## 1 Introduction  
Augmented reality (AR) superimposes rich visual information on the real-world scene, which is intuitively suitable for guiding or training manual operations in the manufacturing industry. However, AR has not fully broken the industrial market yet, because it lacks pervasiveness from the standpoint of industrial users [1], and this paper addresses one related issue that is commonly confronted at the beginning of setting an industrial AR application. To achieve precise user cognition, industrial AR applications require that the rendered virtual information should be spatially consistent with the real scene or target object. In AR applications, the process that aligns the rendered virtual information in the spatially consistent position is defined as the AR registration [2]. Because monocular cameras are the most cost-effective for industrial users, monocular AR registration methods have drawn much attention from the research field. The mechanism of the AR registration is shown in Figure 1, where the registered AR view is generated by projecting the virtual 3D graphics using the same intrinsic and extrinsic parameters as that of the real camera [3]. Therefore, the AR registration accuracy is determined by two matrices, one is the camera's intrinsic matrix that is obtained by camera calibration [4], and the other is the extrinsic matrix that is obtained by 6 DoF (6 degrees of freedom) camera pose estimation or tracking. As reported by several significant reviews on the industrial AR applications [5; 6], the marker-based method is accurate, robust, convenient, and requires no specific knowledge on AR or computer vision, therefore it takes the predominant position in obtaining the extrinsic matrix for industrial AR applications. The marker-based method directly takes the measured 6D camera pose relative to the marker center as the extrinsic matrix, and then project the virtual graphical object based on the assumption that the object's local coordinate system coincides with that of the marker's [7]. However, the assumption does not always hold true in practical conditions, which leads to obvious visible AR registration error as illustrated in Figure 2b. Three reasons have been observed to cause the registration error from our previous work [8; 9]:  
1. Generally, the marker's coordinate system is assumed to coincide with the local coordinate system of the CAD model, but in the practical condition, the marker may be laid on other planar surfaces of the object, which introduces an undetermined transformation between the planned layout position and the real layout position of the marker. 2. Though the transformation could be set manually, it is calculated by multiplying several manually measured transformation matrix, which brings in the systematic measurement error. 3. Given both the manual marker layout and the transformation measurement are accurate, the AR registered CAD model will still not be perfectly aligned with the real object because of the slight structure, shape, or appearance changes caused by the machining or assembling errors. Figure 1: General vision-based augmented reality (AR) registration. When a marker is attached to the target object, all the above errors are fixed as both the marker and the object are rigid in the 3D space. Those errors are combined and defined as the marker-object offset matrix [8], which is independently calibrated before the online tracking to compensate for the extrinsic matrix. The calibration problem is equivalent to the image alignment problem [10] for they both solve the transformation parameters that align two images taken from one target object toward the minimal visual discrepancy. The early stage of the image alignment method [10; 11] employs the homography transformation or affine transformation model between the two images. Such models are linear and only work when the images contain the same static target object and allow little camera view-point changes.