# Improved Direct Linear Transformation for Parameter Decoupling in Camera Calibration
## 1 Introduction  
With the technological developments of digital cameras and microprocessors, computer vision has been widely applied to robot navigation, surveillance, three-dimensional (3D) reconstruction and other fields for its high speed, high accuracy and non-contact nature. To obtain improved 3D information from a two-dimensional (2D) image, it is necessary to calibrate the intrinsic parameters of the camera, such as the focal distance and optical center point, as well its extrinsic parameters, such as rotation and translation, which relate the world coordinate system to the camera coordinate system. Over the last decade, numerous studies have focused on this area. In [1], the authors proposed an efficient approach for the dynamic calibration of multiple cameras. In [2], the authors proposed a calibration algorithm based on line images. In [3], the authors used one-dimensional (1D) information to calibrate the parameters. All of these algorithms make camera calibration faster and more convenient.  
To obtain high accuracy parameter results, high accuracy 3D or 2D templates can be used. These algorithms include direct linear transformation (DLT) [4], the Tsai calibration method [5] and the Zhang calibration method [6]. In space rendezvous and docking, as well as in visual tracking applications, it is necessary to obtain the specific extrinsic and intrinsic parameters of the camera. At the same time, with the wide application of the coordinate measuring machine (CMM) [7; 8], high precision, large range 3D templates have become more readily applied. The DLT algorithm is more suitable for these applications.  
The DLT algorithm is based on the perspective projection between 3D space points and 2D image points. It calculates a transformation matrix and obtains the camera's intrinsic and extrinsic parameters according to the parameter decomposition. With this model, only one 3D template in one position is required for calculation; therefore, the template size, number of feature points and relative distance between the template and camera are critical [9; 10; 11].  
Camera calibration with image noise and distortion remains a difficult task. Algorithms typically solve the camera parameters by analysis; then, they solve the model of image noise and distortion by optimization [6; 12; 13]. All traditional optimization algorithms applied to visual systems, such as the modified Newton algorithm, the Levenberg-Marquardt algorithm and the genetic algorithm, require a good initial solution to optimize. Therefore, the analytic solution is not only robust to image noise, but also effective for addressing distortion.  
When the calibration data contain noise and distortion, coupling errors exist between the camera's extrinsic and intrinsic parameters. This means that an error in an extrinsic parameter may be compensated by an error in an intrinsic parameter [14; 15; 16]. Existing parameter decoupling methods use calibration models without considering extrinsic parameters. These include vanishing points [15; 17; 18], straight lines [16; 19], and cross ratios [20; 21]. In the present study, the image re-projection error from the mathematical model and the camera pinhole geometric model were analyzed. The results show that the coupling error causes a small re-projection error variance and a large calibration parameter error variance. At the same time, this variance is related to the template size, number of feature points and relative distance between the template and camera. To improve the calibration accuracy of the camera parameters, a decoupling algorithm of intrinsic and extrinsic parameters is proposed based on DLT.  
The remainder of this paper is organized as follows. The relationship between the coupling error, template size and relative distance between the template and camera is described in Section 2. The proposed improved DLT (IDLT) algorithm for camera calibration is proposed in Section 3. The experimental results are presented in Section 4, and the conclusions are given in Section 5.