## 2 Concepts and Applications of TMPR  
### Magnification Characteristic of X-Ray Images  
In this section, we formulate the projection between 2D coordinates on an X-ray image and 3D coordinates within a C-arm's coordinate system, utilizing the magnification characteristic of X-ray images. A 2D coordinate of the X-ray image in Figure 1b is a 2D coordinate in the unit of pixels, with the origin located at the top-left corner of the image. A 3D coordinate of a C-arm is a 3D coordinate with a unit of length and defined by the X, Y, and Z axes, as illustrated in Figure 1b. As shown in Figure 1a,b, the source and detector are aligned parallel to each other at the opposite ends of a C-arm [20]. In this study, symbols for the points and lines are written in italic type, while symbols for the coordinates of points and lines are written in italic and bold type. The perpendicular line from the source to the detector is referred to as the principal line (\(L_{0}\)), and its foot of the perpendicular on the image is called the principal point (\(p_{0}\)). The isocenter (\(P_{iso}\)) is a point on the principal line and serves as the rotation center of the C-arm. The 3D coordinate system of the C-arm can be defined by setting \(P_{iso}\) as the origin, \(L_{0}\) as the Z axis, the rotation axis in the RAO/LAO (right/left anterior oblique) direction as the Y axis, and the rotation axis in the CAUD/CRAN (caudal/cranial) direction as the X axis. An object captured by X-rays is magnified on the image. The magnification factor (MF) is the ratio between the image size and object size or between the SID (source to image distance) and SOD (source to object distance), as shown in Figure 1a and Equation (1) [21]. \[\text{MF}=\frac{\text{Image size}}{\text{Object size}}=\frac{\text{SID}}{ \text{SOD}} \tag{1}\]  
The image size, object size, SID, and SOD in (1) have length units. If an object is represented by \(n_{pixel}\) pixels in the image and each pixel has a length of \(d_{pixel}\), then the object size can be expressed as (2) in terms of \(n_{pixel}\), \(d_{pixel}\), \(Z\), \(D_{iso}\), SID, and SOD, which can be seen as follows:  
\[\text{Object size}=\frac{\text{Image size}}{\text{MF}}=n_{pixel}\times d_{ pixel}\times\frac{\text{SOD}}{\text{SID}}=n_{pixel}\times d_{pixel}\times\frac{Z+D_{iso}}{ \text{SID}} \tag{2}\]  
where \(D_{iso}\) represents the distance from the source to the \(P_{iso}\), and \(Z\) is the difference between SOD and \(D_{iso}\). We can obtain \(f(Z)\) in (3) after dividing (2) by \(n_{pixel}\), which is the object size per pixel with a unit of mm/pixel. This shows the ratio between pixel units in the X-ray image and the length unit according to the \(Z\) coordinate in the 3D coordinate system of the C-arm. \[f(Z)=d_{pixel}\times\frac{Z+D_{iso}}{\text{SID}} \tag{3}\]  
For a point \(P\) located in 3D space, the relationship between the position of \(\mathbf{P}\) in the 3D coordinate system of the C-arm and the position of \(\mathbf{p}\) in the 2D coordinate system of the X-ray image is shown in (4) and (5). The \(\mathbf{p}\) is the 2D point where the 3D point \(\mathbf{P}\) is projected onto the X-ray image. Equations (4) and (5) are as follows:  
\[\mathbf{P}= \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} = \begin{bmatrix}(x-x_{0})\times f(Z)\\ (y-y_{0})\times f(Z)\\ Z\end{bmatrix}  \tag{4}\]  
\[\mathbf{p}= \begin{bmatrix}x\\ y\end{bmatrix} =\frac{1}{f(Z)} \begin{bmatrix}X\\ Y\end{bmatrix} + \begin{bmatrix}x_{0}\\ y_{0}\end{bmatrix}  \tag{5}\]  
where \(x_{0}\) and \(y_{0}\) represent the respective x and y coordinates of the principal point (\(\mathbf{p}_{0}\)) in the 2D coordinate system of the X-ray image. Equation (4) shows that \(\mathbf{P}\) in the 3D coordinate system of the C-arm corresponding to \(\mathbf{p}\) in the 2D coordinate system of the X-ray image can be expressed as a function of \(Z\) when \(\mathbf{p}\) is given. Currently, the feasible range of \(Z\) is \(-D_{iso}\leq Z\leq\text{SID}\)\(-D_{iso}\).