Personalized Marker Coordinate System to 2D X-Ray Image Transformation

2.2 Transformation of Personalized Marker Coordinate System to Two-Dimensional X-Ray Image Coordinate System

This study proposes personalized markers as shown in Figure 2, which can be detected using X-rays and monocular cameras to obtain the transformation matrix from the personalized marker coordinate system to the two-dimensional X-ray image coordinate system, as well as from the personalized marker coordinate system to the world space coordinate system.

For this study, Aruco was used to design personalized markers. Aruco[3] is a binary-coded square marker that can be arbitrarily sized and numbered. As shown in Figure 2(a), each Aruco reference marker consists of internal coding and a black border. The internal coding contains basic information such as the four corner points' data and the marker's dictionary number. Given the reference marker's size and dictionary type, the corner points' three-dimensional coordinates in the marker's own coordinate system and the marker's pose can be calculated using the camera's imaging model and parameters.

To ensure the markers are clear and distinguishable in optical and X-ray spectral imaging, this study printed the Aruco marker's internal coding portion using resin material and connected it to a white outer frame. The Aruco black part was made using stainless steel material, which strongly attenuates X-ray radiation, as shown in Figure 2(b). Due to the metal's high attenuation, the marker is equally visible in X-ray spectra and monocular camera images.

Since the personalized marker's shape and size are precisely known in 3D, if the marker can be detected in the 2D image, the 3D to 2D point pair relationship can be calculated using the Perspective-n-Point (PnP) method, as shown in Equation (2):

(2)

where 'x' is the coordinate of the point in the pixel coordinate system, 'X' is the coordinate of the point in the camera coordinate system, 'M' is the coordinate of the point in the marker coordinate system, 'Z' is the depth of the point, 'K' is the camera's intrinsic matrix, and '[R|t]' is the pose transformation from the marker coordinate system to the camera coordinate system.

Taking the Aruco coordinate system as the marker coordinate system and placing the marker's initial position on the z=0 plane, given the marker's true physical size, the three-dimensional spatial coordinates of the marker's four corner points can be obtained as 'A', 'B', 'C', and 'D'. These points' corresponding coordinates in the pixel coordinate system can be obtained through monocular camera detection and recognition, and the camera's intrinsic matrix K was obtained in the previous camera calibration work[25].

As shown in Equation (3), taking point A as an example, where '[R|t]' is the rotation matrix and translation vector we want to find, i.e., the external parameters of the camera.

According to the principle of C-arm X-ray image acquisition, the C-arm X-ray radiation source is treated as a monocular camera to obtain the pixel coordinates of the personalized marker's four vertices. After obtaining the 2D and 3D point pairs, the transformation relationship between the marker coordinate system and the camera coordinate system can be calculated, and the transformation matrix from the personalized marker coordinate system to the two-dimensional X-ray image coordinate system can be obtained.