Camera Calibration with One-Dimensional Objects Zhengyou Zhang, Senior Member, IEEE Abstract—Camera calibration has been studied extensively in computer vision and photogrammetry and the proposed techniques in the literature include those using 3D apparatus (two or three planes orthogonal to each other or a plane undergoing a pure translation, etc.), 2D objects (planar patterns undergoing unknown motions), and 0D features (self-calibration using unknown scene points). Yet, this paper proposes a new calibration technique using 1D objects (points aligned on a line), thus filling the missing dimension in calibration. In particular, we show that camera calibration is not possible with free-moving 1D objects, but can be solved if one point is fixed. A closed-form solution is developed if six or more observations of such a 1D object are made. For higher accuracy, a nonlinear technique based on the maximum likelihood criterion is then used to refine the estimate. Singularities have also been studied. Besides the theoretical aspect, the proposed technique is also important in practice especially when calibrating multiple cameras mounted apart from each other, where the calibration objects are required to be visible simultaneously. Index Terms—Camera calibration, calibration taxonomy, calibration apparatus, 1D objects, singularity, degenerate configuration. æ 1 INTRODUCTION C AMERA calibration is a necessary step in 3D computer vision in order to extract metric information from 2D images. Much work has been done, starting in the photogrammetry community (see [1], [3] to cite a few) and, more recently, in computer vision ([8], [7], [20], [6], [22], [21], [15], [5] to cite a few). According to the dimension of the calibration objects, we can classify those techniques roughly into three categories. 3D reference object-based calibration. Camera calibration is performed by observing a calibration object whose geometry in 3D space is known with very good precision. Calibration can be done very efficiently [4]. The calibration object usually consists of two or three planes orthogonal to each other. Sometimes, a plane undergoing a precisely known translation is also used [20], which equivalently provides 3D reference points. This approach requires an expensive calibration apparatus and an elaborate setup. 2D plane-based calibration. Techniques in this category are required to observe a planar pattern shown at a few different orientations [23], [18]. Different from Tsai’s technique [20], the knowledge of the plane motion is not necessary. Because almost anyone can make such a calibration pattern by him/herself, the setup is easier for camera calibration. Self-calibration. Techniques in this category do not use any calibration object and can be considered as 0D approach because only image point correspon- dences are required. Just by moving a camera in a static scene, the rigidity of the scene provides, in general, two constraints [15], [14] on the cameras’ internal parameters from one camera displacement by using image information alone. Therefore, if images are taken by the same camera with fixed internal para- meters, correspondences between three images are sufficient to recover both the internal and external parameters which allow us to reconstruct 3D structure up to a similarity [13], [10]. Although no calibration objects are necessary, a large number of parameters need to be estimated, resulting in a much harder mathematical problem. A recent overview of this area can be found in [11]. Other techniques exist: vanishing points for orthogonal directions [2], [12] and calibration from pure rotation [9], [17]. To our knowledge, there does not exist any calibration technique reported in the literature which uses one-dimen- sional (1D) calibration objects and this is the topic we will investigate in this paper. In particular, we will consider 1D objects composed of a set of collinear points. Unlike techniques using 3D reference objects, other techniques requires taking several snapshots of calibration objects or the environment. This is the price we pay, although insignif- icant, in practice, by using poorer knowledge of the observa- tion. This is also the case with calibration using 1D objects. Besides the theoretical aspect of using 1D objects in camera calibration, it is also very important, in practice, especially when multicameras are involved in the environment. To calibrate the relative geometry between multiple cameras, it is necessary for all involving cameras to simultaneously observe a number of points. It is hardly possible to achieve this with 3D or 2D calibration apparatus 1 if one camera is mounted in the front of a room while another in the back. This is not a problem for 1D objects. We can, for example, use a string of balls hanging from the ceiling. The paper is organized as follows: Section 2 examines possible setups with 1D objects for camera calibration. Section 3 describes in detail how to solve camera calibration with 1D objects. Both closed-form solution and nonlinear 892 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 7, JULY 2004 . The author is with Microsoft Research, One Microsoft Way, Redmond, WA 98052. E-mail: zhang@microsoft.com. Manuscript received 18 Nov. 2002; revised 7 Dec. 2003; accepted 9 Dec. 2003. Recommended for acceptance by L. Quan. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number 117790. 1. An exception is when those apparatus are made transparent, then the cost would be much higher. 0162-8828/04/$20.00 ß 2004 IEEE Published by the IEEE Computer Society