Hopfield neural networks for affine invariant matching

The affine transformation, which consists of rotation, translation, scaling, and shearing transformations, can be considered as an approximation to the perspective transformation. Therefore, it is very important to find an effective means for establishing point correspondences under affine transformation in many applications. In this paper, we consider the point correspondence problem as a subgraph matching problem and develop an energy formulation for affine invariant matching by a Hopfield type neural network. The fourth-order network is investigated first, then order reduction is done by incorporating the neighborhood information in the data. Thus we can use the second-order Hopfield network to perform subgraph isomorphism invariant to affine transformation, which can be applied to an affine invariant shape recognition problem. Experimental results show the effectiveness and efficiency of the proposed method.     

Part I: Modeling image curves using invariant 3-D object curvemodels-a path to 3-D recognition and shape estimation from imagecontours

This paper and its companion are concerned with the problems of 3-D object recognition and shape estimation from image curves using a 3-D object curve model that is invariant to affine transformation onto the image space, and a binocular stereo imaging system. The objects of interest here are the ones that have markings (e.g., characters, letters, special drawings and symbols, etc.) on their surfaces. The 3-D curves on the object are modeled as B-splines, which are characterized by a set of parameters (the control points) from which the 3-D curve can be totally generated. The B-splines are invariant under affine transformations. That means that the affine projected object curve onto the image space is a B-spline whose control points are related to the object control points through the affine transformation. Part I deals with issues relating to the curve modeling process. In particular, the authors address the problems of estimating the control points from the data curve, and of deciding on the “best” order B-spline and the “best” number of control points to be used to model the image or object curve(s). A minimum mean-square error (mmse) estimation technique which is invariant to affine transformations is presented as a noniterative, simple, and fast approach for control point estimation. The “best” B-spline is decided upon using a Bayesian selection rule. Finally, we present a matching algorithm that allocates a sample curve to one of p prototype curves when the sample curve is an a priori unknown affine transformation of one of the prototype curves stored in the data base. The approach is tried on a variety of images of real objects     

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.     

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。     

Affine Curvature Scale Space with Affine Length Parametrisation

The maxima of Curvature Scale Space (CSS) image have been used to represent 2D shapes under affine transforms. The CSS image is expected to be in the MPEG-7 package of standards. Since the CSS image employs the arc length parametrisation which is not affine invariant, we expect some deviations in the maxima of the CSS image under general affine transforms. Affine length and affine curvature have already been introduced and used as alternatives to arc length and conventional curvature in affine transformed environments. The utility of using these parameters to enrich the CSS representation is addressed in this paper. We use arc length to parametrise the curve prior to computing its CSS image. The parametrisation has been proven to be invariant under affine transformation and has been used in many affine invariant shape recognition methods. Since the organisation of the CSS image is based on curvature zero crossings of the curve, in this paper, we also investigate the advantages and shortcomings of using affine curvature in computation of the CSS image. The enriched CSS representations are then used to find similar shapes from a very large prototype database, and also a small classified database, both consisting of original as well as affine transformed shapes. An improvement is observed over the conventional CSS image.     

Quasi-Invariant Parameterisations and Matching of Curves in Images

In this paper, we investigate quasi-invariance on a smooth manifold, and show that there exist quasi-invariant parameterisations which are not exactly invariant but approximately invariant under group transformations and do not require high order derivatives. The affine quasi-invariant parameterisation is investigated in more detail and exploited for defining general affine semi-local invariants from second order derivatives only. The new invariants are implemented and used for matching curve segments under general affine motions and extracting symmetry axes of objects with 3D bilateral symmetry.     

Statistical Image Object Recognition using Mixture Densities

In this paper, we present a mixture density based approach to invariant image object recognition. To allow for a reliable estimation of the mixture parameters, the dimensionality of the feature space is optionally reduced by applying a robust variant of linear discriminant analysis. Invariance to affine transformations is achieved by incorporating invariant distance measures such as tangent distance. We propose an approach to estimating covariance matrices with respect to image variabilities as well as a new approach to combined classification, called the virtual test sample method. Application of the proposed classifier to the well known US Postal Service handwritten digits recognition task (USPS) yields an excellent error rate of 2.2%. We also propose a simple, but effective approach to compensate for local image transformations, which significantly increases the performance of tangent distance on a database of 1,617 medical radiographs taken from clinical daily routine.     

An affine invariant curve matching method for photo-identification of marine mammals

Individual identification of marine mammals is of interest to marine biologists. This paper aims at the recognition of the edges associated with marine mammals whose pictures are taken under affine transformations. The introduced affine curve matching method uses the area of mismatch between a query and a database curve. This area is obtained by optimally aligning the curves based on the minimum affine distance involving their distinguishing points. The method is applied to databases of sea lions, gray whales, and dolphins, and its performance is compared with two other affine curve matching methods. The results show that the introduced curve matching approach outperforms the other approaches and thus further reduces the search time for identifying an individual. The developed method is of general purpose as it can be used for other affine curve matching applications.     

一种基于目标边界的不变特征提取方法

提出了一种基于目标边界的不变特征提取方法。导出了用物体角点坐标表示的低阶边界矩的闭合形式,构造了基于边界矩的仿射变换不变量。该方法只需要对物体角点进行简单的代数运算,因此,该方法简单明了,计算量很小。实验结果证明了该方法的有效性。     

Blind digital watermarking of rational Bézier and B-spline curves and surfaces with robustness against affine transformations and Möbius reparameterization

We present a blind watermarking scheme for rational Bézier and B-spline curves and surfaces which is shape-preserving and robust against the affine transformations and Möbius reparameterization that are commonly used in geometric modeling operations in CAD systems. We construct a watermark polynomial with real coefficients of degree four which has the watermark as the cross-ratio of its complex roots. We then multiply the numerator and denominator of the original curve or surface by this polynomial, increasing its degree by four but preserving its shape. Subsequent affine transformations and Möbius reparameterization leave the cross-ratio of these roots unchanged. The watermark can be extracted by finding all the roots of the numerator and denominator of the curve or surface: the cross-ratio of the four common roots will be the watermark. Experimental results confirm both the shape-preserving property and its robustness against attacks by affine transformations and Möbius reparameterization.     

Shape-based Invariant Texture Indexing

This paper introduces a new texture analysis scheme, which is invariant to local geometric and radiometric changes. The proposed methodology relies on the topographic map of images, obtained from the connected components of level sets. This morphological tool, providing a multi-scale and contrast-invariant representation of images, is shown to be well suited to texture analysis. We first make use of invariant moments to extract geometrical information from the topographic map. This yields features that are invariant to local similarities or local affine transformations. These features are invariant to any local contrast change. We then relax this invariance by computing additional features that are invariant to local affine contrast changes and investigate the resulting analysis scheme by performing classification and retrieval experiments on three texture databases. The obtained experimental results outperform the current state of the art in locally invariant texture analysis.     

Geometric Invariant Shape Representations Using Morphological Multiscale Analysis

In this paper, we present a new geometric invariant shape representation using morphological multiscale analysis. The geometric invariant is based on the area and perimeter evolution of the shape under the action of a morphological multiscale analysis. First, we present some theoretical results on the perimeter and area evolution across the scales of a shape. In the case of similarity transformations, the proposed geometric invariant is based on a scale-normalized evolution of the isoperimetric ratio of the shape. In the case of general affine geometric transformations the proposed geometric invariant is based on a scale-normalized evolution of the area. We present some numerical experiments to evaluate the performance of the proposed models. We present an application of this technique to the problem of shape classification on a real shape database and we study the well-posedness of the proposed models in the framework of viscosity solution theory.     

Invariant feature extraction for 3D texture analysis using the autocorrelation function

A new method for invariant feature extraction on textured images undergoing affine transformations is presented. This is performed by transformation of the autocorrelation function (ACF) followed by determination of an invariant criterion which is the sum of the coefficients of the discrete correlation matrix. Experimental results support the effectiveness of the proposed approach.     

基于小波变换的局部形状匹配

为解决仿射变换下的局部形状匹配问题,提出了一种新的基于小波描述子的局部形状匹配方法。算法首先基于分割点将轮廓曲线分段,为了更精确地描述每段子曲线,定义一种新的特征点--等面积分割点,并在子曲线上提取,基于该特征点构造一种新的具有局部特征的小波描述子。新定义的等面积分割点有比一般的特征点（角点、拐点、切点）更精确描述曲线的特性,能解决轮廓曲线平滑特征点少而不能被精确描述的问题;定义的等面积分割点和提取的小波描述子都具有仿射不变性,且均为局部描述符,因而该方法适合于仿射变换且在轮廓局部遮挡和缺失的情况下仍然有效。理论分析和实验结果都证明了该算法的有效性。     

结合轮廓与形状特征的仿射形状匹配

目的 针对仿射变换下形状匹配中存在的描述子对形状的描述能力不足,以及描述子计算耗时大的问题,改进基于所有图像点投影的方法,提出一种利用轮廓计算投影面积的仿射形状匹配算法。方法 该算法分为粗匹配和精匹配两个阶段。粗匹配阶段以CSS角点作为备选特征点,首先统计轮廓投影面积分布作为特征点描述子;然后利用动态规划蚁群算法匹配两幅图片公共特征点序列,并将匹配好的特征点序列记为对应的新特征点;最后采用该新特征点划分目标曲线,得到对应的轮廓曲线;这一阶段的目的是对形状的筛选以及寻找一致的轮廓特征点,同时完成轮廓曲线的划分。精匹配阶段,采用小波仿射不变描述子,对粗匹配阶段匹配代价最小的5%的目标进行对应曲线匹配,得到精匹配阶段的匹配代价,从而实现对仿射目标的识别;精匹配弥补了描述子对轮廓细节描述不足的问题。结果 算法的平均检索速度比传统基于形状投影分布描述子提高44.3%,在MPEG-7图像库上的检索效果为98.65%,在MPEG-7仿射图像库上的查准率与查全率综合评价指标比传统的基于形状投影分布描述子高3.1%,比形状上下文高25%。结论 本文算法匹配效果好,效率高,抗噪性强,解决了仿射描述子计算速度慢、描述能力不足的问题,能有效地应用于仿射形状匹配与检索领域。     

Extracting Group Transformations from Image Moments

Image distortion induced by the relative motion between an observer and the scene is an important cue for recovering the motion and the structure of the scene. It is known that the distortion in images can be described by transformation groups, such as Euclidean, affine, and projective groups. In this paper, we investigate how the moments of image curves are changed by group transformations, and we derive a relationship between the change in image moments and the invariant vector fields of the transformation groups. The results are used to formalize a method for extracting invariant vector fields of affine transformations from changes in the moments of orientation of curve segments in images. The method is applied to a realtime robot visual navigation task.