Projective Splines and Estimators for Planar Curves

Recognizing shapes in multiview imaging is still a challenging task, which usually relies on geometrical invariants estimations. However, very few geometric estimators that achieve projective invariance have been devised. This paper proposes a projective length and a projective curvature estimators for plane curves, when the curves are represented by points together with their tangent directions. In this context, the estimations can be performed with only three point-tangent samples for the projective length and five samples for the projective curvature. The proposed length and curvature estimator are based on projective splines built by fitting logarithmic spirals to the point-tangent samples. They are projective invariant and convergent.     

Projective Fourier analysis for patterns

Identifying as a projective group for patterns in the conformal camera model, the projective harmonic analysis on its double covering group is presented in the noncompact and compact pictures — the pictures used to study different aspects of irreducible unitary representations of semisimple Lie groups. Bypassing technicalities of representation theory, but stressing the motivation and similarities with Euclidean Fourier analysis, each constructed picture of the projective Fourier analysis includes the Fourier transform, Plancherel's theorem and convolution property. Projectively covariant characteristics of the analysis in the noncompact picture allow rendering any of image projective transformations of a pattern (after removing conformal distortions) by using only one projective Fourier transform of the original pattern, what is demonstrated in a computer simulation. The convolution properties in both pictures must by used to develop algorithms for projectively invariant matching of patterns. Work in progress on fast algorithms for computing with projective Fourier transforms and for rendering image projective transformations is discussed. Efficient computations of the convolutions would follow from the both fast projective Fourier transforms and their inverses.     

计算射影与排列不变量的新方法

给出了一种表示和计算离散有限点集的射影与排列不变量的简单有效方法.该不变量在计算机视觉、模式识别中有重要应用.首先导出了射影直线上4个点的基于一种对称函数的射影与排列不变量,该不变量等于这4个点的某个原始交比值,具有计算量低,不丢失分辨力等优点.然后根据这个简单的对称函数,结合基本的多项式对称函数,推导出了平面上5个点的两个函数无关的射影与排列不变量,以及空间中6个点的3个函数无关的射影与排列不变量.     

Fundamental limitations on projective invariants of planar curves

In this paper, some fundamental limitations of projective invariants of non-algebraic planar curves are discussed. It is shown that all curves within a large class can be mapped arbitrarily close to a circle by projective transformations. It is also shown that arbitrarily close to each of a finite number of closed planar curves there is one member of a set of projectively equivalent curves. Thus a continuous projective invariant on closed curves is constant. This also limits the possibility of finding so called projective normalisation schemes for closed planar curves     

The Nut in Screw Theory

This study in projective geometry reveals that the principle of duality applies to the screw. Here, the screw is demonstrated to be an element of a projective three‐dimensional space (P³), right alongside the line. Dual elements for the screw and line are also revealed (the nut and spline). Reciprocity is demonstrated for a pair of screws, and incidence is demonstrated for screw and its dual element. Reciprocity and incidence are invariant for projective transformations of P³, but only incidence is invariant for the more general linear transformations of screws. This latter transformation is analogous to a projective transformation of a projective five‐dimensional space (P⁵), which is shown to induce a contact transformation of the original P³, where some points lying on a Kummer surface are directly mapped. © 2003 Wiley Periodicals, Inc.     

An experimental study of projective structure from motion

We describe an essentially algorithm-independent experimental comparison of projective versus Euclidean reconstruction. The Euclidean approach is as accurate as the projective one, even with significant calibration error and for the pure projective structure. Projective optimization has less of a local-minima problem than its Euclidean equivalent. We describe techniques that enhance the convergence of optimization algorithms     

一种GF(2m)上椭圆曲线点运算的混合坐标系

椭圆曲线密码体制(ECC)是一种基于代数曲线的公钥密码体制。椭圆曲线上点运算是该密码体制核心运算，而坐标系的选取决定了点运算速度。为了提高椭圆曲线标量乘速度，在对已有仿射坐标系、Standard投影坐标系、Jacobian投影坐标系和Lopez & Dahab投影坐标系研究的基础上，提出了一种Lopez & Dahab投影坐标系扩展形式，并基于此构建了一种混合坐标系。算法复杂度分析表明，在该混合坐标系下，椭圆曲线标量乘运算时间复杂度比已有坐标系下运算时间复杂度要小。     

基于改进的超混沌系统修正射影同步算法研究

研究超混沌修正射影同步算法问题。由于修正射影同步不可预测的伸缩矩阵因子能够有效地提高保密通信的安全性,所以它在保密通信中有着非常诱人的应用前景,但是传统的同步射影算法由于较高的计算复杂度,难以得到广泛应用。针对上述问题,提出了一种新的基于超混沌系统修正射影同步算法。该方法首先给出了一个比已知的射影同步更一般形式的新同步,之后采用激活控制方法使两个一致的超混沌系统整体同步。最后算法在基于软件Maple和稳定性理论基础上,通过数值仿真同步的结果表明,提出的算法能够自主选择调节缩放矩阵中的比例因子,可以较好地满足状态控制器设计的射影同步,降低了算法的复杂性。     

一种有效的人脸识别方法

人脸识别是模式识别领域中一个相当困难而又有重要理论价值的研究课题.本文首先引入表示人脸的特征图像和投影图像的概念,提出利用投影图像在特征图像上投影的坐标作为描述人像本质属性的特征矢量,并导出了基于投影图像的人脸特征抽取方法.最后构造了一个层次的距离分类器进行人脸的识别.实验结果表明基于投影图像的特征矢量具有很好的稳定性、鉴别能力和识别率.     

The Fisher-Rao Metric for Projective Transformations of the Line

A conditional probability density function is defined for measurements arising from a projective transformation of the line. The conditional density is a member of a parameterised family of densities in which the parameter takes values in the three dimensional manifold of projective transformations of the line. The Fisher information of the family defines on the manifold a Riemannian metric known as the Fisher-Rao metric. The Fisher-Rao metric has an approximation which is accurate if the variance of the measurement errors is small. It is shown that the manifold of parameter values has a finite volume under the approximating metric.These results are the basis of a simple algorithm for detecting those projective transformations of the line which are compatible with a given set of measurements. The algorithm searches a finite list of representative parameter values for those values compatible with the measurements. Experiments with the algorithm suggest that it can detect a projective transformation of the line even when the correspondences between the components of the measurements in the domain and the range of the projective transformation are unknown.     

Fast and accurate algorithms for projective multi-image structurefrom motion

We describe algorithms for computing projective structure and motion from a multi-image sequence of tracked points. The algorithms are essentially linear, work for any motion of moderate size, and give accuracies similar to those of a maximum-likelihood estimate. They give better results than the factorization approach of Sturm and Triggs (1996) and are equally fast and they are much faster than bundle adjustment. Our experiments show that the (iterated) Sturm-Triggs approach often fails for linear camera motions. In addition, we study experimentally the common situation where the calibration is fixed and approximately known, comparing the projective versions of our algorithms to mixed projective/Euclidean strategies. We clarify the nature of dominant-plane compensation, showing that it can be considered a small-translation approximation rather than an approximation that the scene is planar. We show that projective algorithms accurately recover the (projected) inverse depths and homographies despite the possibility of transforming the structure and motion by a projective transformation     

Projective lag synchronization and parameter identification of a new hyperchaotic system

In this paper, the projective lag synchronization of a new hyperchaotic system with certain/uncertain parameters is addressed. Based on Lyapunov stability theory, a generic and simple controller is designed for the projective lag synchronization. Furthermore, with LaSalle s invariance principle, an adaptive method is proposed to identify the unknown parameters of the new hyperchaotic system based on the projective lag synchronization. Finally, numerical simulations are given to support the analytical approach.     

基于遗传算法的射影重构

在实现分层重构的过程中，射影重构是关键的第1步。目前，大多已有算法对模拟数值是非常有效的，但对于真实图象效果并不理想。为了寻求更为鲁棒的算法，提出了一种基于遗传算法的射影重构算法。该算法对于射影深度采用十进制编码，并以测量矩阵的秩为4作为约束，来定义适应度函数，然后利用遗传算法，并结合奇异值分解（SVD）技术来迭代估计射影深度，进而实现射影重构，该算法是行之有效的，且鲁棒性较好。     

基于度量误差模型的摄像机参数求解算法

给出了基于度量误差模型的摄像机定标方案,即由最小二乘法计算投影矩阵,从投影矩阵中分解出参数作为进一步计算的初始值,然后考虑投影矩阵中约束条件及图像坐标的误差结构,由度量误差模型优化参数。结果表明,该方案不仅保证了旋转矩阵的正交性,同时提高了定标精度。     

Invariant computations for analytic projective geometry

This paper focuses on two underlying questions for symbolic computations in projective geometry:

I How should a projective geometric property be written analytically? A first order formula in the language of fields which expresses a “projective geometric property” is translated, by an algorithm, into a restricted class of formulas in the analytic geometric language of brackets (or invariants). This special form corresponds to statements in synthetic projective geometry and the algorithm is a basic step towards translation back into synthetic geometry.

II How are theorems of analytic geometry proven? Axioms for the theorems of analytic projective geometry are given in the invariant language. Identities derived form Hubert's Nullstellensatz then play a central role in the proof. Prom a proof of an open theorem about “geometric properties”, over all fields, or over ordered fields, an algorithm derives Nullstellensatz identities — giving maximal algebraic simplicity, and maximal information in the proof.

The results support the proposal that computational analytic projective geometry should be carried out directly with identities in the invariant language.     

A Geometric Approach for the Theory and Applications of 3D Projective Invariants

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing position. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes in the observation parameters. The study of such geometric invariance is a field of active research. This paper presents the theory and computation of projective invariants formed from points and lines using the geometric algebra framework. This work shows that geometric algebra is a very elegant language for expressing projective invariants using n views. The paper compares projective invariants involving two and three cameras using simulated and real images. Illustrations of the application of such projective invariants in visual guided grasping, camera self-localization and reconstruction of shape and motion complement the experimental part.     

Group recursive discriminant subspace learning with image set decomposition

Discriminant subspace learning is a widely used feature extraction technique for image recognition, since it can extract effective discriminant features by employing the class information and Fisher criterion. A crucial research topic on this technique is how to rapidly extract sufficient and effective features. Recently, recursive discriminant subspace learning technique has attracted lots of research interest because it can acquire sufficient discriminant features. Generally, it recursively decomposes image samples and extracts features from a number of decomposed sample sets. The major drawback of most recursive discriminant subspace learning methods is that they calculate the projective vectors one by one, such that they suffer from big computational costs. The recursive modified linear discriminant method and the incremental recursive Fisher linear discriminant method employ a simple solution for this problem, which calculates the class number minus one projective vectors in each recursion. However, this solution produces the unfavorable projective vectors with poor discriminant capabilities, and it cannot provide the terminating criterion for recursive computation and make the projective vectors orthogonal. In this paper, we propose a novel recursive learning approach that is group recursive discriminant subspace learning, which can rapidly learn multiple orthogonal subspaces with each spanned by a group of projective vectors. And we present a rule to select favorable projective vectors per recursion and provide a matrix-form-based terminating criterion to determine the number of recursions. Experiments on three widely used databases demonstrate the effectiveness and efficiency of the proposed approach.     

A New Method of Image Mosaicking and Its Application to Cultural Heritage Representation

This paper presents an original two-step procedure for estimating projective transformations between pairs of images: first, the transformation between the images is approximated as an affine transformation; second, this estimate is refined into that of a projective transformation. This strategy for matching projective views is computationally very efficient. The proposed method can be applied both to mosaicking of high resolution images of planar textured objects (e.g., frescoes and paintings), with subpixel accuracy, and to construction of panoramic images. Practical examples of mosaicking of cultural heritage imagery obtained by using the presented procedure are discussed in the paper.     

Modeling and computing ternary projective relations between regions

Current spatial database systems offer limited querying capabilities beyond binary topological relations. This paper introduces a model for projective relations between regions to support other qualitative spatial queries. The relations are ternary because they are based on the collinearity invariant of three points under projective geometry. The model is built on a partition of the plane into separate zones that are obtained from projective properties of two reference objects: then, by considering the empty/nonempty intersections of a primary object with these zones, the model is able to distinguish between 34 different projective relations. Then, the paper proposes original algorithms for computing the relations under the assumption that regions of the plane are stored as vector polygons in a spatial database. These algorithms run in optimal O(nlogn) time.