Precise 3D-2D Registration for Spine Surgery Navigation: A Comprehensive Approach

As an expert in the field of medical imaging, I can help you improve the grammar and professional language in the following paragraphs:

Paragraph 1: Given the precise 3D shape and size of the personalized marker, the 3D to 2D point pair relationship can be calculated using the Perspective-n-Point (PnP) method if the marker is detected in the 2D image. This relationship is represented by Equation (2): (2) where represents the coordinate of the point in the pixel coordinate system, represents the coordinate of the point in the camera coordinate system, represents the coordinate of the point in the marker coordinate system, represents the depth of the point, represents the camera's intrinsic matrix, and represents the pose transformation from the marker coordinate system to the camera coordinate system. The ArUco coordinate system is chosen as the marker coordinate system, with the marker's initial position placed on the plane, as shown in Fig. 4. By knowing the marker's true physical size, the three-dimensional spatial coordinates of the marker's four corner points can be obtained as 、、、 . The corresponding coordinates of these points in the pixel coordinate system 、、、 can be obtained through monocular camera detection and recognition, with the camera's intrinsic matrix obtained from previous camera calibration work [26].

Fig. 4 illustrates the transformation of coordinate system in ArUco.

Paragraph 2: Taking point A as an example, Equation (3) shows the rotation matrix and translation vector that we want to find, which represent the external parameters of the camera. (3) Among them:

According to the principle of C-arm X-ray image acquisition, the C-arm X-ray radiation source can be treated as a monocular camera to obtain the pixel coordinates of the personalized marker's four vertices. By 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 can be obtained.

Paragraph 3: To avoid trauma to patients caused by traditional registration methods, we utilize a personalized marker that only appears in intraoperative X-ray images and does not exist in preoperative CT images. Through 2D/3D registration, we can obtain the relationship between the marker and the CT medical images. The process of 2D/3D registration is as follows:

1. 3D modeling of preoperative CT images to obtain a virtual 3D model of the spine.
2. Capture intraoperative lateral X-ray images of the spine.
3. Set the initial transformation parameters to determine the initial guess of the relative position between the X-ray image and the CT data for optimization.
4. Project the virtual 3D model of the spine obtained in step 1 with a point light source using ray tracing technology and obtain the digitally reconstructed radiograph (DRR) image in lateral planes according to the initial transformation parameters.
5. Calculate the similarity between the DRR image and the intraoperative lateral X-ray image.
6. If the similarity calculation result meets the standard, the 3D-2D registration process can be completed to obtain the position of the point light source relative to the 3D model in the virtual space. Otherwise, repeat steps 3 to 6 and update the parameters of the point light source. The 2D/3D registration process is shown in Fig. 5.

Fig. 5 illustrates the 2D/3D registration process.

Paragraph 4: This study utilizes the ray casting algorithm to operate on the CT volume data, simulating the process of X-ray imaging formation as shown in Fig. 6.

Fig. 6 illustrates the principle of digitally reconstructed radiograph (DRR) generation.

After initialization parameters p are given, a virtual X-ray source emits a ray through the object (CT volume data set obtained by CT imaging) to the detector plane perpendicular to the axis of the X-ray source. During this process, the absorption coefficient of each intersection point between the ray and CT volume data is calculated. The CT values obtained along the entire path are accumulated to obtain the pixel value of the DRR image corresponding to the point on the detector. Repeat these steps to obtain DRR images after all ray projections are completed. The DRR image of the spinal model is shown in Fig. 7.

Fig. 7 shows the digitally reconstructed radiograph (DRR) image of the spinal model.

Paragraph 5: The DRR image is compared with the X-ray image to be registered, and the similarity measurement based on pattern intensity (PI) is calculated. This measurement judges whether the registration is successful by measuring whether the pattern in the difference image (the difference in gray values between the two images) has been minimized. The similarity measurement function is represented by Equation (6): (6) where ; , , , and represent pixel coordinates in the image; represents the pixel value of a specific coordinate in the subtracting image of the two images to be registered; represents the pixel value of coordinates in the neighborhood of a pixel coordinate in the subtracting image of the two images to be registered; represents the radius of the effective calculation area of the mode intensity of each pixel; represents the diameter; the constant is the weight of the function to eliminate the interference of noise. represents the final mode intensity value. The pattern intensity considers that when the difference between the value of a pixel and its adjacent pixels is significant, the pixel belongs to a pattern, and the process of image registration aims to eliminate this difference as much as possible. When the image reaches optimal registration, the pattern to be registered disappears, the mode intensity is minimum, and the measured value is maximum. In the process of constantly exploring the optimal solution, we use the Powell algorithm to speed up the search process. The Powell algorithm is a conjugate search method that constructs a conjugate search direction directly using function values, also known as the Powell conjugate direction method or direction acceleration method. It is faster and more effective as it does not require complex calculations such