# Automatic 3D MRI-Ultrasound Registration for Image Guided Arthroscopy
## 4 Results and Discussion
It can therefore be inferred that incorporating the distance transform step in the algorithm helps achieve a more accurate global registration. To demonstrate the step-wise decreasing trend in the registration error from Table 2, the registration output at the end of each of the four stages for subject 5 overlaid on their respective ground truths is shown in Figure 6. Figure 6a shows the initial gap between the initial position of the US image and the ground-truth solution provided by the expert and Figure 6b shows the position after the pre-initialization, and it can be seen that this step provides a good initial alignment. The last two steps, Figure 6c,d, help with further fine-tuning and lowering the registration error. It can be spotted from Figure 6d that the patella, the medial femoral cartilage and the patellar shadow of the output red slice overlap almost perfectly with blue ground truth slice. The same registration result is overlaid onto the respective MRI of subject 5 and shown in Figure 7. Furthermore, to assess the efficiency of LC\({}^{2}\), randomized trials were executed on all 11 data sets, repeating the rigid registration for randomly displaced starting positions, i.e., within the range of \(\pm 10\) mm/\({}^{\circ}\) for all six parameters resulting in a total of 1100 registration attempts. Tracing the RMS error and the value of LC\({}^{2}\) metric, we observed that, in many cases, LC\({}^{2}\) is resilient to perturbations in initialization, and hence is an appropriate measure especially for a multimodal 3D MRI-US registration in a realistic clinical setup. The results for all subjects are shown in Figure 8, including the percentage of converged executions. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Angle**} & \multirow{2}{*}{**Patient No.**} & \multicolumn{4}{c}{**Translation Error (mm)**} & \multicolumn{4}{c}{**Rotation Error (\({}^{\circ}\))**} & \multirow{2}{*}{**Final Error(t,r)**} \\ \cline{3-3} \cline{5-10}  & & **In-Err** & & & & & & & & & \\ \hline
0\({}^{\circ}\) & Subject 1 & 34.72 & 9.83 & 6.04 & 2.00 & 73.70 & 20.13 & 28.67 & 5.50 & 2.00 mm, 5.50\({}^{\circ}\) \\
0\({}^{\circ}\) & Subject 2 & 35.39 & 10.59 & 14.62 & 3.94 & 62.14 & 21.01 & 14.47 & 8.78 & 3.94 mm, 8.78\({}^{\circ}\) \\
0\({}^{\circ}\) & Subject 4 & 39.10 & 13.45 & 2.67 & 1.72 & 60.28 & 8.94 & 4.93 & 3.91 & 1.72 mm, 3.91\({}^{\circ}\) \\
0\({}^{\circ}\) & Subject 5 & 32.09 & 7.43 & 3.88 & 1.47 & 65.88 & 13.29 & 17.26 & 4.15 & 1.47 mm, 4.15\({}^{\circ}\) \\
0\({}^{\circ}\) & Subject 6 & 31.68 & 11.04 & 17.74 & 2.43 & 67.83 & 33.15 & 13.41 & 8.67 & 2.43 mm, 8.67\({}^{\circ}\) \\
30\({}^{\circ}\) & Subject 3 & 26.14 & 12.05 & 1.43 & 1.41 & 45.37 & 32.13 & 2.41 & 2.34 & 1.41 mm, 2.34\({}^{\circ}\) \\
30\({}^{\circ}\) & Subject 6 & 31.89 & 30.06 & 9.19 & 4.80 & 25.88 & 35.29 & 12.56 & 9.03 & 4.80 mm, 9.03\({}^{\circ}\) \\
60\({}^{\circ}\) & Subject 3 & 27.99 & 17.75 & 8.70 & 5.09 & 55.21 & 7.81 & 15.80 & 6.43 & 5.09 mm, 6.43\({}^{\circ}\) \\
60\({}^{\circ}\) & Subject 6 & 27.71 & 7.49 & 7.45 & 6.00 & 27.94 & 18.70 & 9.70 & 5.80 & 6.00 mm, 5.80\({}^{\circ}\) \\
90\({}^{\circ}\) & Subject 3 & 33.13 & 22.79 & 5.36 & 3.63 & 25.82 & 33.29 & 26.46 & 15.07 & 3.63 mm, 15.07\({}^{\circ}\) \\
90\({}^{\circ}\) & Subject 6 & 34.50 & 12.15 & 3.58 & 6.03 & 25.09 & 12.53 & 17.62 & 16.04 & 6.03 mm, 16.04\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Table showing a decreasing trend in the translation and rotation error (left to right) for each subject computed after each step of the proposed registration algorithm. In-Err: Initial error, Pre-in: Error after pre-initialization, DT: Error after applying Distance Transform, LC\({}^{2}\): error after final LC\({}^{2}\) registration. Figure 5: Surface plots of LC\({}^{2}\) cost function against rotation parameters when (**a**) registration is performed directly after Pre-in, without including the DT step and (**b**) registration is performed with the DT step.