Camera self-calibration from bivariate polynomials derived from Kruppa's equations

This paper presents a new set of equations for the self-calibration of a moving camera with constant intrinsic parameters. Unlike most existing methods that require solving equations in three or more unknowns, the proposed equations are only bivariate. In particular, we show that the three scale factors appearing in the Kruppa's equations, that are due to a triplet of images, are not independent but rather closely related. This relationship is used to derive sextic bivariate polynomial equations and allow the recovery of the unknown scale factors using a homotopy continuation method. Once the scale factors are calculated, an estimate of Kruppa's coefficients can be linearly retrieved and then refined through a nonlinear least-squares optimization procedure. The results of our experiments conducted on simulated data as well as the three-dimensional structure reconstruction using real images are also presented in the paper.     

A theory of self-calibration of a moving camera

There is a close connection between the calibration of a single camera and the epipolar transformation obtained when the camera undergoes a displacement. The epipolar transformation imposes two algebraic constraints on the camera calibration. If two epipolar transformations, arising from different camera displacements, are available then the compatible camera calibrations are parameterized by an algebraic curve of genus four. The curve can be represented either by a space curve of degree seven contained in the intersection of two cubic surfaces, or by a curve of degree six in the dual of the image plane. The curve in the dual plane has one singular point of order three and three singular points of order two.If three epipolar transformations are available, then two curves of degree six can be obtained in the dual plane such that one of the real intersections of the two yields the correct camera calibration. The two curves have a common singular point of order three.Experimental results are given to demonstrate the feasibility of camera calibration based on the epipolar transformation. The real intersections of the two dual curves are found by locating the zeros of a function defined on the interval [0, 2]. The intersection yielding the correct camera calibration is picked out by referring back to the three epipolar transformations.     

A new method for camera stratified self-calibration under circular motion

We consider the stratified self-calibration (affine and metric reconstruction) problem from images acquired with a camera with unchanging internal parameters undergoing circular motion. The general stratified method (modulus constraints) is known to fail with this motion. In this paper we give a novel constraint on the plane at infinity in projective reconstruction for circular motion, the constant inter-frame motion constraint on the plane at infinity between every two adjacent views and a fixed view of the motion sequences, by making use of the facts that in many commercial systems rotation angles are constant. An initial solution can be obtained by using the first three views of the sequence, and Stratified Iterative Particle Swarm Optimization (SIPSO) is proposed to get an accurate and robust solution when more views are at hand. Instead of using the traditional optimization algorithm as the last step to obtain an accurate solution, in this paper, the whole motion sequence information is exploited before computing the camera calibration matrix, this results in a more accurate and robust solution. Once the plane at infinity is identified, the calibration matrices of the camera and a metric reconstruction can be readily obtained. Experiments on both synthetic and real image sequence are given, showing the accuracy and robustness of the new algorithm.     

Kruppa's equations derived from the fundamental matrix

The purpose of this paper is to give a specific form for Kruppa's equations in terms of the fundamental matrix. Kruppa's equations can be written explicitly in terms of the singular value decomposition (SVD) of the fundamental matrix     

Bayesian self-calibration of a moving camera

In this paper, a Bayesian self-calibration approach using sequential importance sampling (SIS) is proposed. Given a set of feature correspondences tracked through an image sequence, the joint posterior distributions of both camera extrinsic and intrinsic parameters as well as the scene structure are approximated by a set of samples and their corresponding weights. The critical motion sequences are explicitly considered in the design of the algorithm. The probability of the existence of the critical motion sequence is inferred from the sample and weight set obtained from the SIS procedure. No initial guess for the calibration parameters is required. The proposed approach has been extensively tested on both synthetic and real image sequences and satisfactory performance has been observed.     

基于序列图像的摄像机自标定方法

提出了一种新的结合摄影测量和计算机视觉相关理论的摄像机自标定方法。首先通过序列图像的匹配点对,利用计算机视觉理论中的8点法求得摄像机基础矩阵F,通过矩阵F利用Kruppa方程求得矩阵C,对矩阵C进行Cholesky分解得到摄像机的内参数矩阵K,然后将求出的内参数作为初始值,利用摄影测量理论进行相对定向和绝对定向,最小二乘前方交会计算得到匹配点对的三维空间坐标,最后由匹配点对的三维空间坐标及其图像坐标,采用三维直接线性变换和光束法平差方法解算出摄像机内、外参数及畸变系数。该方法不依赖于特定的场景几何约束条件,只要序列图像之间有匹配点对,就可以进行自标定工作,具有广泛的适用性。模拟数据和真实图像的实验结果表明：该方法计算过程简单,标定精度高,是一种值得借鉴的摄像机自标定方法。     

Plane-based camera self-calibration by metric rectification of images

Plane-based self-calibration aims at the computation of camera intrinsic parameters from homographies relating multiple views of the same unknown planar scene. This paper proposes a straightforward geometric statement of plane-based self-calibration, through the concept of metric rectification of images. A set of constraints is derived from a decomposition of metric rectification in terms of intrinsic parameters and planar scene orientation. These constraints are then solved using an optimization framework based on the minimization of a geometrically motivated cost function. The link with previous approaches is demonstrated and our method appears to be theoretically equivalent but conceptually simpler. Moreover, a solution dealing with radial distortion is introduced. Experimentally, the method is compared with plane-based calibration and very satisfactory results are obtained. Markerless self-calibration is demonstrated using an intensity-based estimation of the inter-image homographies.     

基于张量投票的摄像机自标定方法研究

针对传统的基于Kruppa方程摄像机自标定算法的欠鲁棒性,首次提出将鲁棒的张量投票算法用于摄像机自标定方法中。利用基于尺度不变的SIFT算法查找并匹配出每对图像的特征点,其中待匹配图像由摄像机对同一场景从三个不同角度位置拍摄,对图像张量投票后按棒张量特征值降序排序,由此筛选得到具有鲁棒性边缘特征的前八对特征点,利用八点算法求解相应的基础矩阵和极点,根据Kruppa方程和三维重建（SFM）算法求得摄像机参数矩阵。实验结果证明,该方法具有较高标定精度,并通过加入高斯噪声的仿真实验证明该算法是一种鲁棒的摄像机自标定方法。     

An improved algorithm for two-image camera self-calibration and Euclidean structure recovery using absolute quadric

We present an improved algorithm for two-image camera self-calibration and Euclidean structure recovery, where the effective focal lengths of both cameras are assumed to be the only unknown intrinsic parameters. By using the absolute quadric, it is shown that the effective focal lengths can be computed linearly from two perspective images without imposing scene or motion constraints. Moreover, a quadratic equation derived from the absolute quadric is proposed for solving the parameters of the plane at infinity from two images, which upgrades a projective reconstruction to a Euclidean reconstruction.     

Minimal vision system for linear camera self-calibration and 3D reconstruction using translation platform

Multimedia Tools and Applications - Camera calibration and 3D reconstruction are two crucial steps in computer vision. With the progress of robot and autonomous rover systems, zooming cameras are...     

Self-calibration of a 1D projective camera and its application to the self-calibration of a 2D projective camera

We introduce the concept of self-calibration of a 1D projective camera from point correspondences, and describe a method for uniquely determining the two internal parameters of a 1D camera, based on the trifocal tensor of three 1D images. The method requires the estimation of the trifocal tensor which can be achieved linearly with no approximation unlike the trifocal tensor of 2D images and solving for the roots of a cubic polynomial in one variable. Interestingly enough, we prove that a 2D camera undergoing planar motion reduces to a 1D camera. From this observation, we deduce a new method for self-calibrating a 2D camera using planar motions. Both the self-calibration method for a 1D camera and its applications for 2D camera calibration are demonstrated on real image sequences.     

一种用于14 bit SAR ADC的数字自校准算法

为了实现高精度14bit的逐次逼近型SAR(Successive Approximation)模数转换器ADC,提出一种数字自校准算法。该算法通过切换两种电容阵列的工作状态,得到电容之间的失配误差,并在ADC正常工作时,将得到的电容误差加载到电路中达到在转换过程消除失配的目的。最后对一个失配误差为0.5%的14bit的SARADC系统模型进行参数仿真,结果验证了本数字校准算法的正确性和有效性。     

In-field self-calibration of robotic manipulator using stereo camera: application to Humanitarian Demining Robot

A robotic manipulator using a stereo camera mounted on one of its links requires a precise kinematic transformation calibration between the manipulator and the camera coordinate frames, the so-called hand–eye calibration, to achieve high-accuracy end-effector positioning. This paper introduces a new method that performs simultaneous joint angle and hand–eye calibration, based on a traditional method that uses a sequence of pure rotations of the manipulator links. The new method considers an additional joint angle constraint, which improves the calibration accuracy when the circular arc that can be measured by the stereo camera is very limited. Experimental results using a manipulator developed for humanitarian demining demonstrate that relative errors between the end effector and the external points mapped by the stereo camera are greatly reduced compared to traditional methods.     

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.     

一种基于TOF相机与CCD相机的联合标定算法研究

针对基于Time-of-Flight(TOF)相机的彩色目标三维重建需标定CCD相机与TOF相机联合系统的几何参数,在研究现有的基于彩色图像和TOF深度图像标定算法的基础上,提出了一种基于平面棋盘模板的标定方法。拍摄了固定在平面标定模板上的彩色棋盘图案在不同角度下的彩色图像和振幅图像,改进了Harris角点提取,根据棋盘格上角点与虚拟像点的共轭关系,建立了相机标定系统模型,利用Levenberg-Marquardt算法求解,进行了标定实验。获取了TOF与CCD相机内参数,并利用像平面之间的位姿关系估计两相机坐标系的相对姿态,最后进行联合优化,获取了相机之间的旋转矩阵与平移向量。实验结果表明,提出的算法优化了求解过程,提高了标定效率,能够获得较高的精度。     

A self-calibration algorithm for cross array in the presence of mutual coupling

In this paper,a novel blind mutual coupling self-calibration algorithm especially tailored to cross array is proposed.This method compensates for the mutual coupling not only between sensors in each subarray but also between two subarrays.Exploiting the property of coupling matrix,the proposed method requires neither the calibration sources in known locations nor the multidimensional nonlinear search.This method can jointly estimate the DOAs of the incoming signals and the unknown mutual coupling coefficien...     

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.     

A new method for solving interval and fuzzy equations: Linear case

A new approach to the solution of interval and fuzzy equations based on the generalized procedure of interval extension called “interval extended zero” method is proposed. The central for the proposed approach is the treatment of the interval zero as an interval centered around zero. It is shown that such proposition is not of heuristic nature, but is the direct consequence of the interval subtraction operation. Some methodological problems concerned with this definition of interval zero are discussed. It is shown that the resulting solution of interval linear equations based on the proposed method may be naturally treated as a fuzzy number. An important advantage of a new method is that it substantially decreases the excess width effect. On the other hand, we show that it can be used as a reliable practical tool for solving linear interval and fuzzy equations as well as the systems of them. The features of the method are illustrated by the example of the solution of the well known Leontief input-output problem in the interval setting.     

A flexible new technique for camera calibration

We propose a flexible technique to easily calibrate a camera. It only requires the camera to observe a planar pattern shown at a few (at least two) different orientations. Either the camera or the planar pattern can be freely moved. The motion need not be known. Radial lens distortion is modeled. The proposed procedure consists of a closed-form solution, followed by a nonlinear refinement based on the maximum likelihood criterion. Both computer simulation and real data have been used to test the proposed technique and very good results have been obtained. Compared with classical techniques which use expensive equipment such as two or three orthogonal planes, the proposed technique is easy to use and flexible. It advances 3D computer vision one more step from laboratory environments to real world use.     

一种全局指数收敛的摄像机内参数观测器

针对摄像机标定问题,本文从控制理论角度出发设计了一种具有指数收敛特性的摄像机内参数观测器.当摄像机随移动机器人等运动平台一起旋转时,该观测器使用其运动信息和实时拍摄得到的特征点图像信息,可以实现对摄像机内参数的在线估计.具体而言,论文分析了云台摄像机的运动约束,建立了特征点图像坐标变化的运动学模型,随后基于4个或4个以上特征点（其中任意3点不共线）,构造了一种内参数非线性观测器,并通过理论分析证明了其状态估计指数收敛于摄像机的相应内参数,仿真结果验证了这种非线性观测器的良好特性.