## Materials and Methods
### Model-Based Interleaved Biplane Fluoroscopy Image Tracking
#### 2.4.1 Two-Dimensional/2D Template Registration  
A 2D/2D template registration procedure was devised to provide an initial estimation of the transformation of the bone model for each image frame, \(\mathbf{T}_{treg}=\mathbf{T}(\mathbf{R}_{treg}^{k},\mathbf{v}_{treg}^{k})\), consisting of a rotation matrix \(\mathbf{R}_{treg}^{k}(\theta_{X},\theta_{Y},\theta_{Z})\) and a 3D position vector \(\mathbf{v}_{treg}^{k}(T_{X^{\prime}}T_{Y^{\prime}}T_{Z})\). For the first image frame being analyzed, \(l^{k}\), a 2D template of the bone of interest, \(\mathrm{M}_{j}\), \(j=1\) or \(2\), was initialized by extracting a bounding image region for the bone from the preceding image frame \(l^{k-1}\) (Figure 2B). The border of the template was determined by the outermost projection boundary of the registered surface bone model (Figure 2A). The template \(\mathrm{M}_{j}\) was allowed to approximately match the succeeding image frame \(l^{k+1}\) by adjusting its in-plane positions and rotation \(\overset{\rightarrow}{p}^{j}=\left\{t_{x_{j}},t_{y_{j}},r_{z_{j}}\right\}^{T}\), as \(l^{k-1}\) and \(l^{k+1}\) were both acquired from the \(j\)th X-ray unit (Figure 2B). A sampling importance resampling particle filter [47] was employed to perform template matching and motion tracking (number of particles = 3000), in which the state variables of particle \(i\) at frame \(k-1\) involved the planar pose parameters of the template on the 2D image plane and their first derivatives \(\overset{\rightarrow}{S}_{j,i}^{k-1}=\left\{\overset{\rightarrow}{p}_{j,i}^ {k-1},\overset{\rightarrow}{p}_{j,i}^{k-1}\right\}\). The particle state evolved from time steps of \(l^{k-1}\) to \(l^{k+1}\) via a linear kinetic model, which yielded a set of \(\overset{\rightarrow}{S}_{j,i}^{k+1}=\left\{\overset{\rightarrow}{p}_{j,i}^ {k+1},\overset{\rightarrow}{p}_{j,i}^{k+1}\right\}\). Evolved particles were then resampled [47] in accordance with the normalized weight of the particles, which led to discarding particles with smaller weight. The weight of particle \(i\), \(\omega_{i}\), was defined by the probability density function value of a normal distribution function, evaluated using the mean absolute difference between \(\mathrm{M}_{j}\) and \(l^{k+1}\):  
\[\omega_{i}=\mathrm{N}\Big{(}e_{i}\big{|}\mu,\sigma\Big{)}, \tag{1}\]  
\[e_{i}=\frac{1}{\big{|}\mathrm{M}_{j}\big{|}}\sum_{q\in\mathrm{M}_{j}}\left|D \Big{(}\mathrm{M}_{j}(q),\overset{\rightarrow}{p}_{j,i}^{k+1}\Big{)}-l^{k+1 }(q)\right|, \tag{2}\]  
where \(\mid\mid\) denotes the number of pixels, and \(D\) represents the function that displaces template \(\mathrm{M}_{j}\) given a template pose \(\overset{\rightarrow}{p}_{j,i}^{k+1}\). The final template 2D pose \(\overset{\rightarrow}{p}_{j}^{k+1}=\left\{t_{x_{j}}^{k+1},t_{y_{j}}^{k+1},r_{ z_{j}}^{k+1}\right\}\) was estimated by averaging the poses \(\overset{\rightarrow}{p}_{j,i}^{k+1}\) of the resampled particles. After template registration, the bone template was updated by linearly interpolating the intensity values from \(l^{k+1}\) at the grid points of the template. Figure 2: Schematic representation of the 2D/2D template registration procedure. (**A**) The surface model at the registered pose in the preceding frame was projected onto the image plane, from which (**B**) a bone template, M\({}_{j}\), determined by the outermost boundary, was extracted and used to match the succeeding image frame with a particle filter. (**C**) The 2D translations and rotations on the two image planes estimated from the template registration were incorporated to update the pose parameters of the bone model.