in quantum mechanics with nonnegative distribution function

An algorithm for describing operator structures and calculating energy levels for hydrogen-like atoms in the quantum mechanics with nonnegative distribution function (Kuryshkin’s quantum mechanics) is developed. The algorithm is implemented in Maple. An original library of necessary functions and quantities (Coulomb wave functions, Sturm functions, their Fourier images and kernels of integral transformations, Clebsch-Gordan coefficients, products of spherical harmonics, etc.) has been created. In accordance with the quantization rules in Kuryshkin’s quantum mechanics, operators of observable quantities are computed. Based on the Hamiltonian obtained, energy levels of a hydrogen-like atom are calculated by the Ritz method (in the basis of Sturm functions). Numerical values of model parameters are determined by comparing results obtained with experimental data for hydrogen and alkali metals taken from the NIST Atomic Spectra Database Levels Data library.     

边缘检测算子研究及其在医学图像中的应用

阐述了边缘检测在医学图像处理中的重要作用,分析了几种常用算子(Roberts,Sobel,Prewitt,Laplacian,Canny)基本原理及算法实现,并对比各算子的优缺点,并用VC .NET工具编程实现出以上各种算子对一幅人体头部的MRI图进行边缘提取。对实验结果进行分析得出结论,各种不同的算子对同一幅图片进行边缘提取,会得到完全不同的结果,Canny算子检测出的边缘比一般微分算子更精确、更细,但也可能平滑掉一些有用的边缘。因此,在不同的条件下,应根据具体情况,选择最适合的边缘检测算子对图像进行处理,才能得到最佳效果。