Geometry of single axis motions using conic fitting

Previous algorithms for recovering 3D geometry from an uncalibrated image sequence of a single axis motion of unknown rotation angles are mainly based on the computation of two-view fundamental matrices and three-view trifocal tensors. We propose three new methods that are based on fitting a conic locus to corresponding image points over multiple views. The main advantage is that determining only five parameters of a conic from one corresponding point over at least five views is simpler and more robust than determining a fundamental matrix from two views or a trifocal tensor from three views. It is shown that the geometry of single axis motion can be recovered either by computing one conic locus and one fundamental matrix or by computing at least two conic loci. A maximum likelihood solution based on this parametrization of the single axis motion is also described for optimal estimation using three or more loci. The experiments on real image sequences demonstrate the simplicity, accuracy, and robustness of the new methods.     

曲率连续的有理二次样条插值的一种优化方法

人们通常用有理三次曲线样条来构造整体曲率连续的曲线.提出利用有理二次样条曲线插值整体曲率连续的曲线的一种方法.首先导出了两相邻二次曲线段间曲率连续的拼接条件,然后提出了求解平面上一个闭的点列中每一点处的切线的最优算法.最后给出了闭曲线插值的一些实例以检验方法的有效性.     

场景无关约束下的特征匹配算法

鉴于对极约束是立体图像中完全不依赖于场景的重要几何约束，因此在特征匹配中起着很重要的作用，而且由于同形映射描述了平面场景的立体图像之间的对应关系，故大量文献中利用它对平面场景的立体图像对进行特征匹配。为了提高立体图像匹配精度和速度，提出了一种改进的场景无关约束下的特征匹配算法，该算法针对用对极约束和同形映射来进行曲面场景匹配的过程中同形估计容易出现降阶的情况，通过引入区域面积检测法来避免降阶情况的发生，以改善匹配结果；同时，由于在同形矩阵估计中，通过加入基础矩阵和同形矩阵本质上的约束关系，可使得原本独立的同形约束和对极约柬关系很好地融入到匹配的整个过程中，从而快速有效地抑制了错误匹配的发生。对真实图像的实验分析证明，该改进算法具有迭代次数少、速度更快和匹配精度高的良好性能。     

Fitting conics to paracatadioptric projections of lines

The paracatadioptric camera is one of the most popular panoramic systems currently available in the market. It provides a wide field of view by combining a parabolic shaped mirror with a camera inducing an orthographic projection. Previous work proved that the paracatadioptric projection of a line is a conic curve, and that the sensor can be fully calibrated from the image of three or more lines. However, the estimation of the conic curves where the lines are projected is hard to accomplish because of the partial occlusion. In general only a small arc of the conic is visible in the image, and conventional conic fitting techniques are unable to accurately estimate the curve. The present work provides methods to overcome this problem. We show that in uncalibrated paracatadioptric views a set of conic curves is a set of line projections if and only if certain properties are verified. These properties are used to constrain the search space and correctly estimate the curves. The conic fitting is solved naturally by an eigensystem whenever the camera is skewless and the aspect ratio is known. For the general situation the line projections are estimated using non-linear optimization. The set of paracatadioptric lines is used in a geometric construction to determine the camera parameters and calibrate the system. We also propose an algorithm to estimate the conic locus corresponding to a line projection in a calibrated paracatadioptric image. It is proved that the set of all line projections is a hyperplane in the space of conic curves. Since the position of the hyperplane depends only on the sensor parameters, the accuracy of the estimation can be improved by constraining the search to conics lying in this subspace. We show that the fitting problem can be solved by an eigensystem, which leads to a robust and computationally efficient method for paracatadioptric line estimation.     

基于扩展摄像机成像模型的自标定方法*

提出了一种基于扩展摄像机成像模型的自标定方法,将采用不同视点成像的观念转换为不同方向透视投影下分析,得到在一幅图像中同时建立三对不同方向的空间面与图像面的单应关系,进而建立不同面的圆环点绝对二次方程,实现单幅图像标定。采用长方体进行实验,模拟实验和真实图像实验均验证了这种思维方法的可行性和正确性。     

Multiple View Geometry of General Algebraic Curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus.We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d – 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.     

空间二次曲线射影重建计算方法研究

利用二次曲线上点和直线的对应以及利用二次曲线光流对二次曲线进行解释,需要寻求多元方程组的最优解,计算过程较为复杂。将空间二次曲线表示为一个空间二次曲面和一个平面的交线,提出了一对对应二次曲线存在两个独立的多项式条件;利用两个二次曲面的基曲线推导出二次曲线重建的两个对应条件,在此基础上,给出了空间二次曲线射影重建的计算方法和计算步骤。实验验证表明该算法实用和可靠。     

Discrete contours in multiple views: approximation and recognition

Recognition of discrete planar contours under similarity transformations has received a lot of attention but little work has been reported on recognizing them under more general transformations. Planar object boundaries undergo projective or affine transformations across multiple views. We present two methods to recognize discrete curves in this paper. The first method computes a piecewise parametric approximation of the discrete curve that is projectively invariant. A polygon approximation scheme and a piecewise conic approximation scheme are presented here. The second method computes an invariant sequence directly from the sequence of discrete points on the curve in a Fourier transform space. The sequence is shown to be identical up to a scale factor in all affine related views of the curve. We present the theory and demonstrate its applications to several problems including numeral recognition, aircraft recognition, and homography computation.     

一类三焦点曲线

针对三焦点曲线进行研究。首先提出一类三焦点曲线的概念和数学表达式,对该类 三焦点曲线表达式的性质进行分析。其次,通过变化三焦点曲线参数,用计算机编程方法对三焦 点曲线进行定量化分析和形状分析,分析和总结得到了曲线性质和规律。最后,通过对实际物体 图像进行处理、提取边缘、曲线拟合等图像处理方法获得物体图像的三焦点曲线表示,并对实验 结果进行了误差分析。实验结果表明三焦点曲线能够精确表示某些自然物体图像的轮廓。     

Tracking While Zooming Using Affine Transfer and Multifocal Tensors

This paper presents algorithms for tracking unknown objects in the presence of zoom. Since prior models are unavailable, point and line matches in affine views are used to characterize the structure and to transfer a fixation point into new images in a sequence. Because any affine projection matrix is permitted, the intrinsic camera parameters such as focal length may change freely. Also, since the techniques do not require long feature tracks, a further desirable property is insensitivity to partial occlusion caused, for instance, by part of the object falling off the image plane while zooming in. If only point matches are available, a previous method based on factorization is applied. When also incorporating lines, the affine trifocal and quadrifocal tensors are used for tracking in monocular and stereo systems respectively. Methods for computing the tensors, minimizing algebraic error, are developed. In comparison with their projective counterparts, the affine tensors offer significant advantages in terms of computation time and convenience of parameterization, and the relations between the different tensors are shown to be much simpler. Successful tracking is demonstrated on several real image sequences.     

一种评价仿射不变性特征定位误差的新方法

特征点匹配是计算机视觉中的一个基本问题。不同视点图像中相应特征点邻域窗口之间存在几何上的透视畸变,这可用平面单应映射来表示,而目前大多匹配算法将该映射用仿射变换模型来近似,即用具有仿射不变性的特征进行图像的匹配。仿射变换的线性特点不仅能降低算法复杂度,还能保证迭代过程收敛的稳定性,然而并没有人对该近似的可行性及如何减小近似误差给出定量分析。本文首先回顾了各种几何层次上的特征点匹配策略,重点针对具有仿射不变性特征点的定位误差进行研究和定量分析,通过本文提出的椭圆曲线规范化法推导出该近似所造成相应特征点定位误差的解析表达;然后用真实图像的实验结果验证了本文分析方法的必要性和正确性;最后给出相应的分析结果和结论,并指出提高大基线图像特征点匹配精度的相应措施。     

Circular motion geometry using minimal data

Circular motion or single axis motion is widely used in computer vision and graphics for 3D model acquisition. This paper describes a new and simple method for recovering the geometry of uncalibrated circular motion from a minimal set of only two points in four images. This problem has been previously solved using nonminimal data either by computing the fundamental matrix and trifocal tensor in three images or by fitting conics to tracked points in five or more images. It is first established that two sets of tracked points in different images under circular motion for two distinct space points are related by a homography. Then, we compute a plane homography from a minimal two points in four images. After that, we show that the unique pair of complex conjugate eigenvectors of this homography are the image of the circular points of the parallel planes of the circular motion. Subsequently, all other motion and structure parameters are computed from this homography in a straighforward manner. The experiments on real image sequences demonstrate the simplicity, accuracy, and robustness of the new method.     

Globally Optimal Algorithms for Stratified Autocalibration

We present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality. Given a projective reconstruction, we first upgrade it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step.     

影像匹配误匹配点的剔除算法

景像匹配过程的复杂性不可避免地会产生误匹配点。基于巡航导弹机动性能的规律，巡航导弹飞越一个匹配区得到的各个匹配点的似合曲线具有一定的变化规律。据此，提出了影像匹配误匹配点剔除的3种算法：⑴基于匹配点拟合曲线曲率较小的算法；⑵基于匹配点拟合曲线为直线的算法；⑶利用惯性导航短时间测量距离的精确性剔除误匹配点的方法。采用试飞试验数据对这3种算法进行仿真验证的结果表明，这3种算法在相应条件下，均能有效地剔除误匹配点，并能满足影像匹配实时怀的要求。     

Lines and Points in Three Views and the Trifocal Tensor

This paper discusses the basic role of the trifocal tensor in scene reconstruction from three views. This 3× 3× 3 tensor plays a role in the analysis of scenes from three views analogous to the role played by the fundamental matrix in the two-view case. In particular, the trifocal tensor may be computed by a linear algorithm from a set of 13 line correspondences in three views. It is further shown in this paper, that the trifocal tensor is essentially identical to a set of coefficients introduced by Shashua to effect point transfer in the three view case. This observation means that the 13-line algorithm may be extended to allow for the computation of the trifocal tensor given any mixture of sufficiently many line and point correspondences. From the trifocal tensor the camera matrices of the images may be computed, and the scene may be reconstructed. For unrelated uncalibrated cameras, this reconstruction will be unique up to projectivity. Thus, projective reconstruction of a set of lines and points may be carried out linearly from three views.     

G 2 curve design with planar quadratic rational Bézier spiral segments

Spiral segments are useful in the design of fair curves. They are important in computer-aided design (CAD) and manufacturing applications, the design of highway and railway routes, trajectories of mobile robots, and other similar applications. Quadratic rational Bézier curves are often used in CAD applications because they can be used to draw conic sections exactly. This paper shows how curvature continuous curves can be designed using quadratic rational Bézier curve segments of monotone curvature.     

一种基于平面模板的摄像机自定标方法

给出仿射坐标系下场景中平面与像平面的单应关系、绝对二次曲线及其图像的表示．通过对场景中一个平面模板获取3幅图像(该模板是由含内切圆的等边三角形构成)，利用上述单应关系并结合圆环点对摄像机内参数的约束，获得一组线性方程，进而确定摄像机的内参数．实验结果表明，所给出的方法切实可行，且具有较高的求解精度．     

Three-view multibody structure from motion

We propose a geometric approach to 3-D motion segmentation from point correspondences in three perspective views. We demonstrate that after applying a polynomial embedding to the point correspondences they become related by the socalled multibody trilinear constraint and its associated multibody trifocal tensor, which are natural generalizations of the trilinear constraint and the trifocal tensor to multiple motions. We derive a rank constraint on the embedded correspondences, from which one can estimate the number of independent motions as well as linearly solve for the multibody trifocal tensor. We then show how to compute the epipolar lines associated with each image point from the common root of a set of univariate polynomials and the epipoles by solving a pair of plane clustering problems using Generalized PCA (GPCA). The individual trifocal tensors are then obtained from the second order derivatives of the multibody trilinear constraint. Given epipolar lines and epipoles, or trifocal tensors, one can immediately obtain an initial clustering of the correspondences. We use this clustering to initialize an iterative algorithm that alternates between the computation of the trifocal tensors and the segmentation of the correspondences. We test our algorithm on various synthetic and real scenes, and compare with other algebraic and iterative algorithms.     

有理q-Bernstein-Bézier曲线的构造及其应用

有理Bernstein-Bézier曲线在计算机辅助设计和计算机图形学上具有广泛的应用。在研究了经典的Bernstein-Bézier曲线及de Casteljau算法的基础上,结合q-Bernstein多项式,给出了有理q-Bernstein-Bézier曲线的构造方法、性质和计算有理曲线的de Casteljau算法,并讨论了曲线的细分和升阶的方法,通过改变q的取值,可以获得有理曲线族,在曲线造型上具有较强的灵活性。最后通过表示圆锥曲线和数字图像插值证明有理q-Bernstein-Bézier曲线的推广是有效的。     

基于三焦点张量的树木图像匹配

计算机视觉技术在林业上的应用刚刚起步,但有很广阔的应用前景。在实际应用中由于自然环境的复杂或者遮挡等问题,使得树木图像容易产生误匹配,该文提出了基于三焦点张量解决在树木图像匹配中树木对应问题,这是进一步点对点匹配,计算树木空间位置的前题。首先通过三焦点张量建立三张图像的关联关系,然后由两张图像上的树干线,通过张量的直线转移求得第三张图像上的树干对应位置。这有效地解决了树木图像因拍照角度不同引起的树木在图像上位置的变换,以及遮挡等问题。