平面单应矩阵对摄像机内参数约束的一些性质

空间x-y坐标平面与图像平面之间的单应矩阵可以提供关于摄像机内参数的2个线性约束.文中研究表明:给定一个一般空间平面与图像平面之间的单应矩阵,如果空间平面在世界坐标系的坐标已知,则该单应矩阵同样可以提供关于摄像机内参数的2个线性约束;如果空间平面在世界坐标系的坐标未知,则该单应矩阵不能构成对摄像机内参数的任何约束.另外,仅仅知道某一个场景中的2幅图像间的单应矩阵不能对摄像机内参数构成任何约束.该结果对从事摄像机标定和三维重建的研究人员有一定的参考作用.     

由平行平面的投影确定无穷远平面的单应矩阵

在三维计算机视觉中,无穷远平面的单应矩阵扮演了极其重要的角色,可使众多视觉问题的求解得到简化.主要讨论如何利用平行平面的投影来求解两个视点间的无穷远平面的单应矩阵,用代数方法构造性地证明了下述结论:(1) 如果场景中含有一组平行平面,则可以通过求解一个一元4次方程来确定两个视点间的无穷远平面对应的单应矩阵;(2) 如果场景中含有两组平行平面,则可以线性地确定两个视点间的无穷远平面对应的单应矩阵.并对上述结果给出了相应的几何解释和具体算法.所给出的结果在三维计算机视觉,特别是摄像机自标定中具有一定的理论意义和应用价值.     

再论一维摄像机标定

基于一维标定杆的摄像机标定是一种重要的标定方法.在已有的文献中,几乎所有的一维标定方法均需要一维标定杆在运动前后处于同一平面内,从本质上说,运动前后的一维标定杆等价于一种二维标定物.从射影几何的观点看,由于空间平面与图像平面之间的单应矩阵包含了所有的摄像机的内外参数信息,因此从理论上说,已有的一维标定方法所得到的关于对摄像机内参数的约束应该从单应矩阵的角度同样可以得到.文中对上述问题进行了尝试,得到了与文献中等价的关于内参数的约束.     

基于正交平面的摄像机自定标

提出了一种基于正交平面摄像机定标的新算法。它利用场景中的正交平面,摄像机作五次以上的平移运动,根据每次运动关于平面的单应矩阵建立内参数的线性约束方程组,从而线性地确定内参数。与以往的定标方法相比,文章对摄像机的运动不苛刻,只需控制摄像机作平移运动,这在一般的实验平台上可以很容易地实现,并且线性地确定摄像机所有的五个内参数。模拟实验和真实图象实验表明,文章给出的方法在机器人视觉中具有一定的实用价值。     

基本矩阵的5点和4点算法

基本矩阵(Fundamental Matrix)是两幅图像之间的基本约束,在摄像机标定和三维重建中起着至关重要的作用.本文证明,当摄像机在两幅图像之间的运动为纯平移运动时,给定5对图像对应点,如果其中的4对对应点为共面空间点的投影(称为共面对应点),则可以线性确定基本矩阵.另外,如果摄像机不是5参数模型(完全针孔模型),而是4参数模型(畸变因子为零),则此时仅使用该4对共面对应点即可线性确定基本矩阵.据我们所知,这些结果在文献中还没有类似的报导.     

基于单应矩阵的摄像机标定方法及应用

提出了一种基于单应矩阵的摄像机标定方法,并应用标定结果成功完成了移动机器人的视觉伺服任务。该方法首先根据图像平面和标定板平面之间特征点的对应关系,对单应矩阵进行了估计,进而利用旋转矩阵的单位正交性得到了其对摄像机内参数的约束条件。然后把摄像机内参数矩阵分解为有效焦距与主点位置两部分,并利用最小二乘法分别对其进行求解。针对镜头的径向畸变,恰当地选取了一种畸变模型,并由此得到了一种新的目标函数来对摄像机的所有参数进行非线性优化,从而使获得的畸变系数更适合于从二维图像信号中提取三维位姿信息。最后将标定结果成功应用于移动机器人视觉伺服系统之中,实验结果验证了该标定算法具有简单易用、精度较高等优良性能。     

基于两条平行线段的摄像机标定

提出了一种基于空间平行线段的摄像机标定算法和理论. 1)当两条平行线段的比值已知时: a)该平行线段的$n$幅图像可提供关于摄像机内参数的2(n-1)个二次约束方程; b)如果图像极点同时也是已知的,则该平行线段的n幅图像可提供关于摄像机内参数的5n-6个约束方程,其中3(n-1)个为二次约束,n-2个为三次约束,n-1个为四次约束. 2)当两条平行线段的比值未知时,则该平行线段的n幅图像可提供关于摄像机内参数的2n-3个约束方程,其中n-2个为四次约束, n-1个为六次约束.在理论分析的基础上,本文给出了摄像机标定的具体算法.模拟实验和真实图像实验均证明了本文方法的可行性.另外,鉴于在很多真实场景中均存在平行线段,因此本文所得到的结果不仅具有理论意义而且也有一定的实用价值.     

基于平面与直线的仿射重建

首先给出了无穷远平面的单应矩阵以及仿射重建算法，然后从数学上严格证明了下述命题：在变参数模型下，如果场景中含有一张平面和一对平行直线，或者场景中含有两张平行平面，则从两个平移视点下的图像均可以线性地对场景进行仿射重建；文章同时指出：如果场景中包含一对平行平面和一对平行直线，则从两个一般运动视点也可以线性地重建场景的仿射几何．大量的模拟和真实图像实验表明，该线性仿射重建算法是正确的，同时具有较高的重建精度和鲁棒性．     

基于主动视觉系统的摄像机自定标方法研究

提出了一种新的基于主动视觉系统的摄像机自定标方法,其主要特点是可以线性求解摄像机的所有5个内参数.该方法的基本原理是利用图像中平面场景的信息,通过控制摄像机作多组平面正交平移运动,由平面场景图像的单应性(homography)矩阵建立摄像机内参数线性约束方程组,来求解摄像机的内参数,同时给出约束方程组解的唯一性与平移运动组之间的关系.     

基于一组对应消失线的度量重建

提出了一种由3幅图像间一组对应消失线进行度量重建的方法.首先利用对应的消失线和模值约束计算出无穷远平面的单应矩阵,然后根据无穷远平面的单应矩阵保持绝对二次曲线的像不动的性质,线性求解摄像机内参数,最后得到度量重建.模拟实验和真实图像实验均验证了这种度量重建方法的可行性和正确性.     

Multiview constraints on homographies

The image motion of a planar surface between two camera views is captured by a homography (a 2D projective transformation). The homography depends on the intrinsic and extrinsic camera parameters, as well as on the 3D plane parameters. While camera parameters vary across different views, the plane geometry remains the same. Based on this fact, we derive linear subspace constraints on the relative homographies of multiple (⩾ 2) planes across multiple views. The paper has three main contributions: 1) We show that the collection of all relative homographies (homologies) of a pair of planes across multiple views, spans a 4-dimensional linear subspace. 2) We show how this constraint can be extended to the case of multiple planes across multiple views. 3) We show that, for some restricted cases of camera motion, linear subspace constraints apply also to the set of homographies of a single plane across multiple views. All the results derived are true for uncalibrated cameras. The possible utility of these multiview constraints for improving homography estimation and for detecting nonrigid motions are also discussed     

Extensions of Plane-Based Calibration to the Case of Translational Motion in a Robot Vision Setting

In this paper, a technique for calibrating a camera using a planar calibration object with known metric structure, when the camera (or the calibration plane) undergoes pure translational motion, is presented. The study is an extension of the standard formulation of plane-based camera calibration where the translational case is considered as degenerate. We derive a flexible and straightforward way of using different amounts of knowledge of the translational motion for the calibration task. The theory is mainly applicable in a robot vision setting, and the calculation of the hand–eye orientation and the special case of stereo head calibration are also being addressed. Results of experiments on both computer-generated and real image data are presented. The paper covers the most useful instances of applying the technique to a real system and discusses the degenerate cases that needs to be considered. The paper also presents a method for calculating the infinite homography between the two image planes in a stereo head, using the homographies estimated between the calibration plane and the image planes. Its possible usage and usefulness for simultaneous calibration of the two cameras in the stereo head are discussed and illustrated using experiments.     

Multiple homographies with omnidirectional vision for robot homing

Two relevant issues in vision-based navigation are the field-of-view constraints of conventional cameras and the model and structure dependency of standard approaches. A good solution of these problems is the use of the homography model with omnidirectional vision. However, a plane of the scene will cover only a small part of the omnidirectional image, missing relevant information across the wide range field of view, which is the main advantage of omnidirectional sensors. The interest of this paper is in a new approach for computing multiple homographies from virtual planes using omnidirectional images and its application in an omnidirectional vision-based homing control scheme. The multiple homographies are robustly computed, from a set of point matches across two omnidirectional views, using a method that relies on virtual planes independently of the structure of the scene. The method takes advantage of the planar motion constraint of the platform and computes virtual vertical planes from the scene. The family of homographies is also constrained to be embedded in a three-dimensional linear subspace to improve numerical consistency. Simulations and real experiments are provided to evaluate our approach.     

同形映射下的曲面场景的匹配算法的性能分析

针对移动机器人在曲面场景的匹配问题中,同形约束用于解决极约束产生的匹配模糊性问题和发现新的匹配点.实际上是平面块对曲面进行近似逼近的过程.逼近程度和逼近性能需要有指标进行定性和定量的衡量.故提出了两种性能评价指标:平均映射误差和平均映射匹配对.仿真实验结果的分析证明,场景深度变化或者场景距离摄像机的距离变化,对立体匹配算法性能本身不受影响,但映射和建立匹配关系时所需要的同形矩阵的数量不同.而且,随着特征点的稠密度提高,曲面场景的稳定性降低.可随着迭代过程的进行,算法本身结果还趋于稳定.     

From lines to epipoles through planes in two views

This paper addresses the computation of the fundamental matrix between two views, when camera motion and 3D structure are unknown, but planar surfaces can be assumed. We use line features which are automatically matched in two steps. Firstly, with image based parameters, a set of matches are obtained to secondly compute homographies, which allows to reject wrong ones, and to grow good matches in a final stage. The inclusion of projective transformations gives much better results to match features with short computing overload. When two or more planes are observed, different homographies can be computed, segmenting simultaneously the corresponding planar surfaces. These can be used to obtain the fundamental matrix, which gives constraints for the whole scene. The results show that the global process is robust enough, turning out stable and useful to obtain matches and epipolar geometry from lines in man made environments.     

Estimation of the epipole using optical flow at antipodal points

We present algorithms for estimating the epipole or direction of translation of a moving camera. We use constraints arising from two points that are antipodal on the image sphere in order to decouple rotation from translation. One pair of antipodal points constrains the epipole to lie on a plane, and two such pairs will correspondingly give two planes. The intersection of these two planes is an estimate of the epipole. This means we require image motion measurements at two pairs of antipodal points to obtain an estimate. Two classes of algorithms are possible and we present two simple yet extremely robust algorithms representative of each class. These are shown to have comparable accuracy with the state of the art when tested in simulation under noise and with real image sequences.     

Mirror uncertainty and uniqueness conditions for determining shape and motion from orthographic projection

This paper presents new forms of necessary and sufficient conditions for determining shape and motion to within a mirror uncertainty from monocular orthographic projections of any number of point trajectories over any number of views. The new forms of conditions use image data only and can therefore be employed in any practical algorithms for shape and motion estimation. We prove that the mirror uncertainty for the three view problem also exists for a long sequence: if shapeS is a solution, so is its mirror imageS which is symmetric toS about the image plane. The necessary and sufficient conditions for determining the two sets of solutions are associated with the rank of the measurement matrixW.If the rank ofW is 3, then the original 3D scene points cannot be coplanar and the shape and motion can be determined to within a mirror uncertainty from the image data if and only if there are three distinct views. This condition is different from Ullman's theorem (which states thatthree distinct views of four noncoplanar points suffice to determine the shape and motion up to a reflection) in two aspects: (1) it is expressed in terms of image data; (2) it applies to a long image sequence in a homogeneous way.If the rank ofW is 2 and the image points in at least one view are not colinear in the image plane, then there are two possibilities: either the motion is around the optical axis, or the 3-D points all lie on the same plane. In the first case, the motion can be determined uniquely but the shape is not determined. In the second case, a necessary and sufficient condition is to be satisfied and at least 3 point trajectories over at least 3 distinct views are needed to determine the shape in each view to within a mirror uncertainty, and the number of motion solutions is equal to the combinatorial number of the possible positions of the plane in different views. The necessary and sufficient condition is associated with the rank of a matrixC: ifC has a rank of 1, the plane is undetermined; ifC has a rank of 2 (implying there are exactly 3 distinct views), then a necessary and sufficient condition, whose physical meaning is not completely clear, is to be satisfied to determine the plane to within 2 sets; ifC has a rank of 3 (implying there are 4 or more distinct views), then the plane can always be determined to within two sets.If the rank ofW is 2 or 1 and the image points in each view are colinear in the image plane, then the three dimensional motion problem reduces to a two dimensional motion problem. In this case, the uniqueness condition is associated with the rank of the reduced measurement matrix . If has a rank of 2, then the original 3D points cannot be colinear in the space and the shape and motion can be determined to within two sets if and only if three or more views are distinct. If has a rank of 1, there are two possibilities: if the rows of are identical, then either the original 3D points are not colinear and the motion is zero, or the points are colinear and possibly move between two mirror symmetric positions; if the rows of are not identical, then the motion is not determined.All proofs are constructive and thus define an algorithm for determining the uniqueness of solution as well as for estimating shape and motion from point trajectories.     

基于一维移动物体的双目装置自标定

给出了平移运动的一维物体所在平面的虚圆点图像及其对摄像机内参数的约束,和约束方程的数值求解方法,从而获得摄像机的内参数. 进一步通过恢复空间点在摄像机坐标系中的坐标,求解出双目摄像机之间的方位,即摄像机的外参数．对于一维物体的一般刚体运动,给出了把它转化为平移运动的方法．模拟实验和真实图像实验结果表明该方法具有较高的求解精度,同时也有一定的应用价值．     

1D camera geometry and its application to the self-calibration of circular motion sequences

This paper proposes a novel method for robustly recovering the camera geometry of an uncalibrated image sequence taken under circular motion. Under circular motion, all the camera centers lie on a circle and the mapping from the plane containing this circle to the horizon line observed in the image can be modelled as a 1D projection. A 2x2 homography is introduced in this paper to relate the projections of the camera centers in two 1D views. It is shown that the two imaged circular points of the motion plane and the rotation angle between the two views can be derived directly from such a homography. This way of recovering the imaged circular points and rotation angles is intrinsically a multiple view approach, as all the sequence geometry embedded in the epipoles is exploited in the estimation of the homography for each view pair. This results in a more robust method compared to those computing the rotation angles using adjacent views only. The proposed method has been applied to self-calibrate turntable sequences using either point features or silhouettes, and highly accurate results have been achieved.