# Ultrasound Navigation for Transcatheter Aortic Stent Deployment Using Global and Local Information
## 2 Materials and Methods
### A Global 3D Navigation Map Based on 2D US-3D MR Image Registration
#### 2.2.1 Multi-Feature Particle Filter Tracking Method for the 3D Trajectory of the Catheter  
The calibrated 2D US probe scans the aortic phantom in consecutive cross-sections and results in a series of axis cross-sections of aorta. At the beginning of tracking, an initial tracking region on  
Figure 3: The workflow of local navigation information (3D aortic model including morphological information) based on US images tracking and IVUS image segmentation. MA: media-adventitia. Figure 2: 2D (two-dimensional) US-3D (three-dimensional) MR images registration based on calibrations information of 2D and 3D US probes. the first US image is manually delineated. The delineated region is described as a circle because the outline of aortic cross-section is circular. In addition, because the fixed radius of the aortic phantom is 4.5000 mm and the resolution of 2D US image is 0.2382 mm/pixel, the radius of initial circle tracking region is set as 19 pixels. The cross-sections of the catheter are contained in aortic cross-sections. The intensity of the catheter's cross-section is much higher than for the other region inside the aortic cross-section. Therefore, once the aortic cross-section is tracked in US images, the position of the cross-section of the catheter can be acquired by using a threshold (in our application, the threshold value is set as 220 on a 256 step scale). The tip of interventional catheter is located in the last US image frame where the catheter's cross-section can be detected. Then the positions of catheter's tip in 2D US images are transformed to the optical tracking system by using transformation \(\mathrm{T}^{\mathrm{TS}}_{\mathrm{2 DUS}}\). The 3D location points of the tip of the catheter in the optical tracking system are connected into a curve, which is the 3D trajectory. As a key step, tracking aortic cross-sections in 2D US images is difficult because there are various noises in US images. Particle filter method provides a robust predicting and tracking framework, because it approximates the object position with a finite set of weighted samples (or particles). Conventional particle filter method [21] uses a single intensity feature to calculate the weight of particle. To achieve more stable tracking of aortic cross-section in US images, we propose a multi-feature particle filter tracking algorithm as follows:  
``` Input: US scanning images \(\left\{\mathbf{I}_{1}\right\}_{t=1,2,3...}\) Output: Tracked cross-sections of aorta \(\left\{\mathbf{p}_{1}\right\}_{t=1,2,3...}\) (a) Initialization:(t = 0)  1: Generate initial target position \(\mathbf{p}_{0}\) (\(\mathbf{p}_{0}\) denotes the center of aortic cross-section in US image \(\mathbf{I}_{0}\)). 2: Scatter initial particles \(\left\{\,\mathbf{{}^{s}}\mathbf{p}_{0,m}\right\}_{m=1...M}\) (\(M\) is the number of particles) ( \(\,\mathbf{{}^{s}}\mathbf{p}_{0,m}=\mathbf{p}_{0}+\mathbf{g}_{m^{\prime}}\mathbf{g}_{m}\) is zero mean Gaussian random variables). (b) Particle state propagate:  1: Calculate new particle set \(\left\{\,\mathbf{{}^{s}}\mathbf{p}_{t,m}\right\}_{m=1...M}\) according to state propagation model and \(\left\{\,\mathbf{{}^{s}}\mathbf{p}_{t-1,m}\right\}_{m=1...M}\) (State propagation model is random drift model \(\,\mathbf{{}^{s}}\mathbf{p}_{t,m}=\,\mathbf{{}^{s}}\mathbf{p}_{t-1,m}+\mathbf{U}_{t}\), where \(\mathbf{U}_{t}\) is the Gaussian white noise). 2: Acquire region of the particles \(\left\{\,\mathbf{{}^{s}}\mathbf{p}_{t,m}\right\}_{m=1...M}\) (circle regions centered on position of the particles \(\left\{\,\mathbf{{}^{s}}\mathbf{p}_{t,m}\right\}_{m=1...M}\)) and template region \(\mathbf{R}_{t}\) (a circle region centered on the target position \(\mathbf{p}_{t-1,t}^{s}\)).