## 1 Introduction  
Magnetic medical robots (MMRs) contain permanent magnets and perform medical functions in the human tubular system by controlling an external magnetic field generated by a magnetic navigation system (MNS). MMRs utilize the magnetic characteristics in which the permanent magnet tends to be aligned along the direction of the external magnetic field and propelled along its gradient. Many researchers have developed various MMRs that can perform medical treatments and diagnoses through diverse designs and control mechanisms [1, 2, 3, 4]. The MNS generates a magnetic field through permanent magnets, coils, and electromagnets which are placed around the target lesion [5, 6, 7]. The workspace of an MNS can be defined as a certain volume in which the magnetic field can be controlled to perform the medical functions of an MMR. To precisely control an MMR using an MNS, the 3D coordinate system of an MNS workspace has to be defined and the 3D position of an MMR in this coordinate system has to be identified to precisely generate the magnetic field, ensuring precise control of an MMR and increasing the success rate of medical functions. In many studies on robotic endovascular intervention using MMRs and MNSs, the 3D coordinate system of an MNS workspace was predefined utilizing multiple cameras fixed to MNSs or jigs [8; 9; 10; 11]. However, in real clinical operation, medical doctors utilize a C-arm X-ray device to identify the blood vessel and endovascular devices such as guide wires and catheters along the blood vessel. X-ray images are 2D images and they cannot be directly applied to robotic endovascular intervention, which requires 3D real-world coordinates of both the location of an MMR and the blood vessel map in order to accurately apply magnetic field. To control an MMR using a C-arm X-ray device and an MNS, it is necessary to estimate a 3D vascular map from 2D X-ray images and to ensure compatibility between the commercial C-arm and the MNS. While [PERSON] et al. extracted 3D positions of blood vessels using a dual C-arm embedded in an MNS [12], the studies to estimate the 3D positions of blood vessels by ensuring the compatibility between commercial C-arm X-ray devices and MNSs are limited. Several researchers have reconstructed a 3D coordinate in X-ray imaging devices [13; 14; 15; 16; 17; 18]. They captured two or more images from various views and calibrated intrinsic and extrinsic parameters of the C-arm. They derived intrinsic parameters such as focal length, principal point, and pixel size before the operation to reflect the geometric properties of the images. However, the extrinsic parameters, such as the rotation angle displayed on the control program of C-arm, may not be as accurate as expected, which can cause errors in the reconstruction of the vascular map. Many researchers have developed various methods to calibrate the extrinsic parameters of a C-arm. [PERSON] et al. [13] conducted a calibration procedure that utilizes calibration templates prior to the coronary modeling. However, it cannot consider the changes in extrinsic parameters during operation, requires a point matching process across multiple X-ray images, and shows a significant increase in error when the angular difference between the two images is less than 35\({}^{\circ}\) or greater than 145\({}^{\circ}\). However, when using a commercial C-arm X-ray device and an MNS, the maximum angular difference between the two X-ray images is approximately 15\({}^{\circ}\), making this method inapplicable. Other researchers proposed a method to derive the transformation between two uncalibrated X-ray images using the Levenberg-Marquardt (LM) algorithm [14]. However, it assumes that the C-arm images are acquired at orthogonal angles, converges slowly, and is sometimes trapped by local optima. To reduce convergence time, [PERSON] et al. [15] proposed a two-stage LM algorithm.