Affine connections, midpoint formation, and point reflection

We describe some differential-geometric structures in combinatorial terms: namely affine connections and their torsion and curvature, and we show that torsion free affine connections may equivalently be presented in terms of some simpler combinatorial structure: midpoint formation, and point reflection (geodesic symmetry). The method employed is that of synthetic differential geometry, which is briefly explained.     

利用仿射几何特性提取图像中的仿射不变特征

利用仿射几何的性质从图像中提取仿射不变特征,提出了扩展质心(extended centroid,EC)和仿射区域划分(affine region cutting,ARC)的概念,通过迭代ARC求得多个仿射区域的扩展质心序列,将扩展质心序列按一定规则组合成一系列三角形,然后根据仿射几何的性质,由各个三角形的面积构造不变特征。该不变特征提取方法具有速度快、简单灵活的特点,所构造的特征量对照度变化、噪声干扰、部分遮挡以及小角度3维旋转具有较好的稳定性,实验结果验证了该方法的有效性。     

Linear Scale-Space Theory from Physical Principles

In the past decades linear scale-space theory was derived on the basis of various axiomatics. In this paper we revisit these axioms and show that they merely coincide with the following physical principles, namely that the image domain is a Galilean space, that the total energy exchange between a region and its surrounding is preserved under linear filtering and that the physical observables should be invariant under the group of similarity transformations. These observables are elements of the similarity jet spanned by natural coordinates and differential energies read out by a vision system.Furthermore, linear scale-space theory is extended to spatio-temporal images on bounded and curved domains. Our theory permits a delay-operation at the present moment which is in agreement with the motion detection model of Reichardt. In this respect our theory deviates from that of Koenderink which requires additional syntactical operators to realise such a delay-operation.Finally, the semi-discrete and discrete linear scale-space theories are derived by discretising the continuous theories following the theory of stochastic processes. The relation and difference between our stochastic approach and that of Lindeberg is pointed out. The connection between continuous and (semi-)discrete sale-space theory for infinitely high scales observed by Lindeberg is refined by applying appropriate scaling limits. It is shown that Lindeberg's requirement of normalisation for one-dimensional discrete Green's functions can be incorporated into our theory for arbitrary dimensional discrete Green's functions, parameter determination can be avoided, and the requirement of operation at even and odd coordinates sum can be guaranteed simultaneously by taking a normalised linear combination of the identity operator and the first step discrete Green's functions. The new discrete Green's functions are still intimately related to the continuous Green's functions and appear to coincide with pyramidal discrete Green's functions.     

仿射不变的自适应局部线性嵌入

目的: 为将流形学习有效应用于图像的降维与识别中,并消除图像的仿射变换对流形结构产生的影响,本文提出一种仿射不变的自适应局部线性嵌入算法。方法: 该算法在局部线性嵌入的基础上,为适应产生各种仿射变换的图像样本,引入切线距离计算各样本之间的相似程度,以此描述样本空间中的距离,并通过图像相似度函数自适应计算样本空间中每一点的邻域数量。结果: 实验结果表明,该算法能够构造出更合理的低维流形结构,并有效提升统计识别的正确率。结论: 本文算法对仿射变换不敏感,表现出更强的稳健性。     

三相磁集成VRM的微分几何非线性控制研究

以三相磁集成VRM为研究对象,应用微分几何理论实现三相磁集成VRM的非线性解耦控制。在统一的开关脉冲函数下,基于微分几何理论得到三相磁集成VRM的状态反馈线性化解耦控制规律。建立三输入三输出仿射非线性模型,仿真实验表明,基于微分几何非线性控制的磁集成VRM具有良好的动态品质和稳态特性。     

Area and length preserving geometric invariant scale-spaces

In this paper, area preserving multi-scale representations of planar curves are described. This allows smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. When the Euclidean geometric heat flow is used, the resulting representation is Euclidean invariant, and similarly it is affine invariant when the affine one is used. The flows are geometrically intrinsic to the curve, and exactly satisfy all the basic requirements of scale-space representations. In the case of the Euclidean heat flow, it is completely local as well. The same approach is used to define length preserving geometric flows. A similarity (scale) invariant geometric heat flow is studied as well in this work     

时空双仿射微分不变量及骨架动作识别

目的 人体骨架的动态变化对于动作识别具有重要意义。从关节轨迹的角度出发,部分对动作类别判定具有价值的关节轨迹传达了最重要的信息。在同一动作的每次尝试中,相应关节的轨迹一般具有相似的基本形状,但其具体形式会受到一定的畸变影响。基于对畸变因素的分析,将人体运动中关节轨迹的常见变换建模为时空双仿射变换。方法 首先用一个统一的表达式以内外变换的形式将时空双仿射变换进行描述。基于变换前后轨迹曲线的微分关系推导设计了双仿射微分不变量,用于描述关节轨迹的局部属性。基于微分不变量和关节坐标在数据结构上的同构特点,提出了一种通道增强方法,使用微分不变量将输入数据沿通道维度扩展后,输入神经网络进行训练与评估,用于提高神经网络的泛化能力。结果 实验在两个大型动作识别数据集NTU（Nanyang Technological University）RGB+D（NTU 60）和NTU RGB+D 120（NTU 120）上与若干最新方法及两种基线方法进行比较,在两种实验设置（跨参与者识别与跨视角识别）中均取得了明显的改进结果。相比于使用原始数据的时空图神经卷积网络（spatio-temporal graph convolutional networks,ST-GCN）,在NTU 60数据集中,跨参与者与跨视角的识别准确率分别提高了1.9%和3.0%;在NTU 120数据集中,跨参与者与跨环境的识别准确率分别提高了5.6%和4.5%。同时对比于数据增强,基于不变特征的通道增强方法在两种实验设置下都能有明显改善,更为有效地提升了网络的泛化能力。结论 本文提出的不变特征与通道增强,直观有效地综合了传统特征和深度学习的优点,有效提高了骨架动作识别的准确性,改善了神经网络的泛化能力。     

一种新的图像局部仿射不变特征提取方法

提出了一种新的图像局部仿射不变特征的提取方法,首先利用阈值分割在图像中确定某个灰度取值范围的连通区域,文中证明了该区域的几何中心在图像仿射变换前后具有不变性,然后将几何中心作为局部仿射不变特征的"     

基于深度信息与尺度空间的仿射不变特征检测*

介绍了一种利用深度信息的仿射区域检测器。这种方法在视角变换的情况下能自动检测出图景中同一物理区域,为后续的识别算法提供了坚实的特征检测基础,在计算机视觉领域有广阔的应用前景。该方法是基于尺度空间理论,这个理论已经在自动尺度选择中有较成熟的应用。提出了利用深度信息估计出3D物体模型的算法,并生成相应的仿射不变的高斯尺度空间,并给出从3D到2D的投射变换的高精度估计方法,以补偿投射变换造成的扭曲形变。因此对特征检测的可靠性将有明显的提高。为了评估本算法的鲁棒性,进行了不同视角的真实图片与合成图片的实验,并与其     

Scale-Space Image Analysis Based on Hermite Polynomials Theory

The Hermite transform allows to locally approximate an image by a linear combination of polynomials. For a given scale σ and position ξ, the polynomial coefficients are closely related to the differential jet (set of partial derivatives of the blurred image) for the same scale and position. By making use of a classical formula due to Mehler (late 19th century), we establish a linear relationship linking the differential jets at two different scales σ and positions ξ involving Hermite polynomials. For multi-dimensional images, anisotropic excursions in scale-space can be handled in this way. Pattern registration and matching applications are suggested.We introduce a Gaussian windowed correlation function K (ν) for locally matching two images. When taking the mutual translation parameter ν as an independent variable, we express the Hermite coefficients of K (ν)interms of the Hermite coefficients of the two images being matched. This new result bears similarity with the Wiener-Khinchin theorem which links the Fourier transform of the conventional (flat-windowed) correlation function with the Fourier spectra of the images being correlated. Compared to the conventional correlation function, ours is more suited for matching localized image features.Numerical simulations using 2D test images illustrate the potentials of our proposals for signal and image matching in terms of accuracy and algorithmic complexity.First online version published in June, 2005     

一种新的统计仿射不变量

本文在充分分析了现有各种仿射不变量的基础上，提出了一种新的统计仿射不变量，它对于发生尺度变化、旋转、扭曲和平移的目标具有不变性。实验结果表明，相对于传统的仿射不变量，在目标轮廓分割不完整或噪声污染的情况下，仍能够保持较高的稳健性，同时克服了基于轮廓的小波方法和傅立叶级数等方法对目标轮廓出现缺陷时的不足，
，在图像目标识别中具有实用价值。     

Forward变换器的精确线性化控制

为了提高Forward变换器非线性系统的控制性能,采用了精确线性化控制方法。首先采用开关函数和开关周期平均算子建立适合微分几何方法的仿射非线性系统模型。从理论上证明了该模型满足系统精确线性化的条件。对非线性坐标变换后得到的线性系统,利用二次型最优控制策略推导出非线性状态反馈控制律。实验结果表明,系统具有良好的静态和动态性能,验证了该控制方法的有效性和正确性。     

Left-Invariant Diffusions on the Space of Positions and Orientations and their Application to Crossing-Preserving Smoothing of HARDI images

HARDI (High Angular Resolution Diffusion Imaging) is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. In this article we study left-invariant diffusion on the group of 3D rigid body movements (i.e. 3D Euclidean motion group) SE(3) and its application to crossing-preserving smoothing of HARDI images. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space of positions and orientations in 3D embedded in SE(3) and can be solved by ℝ³ ⋊ S ²-convolution with the corresponding Green’s functions. We provide analytic approximation formulas and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (nonlinear) convection-diffusions on HARDI data we explain the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan connection. This Cartan connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI data containing crossing-fibers.     

Numerically Invariant Signature Curves

Corrected versions of the numerically invariant expressions for the affine and Euclidean signature of a planar curve introduced by Calabi et al. in (Int. J. Comput. Vision, 26: 107–135, 1998) are presented. The new formulas are valid for fine but otherwise arbitrary partitions of the curve. We also give numerically invariant expressions for the four differential invariants parameterizing the three dimensional version of the Euclidean signature curve, namely the curvature, the torsion and their derivatives with respect to arc length.     

Reliable and Efficient Pattern Matching Using an Affine Invariant Metric

We present a new pattern similarity measure that behaves well under affine transformations. Our similarity measure is useful for pattern matching since it is defined on patterns with multiple components, satisfies the metric properties, is invariant under affine transformations, and is robust with respect to perturbation and occlusion. We give an algorithm, based on hierarchical subdivision of transformation space, which minimises our measure under the group of affine transformations, given two patterns. In addition, we present results obtained using an implementation of this algorithm.     

基于连通区域的仿射不变区域提取方法?

针对现有仿射不变特征提取方法存在计算量大、鲁棒性差的问题,提出一种基于连通区域的仿射不变区域提取方法。对输入的灰度图像进行各向异性高斯滤波,并对滤波后的图像进行灰度直方图均衡化。找到图像中灰度值相同点所组成的连通区域,将灰度值差小于delta的相邻连通区域进行合并,把满足条件的最后一次合并结果作为图像的局部仿射不变区域。实验证明,该方法提取效果好、速度快且鲁棒性强。     

On differential invariants of planar curves and recognizing partially occluded planar shapes

Viewing transformations like similarity, affine and projective maps may distort planar shapes considerably. However, it is possible to associate local invariant signature functions to smooth boundaries that enable recognition of distorted shapes even in the case of partial occlusion. The derivation of signature functions, generalizing the intrinsic curvature versus arc-length representation in the case of rigid motions in the plane, is based on differential invariants associated to viewing transformation.