2D X-ray Image to 3D CT Coordinate System Transformation for Personalized Spine Surgery

2.3 Transformation from 2D X-ray Image Coordinate System to 3D CT Coordinate System

To avoid trauma to patients caused by traditional registration methods, this study used personalized markers that only appear in intraoperative X-ray images and do not exist in preoperative CT images. Through 2D/3D registration, the relationship between the markers and the CT medical images was obtained.

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) intraoperative lateral X-ray images of the spine were taken; 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) use ray tracing technology with a point light source to project the virtual 3D model of the spine obtained in step 1 to obtain the lateral DRR image based on 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 Figure 3.

Figure 3. 2D/3D Registration Process

This study used the ray casting algorithm to operate on the CT volume data, which simulates the process of X-ray imaging formation (see Figure 2).

After given initialization parameters p, a point is defined as a virtual X-ray source, and X-rays are projected through the object (CT volume data set obtained by CT imaging) to the detector plane perpendicular to the axis of the X-ray source. Some simplifications are involved: X-rays are monochromatic radiation that passes through the substance in a straight line (ignoring scattering effects); the X-ray source (focus) is very small (ideally a point) and located at a finite distance from the X-ray imaging plane; the following monochromatic radiation equation describes the attenuation through the body [26]: (4)

where I represents the radiation intensity of X-rays on the radiographic plane, I0 represents the intensity of X-ray radiation at the source, and μ represents the linear attenuation coefficient of voxel x, whose width is Δx (note that the effect of beam hardening is not considered). The relationship between CT data (expressed in Hounsfield units, HU) and the corresponding attenuation coefficient is expressed by Equation 5 [26]: (5)

where, for an X-ray source of 120 kVp, μ is approximately 0.2 cm-1.

In order to reconstruct the X-ray image, a ray is emitted from the virtual point source, and the intersection point of the ray with the plane determines the position of the point in the DRR; during this process, the intersection point of the point source with each CT slice in the CT volume data can be calculated to obtain the corresponding CT value. After a ray projection is completed, 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 the above steps, and after all the ray projections are completed, the CT value of each pixel block of the entire DRR image is accumulated, and the accumulated value is mapped to the pixel gray value to obtain a DRR image, as shown in Figure 4.

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 as follows: (6)

	(7)

where k is a constant; Idiff(x, y) represents the gray value in the difference image. 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 is to eliminate this difference as much as possible. When the image reaches optimal registration, the pattern to be registered disappears, and its pattern intensity is minimal, and the value of this measurement is highest.

In the process of constantly exploring the optimal solution, this study used the Powell algorithm to speed up the search process. It is a conjugate search method that constructs a conjugate search direction directly using function values. It is also called the Powell conjugate direction method or direction acceleration method. The Powell algorithm was formed when studying the problem of minimizing a quadratic function with a positive definite symmetric matrix H. Its basic idea is to construct a conjugate direction about H in the iterative process without using function derivative information. Therefore, its essence is to search for the optimal direction set, and it does not require complex calculations such as gradients and Hessian matrices, so it is faster and more effective. Using the Powell algorithm for optimization, the transformation parameters are output after the measurement value reaches the optimum, and the transformation matrix from the X-ray image coordinate system to the 3D CT image coordinate system is solved.