Generic self-calibration of central cameras

We consider the self-calibration problem for a generic imaging model that assigns projection rays to pixels without a parametric mapping. We consider the central variant of this model, which encompasses all camera models with a single effective viewpoint. Self-calibration refers to calibrating a camera’s projection rays, purely from matches between images, i.e. without knowledge about the scene such as using a calibration grid. In order to do this we consider specific camera motions, concretely, pure translations and rotations, although without the knowledge of rotation and translation parameters (rotation angles, axis of rotation, translation vector). Knowledge of the type of motion, together with image matches, gives geometric constraints on the projection rays. We show for example that with translational motions alone, self-calibration can already be performed, but only up to an affine transformation of the set of projection rays. We then propose algorithms for full metric self-calibration, that use rotational and translational motions or just rotational motions.     

A New Linear Method for Camera Self-Calibration with Planar Motion

We consider the self-calibration (affine and metric reconstruction) problem from images acquired with a camera with unchanging internal parameters undergoing planar motion. The general self-calibration methods (modulus constraint, Kruppa equations) are known to fail with this camera motion. In this paper we give two novel linear constraints on the coordinates of the plane at infinity in a projective reconstruction for any camera motion. In the planar case, we show that the two constraints are equivalent and easy to compute, giving us a linear version of the quartic modulus constraint. Using this fact, we present a new linear method to solve the self-calibration problem with planar motion of the camera from three or more images. This work was partly supported by project BFM2003-02914 from the Ministerio de Ciencia y Tecnología (Spain). Ferran Espuny received the MSc in Mathematics in 2002 from the Universitat de Barcelona, Spain. He is currently a PhD student and associate professor in the Departament d’àlgebra i Geometria at Universitat de Barcelona, Spain. His research, supervised by Dr. José Ignacio Burgos Gil, is focussed on self-calibration and critical motions for both pinhole and generic camera models.     

A case against Kruppa's equations for camera self-calibration

We consider the self-calibration problem for perspective cameras and especially the classical Kruppa equation approach. It is known that for several common types of camera motion, self-calibration is degenerate, which manifests itself through the existence of ambiguous solutions. The author previously (1997, 1999) studied these critical motion sequences and showed their importance for practical applications. Here, we reveal a type of camera motion that is not critical for the generic self-calibration problem, but for which the Kruppa equation approach fails. This is the case if the optical centers of all cameras lie on a sphere and if the optical axes pass through the sphere's center, a very natural situation for 3D object modeling from images. Results of simulated experiments demonstrate the instability of numerical self-calibration algorithms in near-degenerate configurations.     

Self-calibration of hybrid central catadioptric and perspective cameras

Hybrid central catadioptric and perspective cameras are desired in practice, because the hybrid camera system can capture large field of view as well as high-resolution images. However, the calibration of the system is challenging due to heavy distortions in catadioptric cameras. In addition, previous calibration methods are only suitable for the camera system consisting of perspective cameras and catadioptric cameras with only parabolic mirrors, in which priors about the intrinsic parameters of perspective cameras are required. In this work, we provide a new approach to handle the problems. We show that if the hybrid camera system consists of at least two central catadioptric and one perspective cameras, both the intrinsic and extrinsic parameters of the system can be calibrated linearly without priors about intrinsic parameters of the perspective cameras, and the supported central catadioptric cameras of our method can be more generic. In this work, an approximated polynomial model is derived and used for rectification of catadioptric image. Firstly, with the epipolar geometry between the perspective and rectified catadioptric images, the distortion parameters of the polynomial model can be estimated linearly. Then a new method is proposed to estimate the intrinsic parameters of a central catadioptric camera with the parameters in the polynomial model, and hence the catadioptric cameras can be calibrated. Finally, a linear self-calibration method for the hybrid system is given with the calibrated catadioptric cameras. The main advantage of our method is that it cannot only calibrate both the intrinsic and extrinsic parameters of the hybrid camera system, but also simplify a traditional nonlinear self-calibration of perspective cameras to a linear process. Experiments show that our proposed method is robust and reliable.     

摄像机内参数自标定——理论与算法

讨论如何通过摄像机的旋转运动标定其内参数.当摄像机绕其坐标轴旋转时,运用 代数方法给出了计算内参数的公式.该公式在2D投影变换接近理论值P时是非常实用的. 在摄像机绕未知轴旋转时,根据相应的2D投影变换,运用矩阵特征向量理论给出了内参数的 通解公式.通过摄像机绕两个不同未知轴的旋转,摄像机内参数能被唯一地确定.这些结果为 摄像机自标定算法提供了理论基础,同时也给出了实用性算法.模拟实验和真实图像实验的 结果表明本文所给的算法具有一定实用价值.     

Intrinsic and extrinsic active self-calibration of multi-camera systems

We present a method for active self-calibration of multi-camera systems consisting of pan-tilt zoom cameras. The main focus of this work is on extrinsic self-calibration using active camera control. Our novel probabilistic approach avoids multi-image point correspondences as far as possible. This allows an implicit treatment of ambiguities. The relative poses are optimized by actively rotating and zooming each camera pair in a way that significantly simplifies the problem of extracting correct point correspondences. In a final step we calibrate the entire system using a minimal number of relative poses. The selection of relative poses is based on their uncertainty. We exploit active camera control to estimate consistent translation scales for triplets of cameras. This allows us to estimate missing relative poses in the camera triplets. In addition to this active extrinsic self-calibration we present an extended method for the rotational intrinsic self-calibration of a camera that exploits the rotation knowledge provided by the camera’s pan-tilt unit to robustly estimate the intrinsic camera parameters for different zoom steps as well as the rotation between pan-tilt unit and camera. Quantitative experiments on real data demonstrate the robustness and high accuracy of our approach. We achieve a median reprojection error of $0.95$ pixel.     

摄像机内参数自标定——理论与算法1)

讨论如何通过摄像机的旋转运动标定其内参数.当摄像机绕其坐标轴旋转时,运用代数方法给出了计算内参数的公式.该公式在2D投影变换接近理论值P时是非常实用的.在摄像机绕未知轴旋转时,根据相应的2D投影变换,运用矩阵特征向量理论给出了内参数的通解公式.通过摄像机绕两个不同未知轴的旋转,摄像机内参数能被唯一地确定.这些结果为摄像机自标定算法提供了理论基础,同时也给出了实用性算法。模拟实验和真实图像实验的结果表明本文所给的算法具有一定实用价值.     

An Algebraic Approach to Camera Self-Calibration

This paper describes a new self-calibration method for a single camera undergoing general motions. It has the following main contributions. First, we establish new constraints which relate the intrinsic parameters of the camera to the rotational part of the motions. This derivation is purely algebraic. We propose an algorithm which simultaneously solves for camera calibration and the rotational part of motions. Second, we provide a comparison between the developed method and a Kruppa equation-based method. Extensive experiments on both synthetic and real image data show the reliability and outperformance of the proposed method. The practical contribution of the method is its interesting convergence property compared with that of the Kruppa equations method.     

Planar Motion Estimation and Linear Ground Plane Rectification using an Uncalibrated Generic Camera

We address and solve the self-calibration of a generic camera that performs planar motion while viewing (part of) a ground plane. Concretely, assuming initial sets of correspondences between several images of the ground plane as known, we are interested in determining both the camera motion and the geometry of the ground plane. The latter is obtained through the rectification of the image of the ground plane, which gives a bijective correspondence between pixels and points on the ground plane.     

Self-calibration of a stereo rig using monocular epipolar geometries

This paper addresses the problem of self-calibration from one unknown motion of an uncalibrated stereo rig. Unlike the existing methods for stereo rig self-calibration, which have been focused on applying the autocalibration paradigm using both motion and stereo correspondences, our method does not require the recovery of stereo correspondences. Our method combines purely algebraic constraints with implicit geometric constraints. Assuming that the rotational part of the stereo geometry has two unknown degrees of freedom (i.e., the third dof is roughly known), and that the principle point of each camera is known, we first show that the computation of the intrinsic and extrinsic parameters of the stereo rig can be recovered from the motion correspondences only, i.e., the monocular fundamental matrices. We then provide an initialization procedure for the proposed non-linear method. We provide an extensive performance study for the method in the presence of image noise. In addition, we study some of the aspects related to the 3D motion that govern the accuracy of the proposed self-calibration method. Experiments conducted on synthetic and real data/images demonstrate the effectiveness and efficiency of the proposed method.     

摄像机自标定方法的研究与进展

该文回顾了近几年来摄像机自标定技术的发展,并分类介绍了其中几种主要方法.同传统标定方法相比,自标定方法不需要使用标定块,仅根据图像间图像点的对应关系就能估计出摄像机内参数.文中重点介绍了透视模型下的几种重要的自标定方法,包括内参数恒定和内参数可变两种情形;最后还简要介绍了几种非透视模型下的摄像机自标定方法.     

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.     

Self-Calibration of Rotating and Zooming Cameras

In this paper we describe the theory and practice of self-calibration of cameras which are fixed in location and may freely rotate while changing their internal parameters by zooming. The basis of our approach is to make use of the so-called infinite homography constraint which relates the unknown calibration matrices to the computed inter-image homographies. In order for the calibration to be possible some constraints must be placed on the internal parameters of the camera.We present various self-calibration methods. First an iterative non-linear method is described which is very versatile in terms of the constraints that may be imposed on the camera calibration: each of the camera parameters may be assumed to be known, constant throughout the sequence but unknown, or free to vary. Secondly, we describe a fast linear method which works under the minimal assumption of zero camera skew or the more restrictive conditions of square pixels (zero skew and known aspect ratio) or known principal point. We show experimental results on both synthetic and real image sequences (where ground truth data was available) to assess the accuracy and the stability of the algorithms and to compare the result of applying different constraints on the camera parameters. We also derive an optimal Maximum Likelihood estimator for the calibration and the motion parameters. Prior knowledge about the distribution of the estimated parameters (such as the location of the principal point) may also be incorporated via Maximum a Posteriori estimation.We then identify some near-ambiguities that arise under rotational motions showing that coupled changes of certain parameters are barely observable making them indistinguishable. Finally we study the negative effect of radial distortion in the self-calibration process and point out some possible solutions to it.An erratum to this article can be found at     

一种机器人手眼关系自标定方法

设计了一种基于场景中单个景物点的机器人手眼关系标定方法．精确控制机械手末端执行器做 5 次以上平移运动和2 次以上旋转运动,摄像机对场景中的单个景物点进行成像．通过景物点的视差及深度 值反映摄像机的运动,建立机械手末端执行器与摄像机两坐标系之间相对位置的约束方程组,线性求得摄像 机内参数及手眼关系．标定过程中只需提取场景中的一个景物点,无需匹配,无需正交运动,对机械手的运 动控制操作方便、算法实现简洁．模拟数据实验与真实图像数据实验结果表明该方法可行、有效．     

Self-Calibration and Metric Reconstruction Inspite of Varying and Unknown Intrinsic Camera Parameters

In this paper the theoretical and practical feasibility of self-calibration in the presence of varying intrinsic camera parameters is under investigation. The paper's main contribution is to propose a self-calibration method which efficiently deals with all kinds of constraints on the intrinsic camera parameters. Within this framework a practical method is proposed which can retrieve metric reconstruction from image sequences obtained with uncalibrated zooming/focusing cameras. The feasibility of the approach is illustrated on real and synthetic examples. Besides this a theoretical proof is given which shows that the absence of skew in the image plane is sufficient to allow for self-calibration. A counting argument is developed which—depending on the set of constraints—gives the minimum sequence length for self-calibration and a method to detect critical motion sequences is proposed.     

MultiCol Bundle Adjustment: A Generic Method for Pose Estimation,Simultaneous Self-Calibration and Reconstruction for Arbitrary Multi-Camera Systems

In this paper, we present a generic, modular bundle adjustment method for pose estimation, simultaneous self-calibration and reconstruction for multi-camera systems. In contrast to other approaches that use bearing vectors (camera rays) as observations, we extend the common collinearity equations with a general camera model and include the relative orientation of each camera w.r.t to the fixed multi-camera system frame yielding the extended collinearity equations that directly express all image observations as functions of all unknowns. Hence, we can either calibrate the camera system, the cameras, reconstruct the observed scene, and/or simply estimate the pose of the system by including the corresponding parameter block into the Jacobian matrix. Apart from evaluating the implementation with comprehensive simulations, we benchmark our method against recently published methods for pose estimation and bundle adjustment for multi-camera systems. Finally, all methods are evaluated using a 6 degree of freedom ground truth data set, that was recorded with a lasertracker.     

Uncalibrated Motion Capture Exploiting Articulated Structure Constraints

We present an algorithm for 3D reconstruction of dynamic articulated structures, such as humans, from uncalibrated multiple views. The reconstruction exploits constraints associated with a dynamic articulated structure, specifically the conservation over time of length between rotational joints. These constraints admit reconstruction of metric structure from at least two different images in each of two uncalibrated parallel projection cameras. As a by product, the calibration of the cameras can also be computed. The algorithm is based on a stratified approach, starting with affine reconstruction from factorization, followed by rectification to metric structure using the articulated structure constraints. The exploitation of these specific constraints admits reconstruction and self-calibration with fewer feature points and views compared to standard self-calibration. The method is extended to pairs of cameras that are zooming, where calibration of the cameras allows compensation for the changing scale factor in a scaled orthographic camera. Results are presented in the form of stick figures and animated 3D reconstructions using pairs of sequences from broadcast television. The technique shows promise as a means of creating 3D animations of dynamic activities such as sports events.     

一种虚拟物体和真实场景合成的新算法

提出了一个新的合成虚拟物体和真实场景的算法．从两幅照片出发进行摄像机自定标，三维重建，然后在估计表面初始辐射参数的情况下，利用快速层次辐射度的方法实现虚拟景物和真实场景的合成．最后给出了良好的实验结果．     

一种基于Kruppa方程的分步自标定方法

针对摄像机自标定中Kruppa方程求解的非线性优化问题和标定结果的欠鲁棒性,提出一种基于Kruppa方程的分步自标定方法。根据两图像匹配的特征点对采用8点算法求解相应的基本矩阵,其中待匹配图像选用摄像机对同一场景在不同焦距下拍摄的两帧图片,对图片的特征匹配点建立约束关系,采用最小二乘法求出摄像机的主点坐标,然后利用遗传算法优化Kruppa方程的比例因子,最后通过优化后的比例因子完成摄像机的标定。实验表明,该方法可提高标定精度,并通过对特征点坐标加入高斯噪声,验证了算法的鲁棒性。