# 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
Oriented gradient OG is calculated by

\[\mathrm{OG}\left(i,j\right)=\sqrt{\mathrm{G}_{1}\left(i,j\right)^{2}+\mathrm{ G}_{2}\left(i,j\right)^{2}}\cdot\tan^{-1}\left(\mathrm{G}_{1}\left(i,j\right)/ \mathrm{G}_{2}\left(i,j\right)\right) \tag{2}\]

where \(\mathrm{G}_{1}\) and \(\mathrm{G}_{2}\) are gradients along the horizontal and vertical directions. The oriented gradient OG is counted into \(K_{2}\) bins and normalized HOG is

\[\mathbf{h}=\mathbf{h}/\left\|\mathbf{h}\right\|=\left[h_{1}\right/\sqrt{\sum \limits_{k=1}^{K_{2}}{{{h_{k}}^{2}}}},\ \cdots,h_{K_{2}}\Big{/}\sqrt{\sum\limits_{i=1}^{K_{2}}{{{h_{k}}^{2}}}} \tag{3}\]

where \(h_{k}\) is histogram counts of \(k^{\mathrm{th}}\) bin. The fusion feature \(\mathbf{f}\) which combines normalized HOI \(\mathbf{g}\) with HOG \(\mathbf{h}\) can be represented as

\[\mathbf{f}=\left[a\cdot\mathbf{g};\left(1-a\right)\cdot\mathbf{h}\right]\quad \mathrm{a}\in\left[0,1\right] \tag{4}\]

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