1、A compact algorithm for rectification of stereo pairs中文版翻译Machine Vision and Applications (2000) 12: 1622 机器视觉与应用（2000）12:16-22Machine Vision and Applications Springer-Verlag 2000机器视觉与应用施普林格出版社2000Andrea Fusiello1, Emanuele Trucco2, Alessandro Verri31 Dipartimento Scientifico e Tecnologico, Universi
2、ta di Verona, Ca Vignal 2, Strada Le Grazie, 37134 Verona, Italy; e-mail: fusiellosci.univr.it 2 Heriot-Watt University, Department of Computing and Electrical Engineering, Edinburgh, UK3 INFM, Dipartimento di Informatica e Scienze dellInformazione, Universita di Genova, Genova, ItalyReceived: 25 Fe
3、bruary 1999 / Accepted: 2 March 2000收稿日期：1999年2月25日/接受日期：2000年3月2日Abstract. We present a linear rectification algorithm for general, unconstrained stereo rigs. The algorithm takes the two perspective projection matrices of the original cameras,and computes a pair of rectifying projection matrices. I
4、t is compact (22-line MATLAB code) and easily reproducible.We report tests proving the correct behavior of our method,as well as the negligible decrease of the accuracy of 3D reconstruction performed from the rectified images directly.摘要：我们在本篇文章中阐述一个用于通用的不加约束的立体视觉设备的线性修正算法。这个算法输入原始双目视觉 相机投影矩阵图像，计算出一
5、对修正投影矩阵。它是压缩的（22行的MATLAB 代码）并且易于复现的。我们的报告中包含证实这个方法正确工作的测试，和通过直接修正后图像重构3D视觉造成的微小衰减的测试。Key words: Rectification Stereo Epipolar geometry关键字：校正 立体 极线几何 （译者注：极线几何校正）1 Introduction and motivations1简介和研究动机Given a pair of stereo images, rectification determines a transformation of each image plane such tha
6、t pairs of conjugate epipolar lines become collinear and parallel to one of the image axes (usually the horizontal one). The rectified images can be thought of as acquired by a new stereo rig, obtained by rotating the original cameras. The important advantage of rectification is that computing stere
7、o correspondences (Dhond and Aggarwal, 1989) is made simpler,because search is done along the horizontal lines of the rectified images.对于一对双目视觉图像，校正被定义为每张图像平面的旋转变换以至于它们的共轭极线成为共线的并且平行于图像的某一个轴（通常是水平轴）。修正后的图像可以看成由原始摄像机旋转后新的双目摄像机拍摄得到的。修正的重要优势在于立体视觉的相关性（Dhond 和 Aggarwal, 1989）计算更为简单，因为搜索只在修正图像的水平线上进行。We
8、assume that the stereo rig is calibrated, i.e., the cameras internal parameters, mutual position and orientation are known. This assumption is not strictly necessary, but leads to a simpler technique. On the other hand, when reconstructing 3D shape of objects from dense stereo, calibration is mandat
9、ory in practice, and can be achieved in many situations and by several algorithms (Caprile and Torre, 1990;Robert, 1996)我们假设立体摄像机是标准的，意即相机的内部参数、相互之间的位置和旋转角度是已知的。这个假定不是严格要求的，但是可以使技术实现更为简单。另一方面，实际当从密集立体图像中重构3D物体形状时，标准化的摄像机是强制的，而且可以通过在很多种情况中使用几种算法（Caprile 和 Torre, 1990;Robert, 1996）获得这些参数。Correspondenc
10、e to: A. FusielloRectification is a classical problem of stereo vision; however, few methods are available in the computer vision literature, to our knowledge. Ayache and Lustman (1991) introduced a rectification algorithm, in which a matrix satisfying a number of constraints is handcrafted. The dis
11、tinction between necessary and arbitrary constraints is unclear. Some authors report rectification under restrictive assumptions; for instance, Papadimitriou and Dennis (1996) assume a very restrictive geometry (parallel vertical axes of the camera reference frames). Recently, Hartley and Gupta (199
12、3), Robert et al. (1997) and Hartley (1999) have introduced algorithms which perform rectification given a weakly calibrated stereo rig, i.e., a rig for which only points correspondences between images are given.修正是立体视觉中的经典问题，然而我们仅知道很少的几种实现方法。. Ayache and Lustman (1991)介绍了一种手工计算矩阵满足一系列约束的算法。必要的和随意的约
13、束的区别并不明显。有一些学者提出约束假设下的修正方案，例如Papadimitriou 和Dennis (1996)提出一种十分约束的几何（平行于相机参考平面的垂直轴）。最近Hartley 和 Gupta (1993), Robert et al. (1997) 和 Hartley (1999)介绍了一种弱标准化立体摄像机即只给出一些点相关的图像的相机的修正方案。Latest work, published after the preparation of this manuscript includes Loop and Zhang (1999), Isgro and Trucco (1999
14、) and Pollefeys et al. (1999). Some of this work also concentrates on the issue of minimizing the rectified image distortion. We do not address this problem, partially because distortion is less severe than in the weakly calibrated case.最近的修正相关的工作，在整理了Loop 和 Zhang (1999), Isgro 和 Trucco (1999) 和Poll
15、efeys et al. (1999)的一些手稿后得以出版。这些工作中也有些关注最小化修正图像失真的问题。我们并不致力于这些问题，部分原因是标准设备失真没有弱约束中那么重要。This paper presents a novel algorithm rectifying a calibrated stereo rig of unconstrained geometry and mounting general cameras. Our work improves and extends Ayache and Lustman (1991). We obtain basically the sam
16、e results,but in a more compact and clear way. The algorithm is simple and detailed. Moreover, given the shortage of easily reproducible, easily accessible and clearly stated algorithms,we have made the code available on the Web.本篇论文陈述了一种通用的修正算法用来处理校正过的立体设备得到的未加约束的几何和校准通用相机。我们的工作提升和扩展了Ayache 和 Lustm
17、an (1991)的工作。我们基本上得到了相同的结果，但是使用了更为压缩和清晰的方法。给出的算法是简单和细致的。更为重要的是解决了原来算法缺乏复现性，易获取和清晰地陈述，代码可以在互联网上获取到。2 Camera model and epipolar geometry2 相机模型和极几何This section recalls briefly the mathematical background on perspective projections necessary for our purposes. For more details see Faugeras (1993).这一段为我们的
18、目的简单地回顾了透视几何中相关的数学背景知识，可以通过Faugeras (1993)的论文得到更多的细节。2.1 Camera model2.1相机模型A pinhole camera is modeled by its optical center C and its retinal plane (or image plane) R. A 3D point W is projected into an image point M given by the intersection of R with the line containing C and W. The line containi
19、ng C and orthogonal to R is called the optical axis and its intersection with R is the principal point. The distance between C and R is the focal length.一个针孔相机可以由它的光学中心C和它的视网膜平面（或成像平面）R进行建模。一个3D点W成像在点M，点M由一条通过点C和点W的直线与R平面相交得到。包含点C并且垂直于R的直线叫做光轴，光轴与R平面的交点称为投影中心。点C与像平面R之间的距离称为焦距。Let w = x y z T be the
20、coordinates of W in the world reference frame (fixed arbitrarily) and m = u vT the coordinates of M in the image plane (pixels). The mapping from 3D coordinates to 2D coordinates is the perspective projection, which is represented by a linear transformation in homogeneous coordinates. Let = u v 1 an
21、d = x y z 1 be the homogeneous coordinates of M and W, respectively; then, the perspective transformation is given by the matrix :w = x y z T是点W在世界参考系（任意固定的）中的坐标且m = u vT 是点M在像平面（像素）中的坐标。从3D坐标向2D坐标的映射是一个可以用同一坐标系下线性变换表示的透视过程。假设= u v 1 和= x y z 1是点 M和点 W各自相同的坐标，那么，透视变换可以由矩阵给出： , (1)where means equal u
22、p to a scale factor. The camera is therefore modeled by its perspective projection matrix(henceforth PPM), which can be decomposed, using the QR factorization, into the product：这里意味着等于的情况取决于比例因子。因此相机可以由透视矩阵表示，而矩阵则可以通过QR因式分解为下面的形式：= AR | t. (2)The matrix A depends on the intrinsic parameters only, an
23、d has the following form:矩阵A由固有参数决定，并有下面的形式： , (3)where =, are the focal lengths in horizontal and vertical pixels, respectively (f is the focal length in millimeters, and are the effective number of pixels per millimeter along the u and v axes), are the coordinates of the principal point, given by
24、the intersection of the optical axis with the retinal plane, and is the skew factor that models non-orthogonal u v axes.这里=,分别为水平方向和垂直方向上的焦距（f是用毫米表示的焦距，相关系数和分别为 u 和 v 轴上每毫米像素数），是中心点的坐标，中心点即是光轴与像平面的交点，是畸变因子用以表征u v轴的非正交度。The camera position and orientation (extrinsic parameters), are encoded by the 3
25、3 rotation matrix R and the translation vector t, representing the rigid transformation that brings the camera reference frame onto the world reference frame.Let us write the PPM as：相机位置和方向（外部参数）通过3 3旋转矩阵R和转化向量t来表征的，表示把相机参考系转换到世界参考系的刚性变换。我们把PPM定义为： . (4)In Cartesian coordinates, the projection (Eq.
26、1) writes在笛卡尔坐标系中，这个映射形式如下： (5)The focal plane is the plane parallel to the retinal plane that contains the optical center C. The coordinates c of C are given by焦平面是平行于像平面并且包含光学中心C，点C的坐标c形式如下：. (6)Therefore, can be written:因此，可以被写为： = Q| QC. (7)The optical ray associated to an image point M is the l
27、ine M C, i.e., the set of 3D points. In parametricform:与点M相关的光线是直线MC即一系列满足3D点集合。参数形式如下：, R . (8)Fig. 1. Epipolar geometry. The epipole of the first camera E is the projection of the optical center C2 of the second camera (and vice versa)图1 极线几何 第一相机E的核点事第二相机光学中心C2的映射（以此类推）2.2 Epipolar geometry2.2 极线
28、几何Let us consider a stereo rig composed by two pinhole cameras (Fig. 1). Let C1 and C2 be the optical centers of the left and right cameras, respectively. A 3D point W is projected onto both image planes, to points M1 and M2, which constitute a conjugate pair. Given a point M1 in the left image plan
29、e, its conjugate point in the right image is constrained to lie on a line called the epipolar line (of M1). Since M1 may be the projection of an arbitrary point on its optical ray, the epipolar line is the projection through C2 of the opticalray of M1. All the epipolar lines in one image plane pass
30、through a common point (E1 and E2, respectively) called the epipole, which is the projection of the optical center of the other camera. 考虑到由两个相机构成的立体仪器（图1）。C1 和 C2分别是是左右两个相机的光学中心。一个3D点W被投影到两个像平面上构成一对共轭点的M1 和 M2。左图像平面中任意点M1在右图像平面中的共轭点地位置被限制在一条被称为极线（M1的）的直线上。因为M1可以是任意一点在光轴上的投影，极线则是点M1光线过C2的投影。某一像平面中的所
31、有极线都过一个共同点（分别为E1 和 E2），它们是另一个相机光学中心的投影。When C1 is in the focal plane of the right camera, the right epipole is at infinity, and the epipolar lines form a bundle of parallel lines in the right image. A very special case is when both epipoles are at infinity, that happens when the line C1C2 (the basel
32、ine) is contained in both focal planes, i.e., the retinal planes are parallel to the baseline. Epipolar lines, then, form a bundle of parallel lines in both images. Any pair of images can be transformed so that epipolar lines are parallel and horizontal in each image. This procedure is called rectification.当C1在右相机的焦平面上时，右核点就在无穷处，一系列相平行的右极线在右图像中，一种非常特殊的情况是当线C1C2（基线）包含在两个相机的焦平面上时，即像平面平行于基线时，两个核点都在无穷远处。极线都是一
