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Line-based camera calibration with lens distortion correction from a single image

Fuqiang Zhou

a,n , Yi Cui a , He Gao a,b , Yexin Wang a a

Key Laboratory of Precision Opto-mechatronics Technology, Ministry of Education, Beihang University, Beijing 100191, China

b

School of Information & Electrical Engineering, Shangdong Jianzhu University, Ji'nan 250101, Chinaarticle info

Article history:

Received 4 February 2013

Received in revised form

1 April 2013

Accepted 9 May 2013

Available online 2 June 2013

Keywords:

Camera calibration

Lens distortion

Single image

abstract Camera calibration is a fundamental and important step in many machine vision applications. For some

practical situations, computing camera parameters from merely a single image is becoming increasingly

feasible and significant. However, the existing single view based calibration methods have various

disadvantages such as ignoring lens distortion, requiring some prior knowledge or special calibration

environment, and so on. To address these issues, we propose a line-based camera calibration method

with lens distortion correction from a single image using three squares with unknown length. Initially,

the radial distortion coefficients are obtained through a non-linear optimization process which is isolated

from the pin-hole model calibration, and the detected distorted lines of all the squares are corrected

simultaneously. Subsequently, the corresponding lines used for homography estimation are normalized to avoid the specific unstable case, and the intrinsic parameters are calculated from the sufficient restrictions provided by vectors of homography matrix. To evaluate the performance of the proposed method, both simulative and real experiments have been carried out and the results show that the proposed method is robust under general conditions and it achieves comparable measurement accuracy in contrast with the traditional multiple view based calibration method using 2D chessboard target. &2013 Elsevier Ltd. All rights reserved.1. Introduction Camera calibration is absolutely essential in many machine vision applications such as industrial photogrammetry, 3D recon- struction, image-based rendering, and so on. The so-called camera calibration technique is the process of determining the mapping relationship between the 3D world and a 2D image based on a pin- hole camera model[1]. The calibrated model parameters often include two categories: one is the intrinsic parameters that determine the internal camera geometric and optical character- istics; the other is the extrinsic parameters that determine 3D position and orientation of the camera frame relative to a certain world coordinate system[2]. Also, many lens distortion model is included in the calibration process to improve the computed camera model[3], and distortion model can be calibrated alone or together with the pin-hole model[4,5]. Currently, a rich variety of camera calibration algorithms have been reported and the existing methods are suitable for different situations. Tsai[2]presented a radial alignment constraint calibra- tion method based on the measurements of the 3D points in the

template taking afixed reference. Zhang[6]proposed aflexiblecalibration technique by observing a planar 2D target shown freely

at least two different orientations, which is used widely in plenty of vision measurement applications that require high accuracy. Subsequently, Zhang[7]firstly investigated the possibility of camera calibration using 1D objects, and shown that the calibra- tion process is not possible with freely moved 1D objects, but can be solved if one point isfixed. This approach needs six or more observations of such a 1D object to obtain the closed-form solution. Another calibration technique is self-calibration that does not use any reference object and can be considered as 0D approach because only image point correspondences are required[7,8]. As mentioned above, the calibration methods based on 0D/1D/

2D features usually require capturing several images with respect

to the reference object or a static scene since only one image cannot provide enough constraints to solve a set of equations involving camera parameters. Although Tsai's method could cali- brate a camera using non-coplanar points from a single view, it requires a very precise 3D reference object that is difficult and expensive to manufacture. Therefore, it would be more practical and convenient if the calibration method requires only a single image of the reference object. The single image based calibration technique would be quite useful in conditions that recourse to flexible and simple calibration methods, such as real-time 3D reconstruction, surveillance camera, microscopic vision systems with limited depth offield, and so forth.Contents lists available atSciVerse ScienceDirect journal homepage:www.elsevier.com/locate/optlaseng

Optics and Lasers in Engineering

0143-8166/$-see front matter&2013 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.:+86 1082339281.

E-mail address:zfq@buaa.edu.cn (F. Zhou).

Optics and Lasers in Engineering 51 (2013) 1332-1343 Up to now, there are many studies on the problem of single view based calibration in the literature. Most approaches in recent years are trying to use the geometrical information obtained from a single image. For example, Guillou et al.[9]utilized vanishing points to calibrate a camera and to recover the geometry of objects from a single image. The calibration step does not need any calibration target but makes four assumptions. Avinash et al.[10]proposed a calculation method of focal length and the optical center of the camera by employing a rectangular prism as the calibration target to generate vanishing points. The proposed technique only requires a single image containing two vanishing points but the orientation of the rectangular prism has to be known. Wang et al.[11]provided an additional independent constraint to the popular calibration method based on three mutual orthogonal vanishing points. The added constraint can be obtained if the pair of line segments is of equal length or with known length ratio in space. Hence, camera can be calibrated based on four independent constraints from one image. Wilczkowiak et al. [12,13] exploited the full potential of rigidity constraints through parallelepipeds in camera calibration as well as in scene reconstruction from a single image. Similar to Wilczkowiak's work, de la Fraga et al. [14]proposed to solve directly the problem of calibrating a camera from a single image of a cuboid usingdifferential evolution. Unlike the above approaches based on vanishing points or parallelepipeds, a camera calibration method to estimate the extrinsic parameters and the focal length of a camera by using only one image of two coplanar circles with arbitrary radius was described in Refs.[15,16]. Moreover, Miyagawa et al.[17]developed a simple calibration method from a single image usingfive points on two orthogonal 1D objects. Like Refs. [15,16], this method cannot compute the camera's principal point coordinates. Additionally, single view based camera poses estimation methods under the assumption of some known intrinsic parameters were also presented in Refs.[18-20]. However, the above methods of calibrating a camera using a single image presented in the literature mainly have the following drawbacks: (1) lens distortion effect is not considered, so the result of vision measurement or 3D reconstruction would not be accurate enough; (2) some methods only compute partial camera parameters and others need some prior knowledge about intrinsic parameters; (3) in some cases, special reference objects or strict geometric constraints are required in advance. To solve these problems, we propose a line-based calibration method from a single image using at least three squares with unknown length. First, detected lines in the image are corrected by means of projective invariant and separate model constraint. The radial lens distortion coefficients are obtained simultaneously. Second, all the intrinsic parameters are computed using corrected image lines and corresponding space lines from the referred squares. The second step is a linear process since the coupling between the pin-hole and lens distortion models is avoided. The main advantages of our method lie in the following aspects: (1) The complete intrinsic parameters and lens distortion coefficients are acquired from a single image by observing at least three squares with unknown length. To the best of our knowledge, all the single view based camera calibration techniques take no consideration of lens dis- tortion. (2) The line-based calibration method is adopted since it can suppress the image noise well and perform better than conventional point-based approaches[18,21,22]. In addition, the normalized method specially designed for line-based homography estimation proposed by Ref. [21]is employed to prevent the possible unstable case. (3) The lens distortion parameters are computed isolated from the camera intrinsic parameters, and the final measurement accuracy achieved by our method is compar- able to Zhang's method[6]which relies on capturing multiple images of 2D target. Moreover, we want to indicate that our work is inspired by the

research of Hartley and Zisserman[1].InRef.[1],asimplecalibrationdevice composed of three squares provides sufficient constraints to

compute intrinsic parameters. However, the method proposed by Ref. [1]ignores the lens distortion effect, which may lead to significant errors in calibration results, 3D reconstruction, and metric measure- ments[23-27]. The other difference is that our method utilizes the robust line information instead of corner points to compute the homography. Besides, another similar work is described in Ref.[28] which implemented 3D reconstruction using a single image of three planar chessboard patterns. In Ref.[28], the commonly used chess- board target is adopted and the lens distortion is also neglected, which are the main differences compared with our work. The rest of the paper is organized as follows.Section 2gives some preliminaries of camera model with lens distortion. The detailed procedure of lined-based camera calibration method from a single image is described inSection 3.InSection 4, both computer simulative and real data are used to validate the proposed method, and the Zhang'smethod[6]based on multiple calibration views is also compared with ours. The paper ends with some concluding remarks inSection 5.

2. Preliminaries

2.1. Camera pin-hole model

A camera is modeled by the usual pin-hole and the relationship between a 3D pointMand its image projectionmis given by ~m¼AðRjtÞ~M;A¼f x 0u 0 0f y v 0 0012 6 43
7

5ð1Þ

whereλis an arbitrary scale factor,~mand~Mare the homogeneous coordinates of image point and its corresponding space point, (R,t), called the extrinsic parameters, is the rotation and transla- tion which relates the world coordinate system to the camera coordinate system, andAis the camera intrinsic matrix, consisting of four parameters:f x andf y the effective focal length, (u 0 ,v 0 ) the coordinates of the principal point. In this paper, we assume the skew factor is zero and the primary task of camera calibration is to determine these four intrinsic parameters.

2.2. Lens distortion model

Actually, real lenses do not satisfy the ideal pin-hole model and usually exhibit some degree of geometric distortion. Thus, a proper distortion model should be selected. The most commonly used approach proposed by Brown[29,30] is to decompose the distor- tion into radial, decentering, and prism components. In general, the radial distortion is sufficient for a high-accuracy measurement since some more elaborated model do not improve the accuracy significantly[23,31]. In this paper, the lens distortion correction is carried out before the internal camera calibration and undistortedquotesdbs_dbs14.pdfusesText_20

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