The Effect of Noise on Camera Calibration Parameters

Camera calibration is the estimation of parameters (both intrinsic and extrinsic) associated with a camera being used for imaging. Given the world coordinates of a number of precisely placed points in a 3D space, camera calibration requires the measurement of the 2D projection of those scene points on the image plane. While the coordinates of the points in space can be known precisely, the image coordinates that are determined from the digital image are often inaccurate and hence noisy. In this paper, we look at the statistics of the behavior of the camera calibration parameters, which are important for stereo matching, when the image plane measurements are corrupted by noise. We derive analytically the behavior of the camera calibration matrix under noisy conditions and further show that the elements of the camera calibration matrix have a Gaussian distribution if the noise introduced into the measurement system is Gaussian. Under certain approximations we derive relationships between the camera calibration parameters and the noisy camera calibration matrix and compare it with Monte Carlo simulations.     

Implicit and explicit camera calibration: theory and experiments

By implicit camera calibration, we mean the process of calibrating a camera without explicitly computing its physical parameters. Implicit calibration can be used for both three-dimensional (3-D) measurement and generation of image coordinates. In this paper, we present a new implicit model based on the generalized projective mappings between the image plane and two calibration planes. The back-projection and projection processes are modelled separately to ease the computation of distorted image coordinates from known world points. A set of constraints of perspectivity is derived to relate the transformation parameters of the two calibration planes. Under the assumption of the radial distortion model, we present a computationally efficient method for explicitly correcting the distortion of image coordinates in frame buffer without involving the computation of camera position and orientation. By combining with any linear calibration techniques, this method makes explicit the camera physical parameters. Extensive experimental comparison of our methods with the classic photogrammetric method and Tsai's (1986) method in the aspects of 3-D measurement (both absolute and relative errors), the prediction of image coordinates, and the effect of the number of calibration points, is made using real images from 15 different depth values     

基于投影仪和摄像机的结构光视觉标定方法

为实现基于投影仪和摄像机的结构光视觉系统连续扫描,需要计算投影仪投影的任意光平面与摄像机图像平面的空间位置关系,进而需要求取摄像机光心与投影仪光心之间的相对位置关系。求取摄像机的内参数,在标定板上选取四个角点作为特征点并利用摄像机内参数求取该四个特征点的外参数,从而知道四个特征点在摄像机坐标系中的坐标。利用投影仪自身参数求解特征点在投影仪坐标系中的坐标,从而计算出摄像机光心与投影仪光心之间的相对位置关系,实现结构光视觉标定。利用标定后的视觉系统,对标定板上的角点距离进行测量,最大相对误差为0.277%,表明该标定算法可以应用于基于投影仪和摄像机的结构光视觉系统。     

一种简单的基于共面的摄像机参数标定方法

共面摄像机标定就是采用平面式模板来确定摄像机内外参数的过程,在此过程中图像像素和二维特征点是已知的。对于共面标定提出了一种简单及有效的方法去标定摄像机参数。即使用帧缓存中的计算机阵列图像直接来标定参数。首先采用预标定的方法标定出图像中心位置,然后根据帧存图像坐标和世界坐标之间的对应关系使用正交矩阵的约束条件来求解,在此算法中假设尺度因子为1,并且不考虑透镜畸变。所提出的算法用数字仿真图像及真实的图像检验。结果显示,所提出的算法具有较好的精度,是一种简单有效的标定方法。     

Segment Based Camera Calibration

The basic idea of calibrating a camera system in previous approaches is to determine camera parmeters by using a set of known 3D points as calibration reference.In this paper,we present a method of camera calibration in whih camera parameters are determined by a set of 3D lines.A set of constraints is derived on camea parameters in terms of perspective line mapping.Form these constraints,the same perspective transformation matrix as that for point mapping can be computed linearly.The minimum number of calibration lines is 6.This result generalizes that of Liu,Huang and Faugeras^[12] for camera location determination in which at least 8 line correspondences are required for linear computation of camera location.Since line segments in an image can be located easily and more accurately than points,the use of lines as calibration reference tends to ease the computation in inage preprocessing and to improve calibration accuracy.Experimental results on the calibration along with stereo reconstruction are reported.     

直线运动摄像机在线动态标定

为实现AS-R智能机器人在运动情况下摄像机在线动态标定,提出一种新的基于粒子滤波的直线运动摄像机标定方法。用状态空间方法描述直线运动摄像机模型,把摄像机内参数和位置运动参数作为状态量,特征点图像坐标作为观测量,根据粒子滤波算法求得摄像机内参数和位置运动参数的最优估计,并用双线程实现整个标定过程。AS-R机器人在直线运动情况下的摄像机在线动态标定实验结果表明：该算法是合理可行的,并且具有很高的标定精度和良好的鲁棒性。该方法适用于各种类型的系统噪声。     

Self-Calibration of Turntable Sequences from Silhouettes

This paper addresses the problem of recovering both the intrinsic and extrinsic parameters of a camera from the silhouettes of an object in a turntable sequence. Previous silhouette-based approaches have exploited correspondences induced by epipolar tangents to estimate the image invariants under turntable motion and achieved a weak calibration of the cameras. It is known that the fundamental matrix relating any two views in a turntable sequence can be expressed explicitly in terms of the image invariants, the rotation angle, and a fixed scalar. It will be shown that the imaged circular points for the turntable plane can also be formulated in terms of the same image invariants and fixed scalar. This allows the imaged circular points to be recovered directly from the estimated image invariants, and provide constraints for the estimation of the imaged absolute conic. The camera calibration matrix can thus be recovered. A robust method for estimating the fixed scalar from image triplets is introduced, and a method for recovering the rotation angles using the estimated imaged circular points and epipoles is presented. Using the estimated camera intrinsics and extrinsics, a Euclidean reconstruction can be obtained. Experimental results on real data sequences are presented, which demonstrate the high precision achieved by the proposed method.     

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．     

Projection matrix by orthogonal vanishing points

Calculation of camera projection matrix, also called camera calibration, is an essential task in many computer vision and 3D data processing applications. Calculation of projection matrix using vanishing points and vanishing lines is well suited in the literature; where the intersection of parallel lines (in 3D Euclidean space) when projected on the camera image plane (by a perspective transformation) is called vanishing point and the intersection of two vanishing points (in the image plane) is called vanishing line. The aim of this paper is to propose a new formulation for easily computing the projection matrix based on three orthogonal vanishing points. It can also be used to calculate the intrinsic and extrinsic camera parameters. The proposed method reaches to a closed-form solution by considering only two feasible constraints of zero-skewness in the internal camera matrix and having two corresponding points between the world and the image. A nonlinear optimization procedure is proposed to enhance the computed camera parameters, especially when the measurement error of input parameters or the skew factor are not negligible. The proposed method has been run on real and synthetic data for more precise evaluations. The provided experimental results demonstrate the superiority of the proposed method.     

线结构光视觉传感器直接标定方法

分析了传感器测量模型,推导了摄像机畸变模型下的像坐标与像素坐标变换关系,在此基础上,提出了一种基于二元五次多项式拟合的线结构光视觉传感器物像标定模型。在数控平台上,利用锯齿形标定块获取一组标定点,应用最小二乘方法实现参数标定。标定与测量实验表明：本方法实现简单,精度较高,满足测量要求。     

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．     

基于场景几何约束未标定两视图的三维模型重建

提出了一种从两幅未标定图象重建场景三维模型的方法 .这种方法充分利用了人造结构场景中大量存在的平行性和正交性几何约束 ,即利用每幅视图中三组互相垂直的平行线 ,计算出 3个影灭点 ,从而对每幅视图进行标定 .对两幅未标定图象 ,从基本矩阵只能得到射影重构 ,如果每幅图象都已标定 ,则可将基本矩阵转化为本质矩阵 .三维重构过程有两个步骤 :先是恢复相机的位置和运动 ;后是用三角测量法计算出点的三维坐标 .对多平面组成的场景进行三维重构实验 ,所得三维模型产生新的视点图象 ,与所观察的场景一致 ,重构的两个平面夹角与实际值相近 ,实验结果表明 ,该算法是行之有效的     

简化UKF算法在摄像机标定中的应用

提出一种基于简化无迹卡尔曼滤波（UKF）算法的摄像机标定方法。将平面靶标图像上的不同特征点坐标视为同一个特征点在不同时刻的运动坐标。为避免欧拉角描述法带来的奇异问题,用单位四元数描述世界坐标系和摄像机坐标系之间的变换关系,选取摄像机内外参数作为系统状态变量。结合实际应用背景,简化标准UKF算法,将其用于摄像机参数估计,在保证标定精度的前提下降低运算复杂度。仿真结果表明了该方法的有效性。     

单目平行线约束下的空间点坐标恢复

提出一种从单目像机拍摄的二维投影图像中恢复空间点三维坐标的简化算法.假定摄像机与空间中的一对平行线的位置关系已知,同时摄像机坐标系下平行线所在平面中的任意空间点在像平面中的投影也已知,通过射影几何可得图像坐标到目标点三维空间坐标的转换系数,即坐标转换因数,从而估计出三维空间点的坐标.实验中通过对人体行走轨迹进行估计来测试算法性能,结果表明算法简单有效,计算开销小.     

利用平面约束中的反对称性进行相机自定标

回顾了2幅图像中的平面约束，以及一个图像对的基础矩阵和同形矩阵的乘积是一个反对称矩阵的性质，并通过证明展示了这种反对称性质和平面约束之问的关系。给定两幅图像中的一系列对应点，利用反对称性质提出了一种改进的相机自定标算法，将利用平面约束进行相机自定标过程中求取同形矩阵(homography matrix)的问题转化成了方程组约束条件下的二次规划问题，通过解决给定的二次规划问题求解同形矩阵，提高了算法的鲁棒性，然后利用平面约束求解内参数，最后通过本质矩阵(essential matrix)和基础矩阵(fundarnental matrix)之间的关系以及旋转矩阵的性质求解相机外参数。实验结果表明，算法在稳定性方面有了较大程度的提高。     

Decomposition of transformation matrices for robot vision

The relationship between the three-dimensional coordinates of a point and the corresponding two-dimensional coordinates of its image, as seen by a camera, can be expressed in terms of a 3 by 4 matrix using the homogeneous coordinate system. This matrix is known more generally as the transformation matrix and it is well known that such a matrix can be determined experimentally by measuring the image coordinates of six or more points in space, whose three-dimensional coordinates are known.Such a transformation matrix can be derived analytically from knowledge of the camera position, orientation, focal length and scaling and translation parameters in the image plane. However, the inverse problem of computing the camera location and orientation from the transformation matrix involves solution of simultaneous nonlinear equations in several variables and is considered difficult.In this paper we present a new and simple analytical technique that accomplishes this inversion rather easily. This technique is quite powerful and has applications to a wide variety of problems in Computer Vision for both static and dynamic scenes. The technique has been implemented as a C program running under Unix and works well on real data.     

未标定单幅结构场景图像的三维重构

提出了一种单视三维重构方法，该方法是利用用户提供图像点及其对应的三维点之间几何信息。由于结构场景是由大量平面构成的，存在大量的平行性、正交性约束，因此该方法主要应用于结构场景的三维重构。首先，相机定标和计算每个平面的度量信息，即先基于3组互相垂直方向的影灭点，对方形像素相机标定，再利用影灭线和圆环点像，对每个平面度量校正；然后考虑每个校正平面的尺度因子和非正交平面间的相对面向，从而将所有校正后的平面缝合起来。采用真实图像进行实验，实验结果表明，该方法简单易用。     

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。     

Initialization method for self-calibration using 2-views

Recently, 3D structure recovery through self-calibration of camera has been actively researched. Traditional calibration algorithm requires known 3D coordinates of the control points while self-calibration only requires the corresponding points of images, thus it has more flexibility in real application. In general, self-calibration algorithm results in the nonlinear optimization problem using constraints from the intrinsic parameters of the camera. Thus, it requires initial value for the nonlinear minimization. Traditional approaches get the initial values assuming they have the same intrinsic parameters while they are dealing with the situation where the intrinsic parameters of the camera may change. In this paper, we propose new initialization method using the minimum 2 images. Proposed method is based on the assumption that the least violation of the camera's intrinsic parameter gives more stable initial value. Synthetic and real experiment shows this result.     

Creating Architectural Models from Images

We present methods for creating 3D graphical models of scenes from a limited numbers of images, i.e. one or two, in situations where no scene co-ordinate measurements are available. The methods employ constraints available from geometric relationships that are common in architectural scenes – such as parallelism and orthogonality – together with constraints available from the camera. In particular, by using the circular points of a plane simple, linear algorithms are given for computing plane rectification, plane orientation and camera calibration from a single image. Examples of image based 3D modelling are given for both single images and image pairs.