## 1 Introduction
Nevertheless, the convergence speed of the LM algorithm still needs to be improved, and it is also vulnerable to geometric errors such as inaccuracies in the actual rotation angles of the C-arm. Meanwhile, many studies estimated extrinsic parameters using some specific landmark points in vessels such as bifurcations or end points [16; 17]. These methods require additional steps to identify the landmark points and determine the correspondence between them in each image. Also, three or more images are still required to determine the unique correspondence of each landmark point [18]. Also, various methods have been developed to reconstruct a 3D vascular map that represents the centerline of the blood vessel in a 3D coordinate [14; 15; 17; 19]. The LM algorithm used for the calibration of the external parameters was also used for the reconstruction of the centerlines of vessels [14; 15]. However, the limitations of the LM algorithm of slow convergence and trapping in local optima also exist in the reconstruction process of a vascular map. Some researchers proposed a point cloud method to reconstruct a 3D vascular map [17]. This method utilizes multiple images and conducts iterative fitting of B-spline curve. [PERSON] et al. [19] defined an energy function as the objective function which represents the difference between the data displayed in the images and the 3D coordinate, and reconstructed a 3D vascular map by minimizing the energy function through an \(\alpha\)-expansion method. It has low computation cost and fast convergence time. However, the accuracy and computation cost of the \(\alpha\)-expansion method are significantly affected by the initial Z coordinate at the beginning of computation because it iteratively searches for the optimal Z coordinate from the initial Z coordinate to minimize the energy function. In this study, we propose a method to reconstruct a 3D coordinate system and a vascular map for the operation of an MMR controlled by an MNS using two X-ray images with restricted angles and four corners of an MNS. In Section 2.1, we outline the relationship between the 2D coordinates within the coordinate system of an X-ray image and the 3D coordinates within the coordinate system of a C-arm. In Section 2.2, we suggest a different viewpoint to describe the translation and rotation of a focal spot of a C-arm as the rotation of an object in a fixed coordinate system using the motion characteristic of a C-arm. In Section 2.3, we propose a transformation method from two 2D X-ray pixel images to a 3D real-world coordinate (TMPR) to obtain the coordinates of an object in a 3D coordinate system of a C-arm without the point matching process across multiple images. In Section 2.4, we propose a method to estimate the relative rotation angle between two views with four corners of an MNS using the TMPR, thereby compensating for rotational angle errors of a C-arm. In Section 2.5, we suggest a technique to calculate the centerline path of the blood vessel utilizing the TMPR. In Section 3, we validate the TMPR and propose reconstruction methods in an in vitro experiment using an MNS mock-up and a vascular jig. Finally, the effectiveness of the proposed methods is verified through in vivo experiments in the superficial femoral artery (SFA) of a mini pig, where an MMR is remotely operated by an MNS. This confirms that the TMPR can enhance the compatibility between a commercial C-arm X-ray device and an MNS in a constrained environment where the maximum angular difference between the two X-ray images is approximately \(15^{\circ}\).