# Automatic 3D MRI-Ultrasound Registration for Image Guided Arthroscopy
## 4 Results and Discussion  
To evaluate the performance of the proposed algorithm, an experienced orthopaedic surgeon [PERSON], manually registered the MRI-US pairs used in this study. The registrations were performed in ImFusion software. The MRI-US volumes were rotated/translated in a common reference system by selecting the rotation/translation values associated with each volume along three orthogonal planes. The software allowed users to visualize the two volumes simultaneously directly while the rotation/translation values were modified by the user. The MRI volume is kept fixed, while the US volume is rigidly moved to overlay the tibia and the femur surfaces in the two modalities along the three orthogonal planes. The transformation obtained as a result of this process which the surgeon ([PERSON]) performed manually is considered as the ground truth transform. For every registration, the three translation (mm) and rotation (degree) parameters were obtained from the transform. A similar kind of transform is obtained as a result of registering the two modalities using the proposed algorithm. In-order-to calculate the registration error, the MRI volume is fixed in the position selected in the manual registration performed by the surgeon and the transformation parameters of the US volumes (translation and rotation) are compared as in [27]. The norm of the difference vector of the parameters, computed using Equations (5) and (6), give the translation (\(t\)) and rotation (\(r\)) error values for each MRI-US pair,  
\[t=\sqrt{(x_{GT}-x_{o})^{2}+(y_{GT}-y_{o})^{2}+(z_{GT}-z_{o})^{2}} \tag{5}\]  
\[r=\sqrt{(\theta_{GT}-\theta_{o})^{2}_{yz}+(\theta_{GT}-\theta_{o})^{2}_{xz}+( \theta_{GT}-\theta_{o})^{2}_{xy}} \tag{6}\]  
where subscript \(GT\) refers to the ground truth parameter value, and subscript \(o\) refers to the outputted parameter value from the developed registration method. \(x\), \(y\), \(z\) represent translation parameters in each plane and \(\theta\) represents rotation parameter for \(yz\), \(xz\), \(xy\) planes. The proposed algorithm is tested on 11 MRI-US pairs obtained from six volunteers at four different knee flexion angles. The transform obtained at the end of each of the three stages of the algorithm is compared against the ground truth transform, and the translation/rotation error is computed using Equations (5) and (6). Table 2 shows the detailed evaluated error values at each stage from the registration method developed. To calculate the initial error, the transforms of US and MRI volumes are set to identity. It can be clearly seen from Table 2 that there is a decreasing trend in the registration error and that each stage of the registration contributes notably to reduce this error. The pre-initialization step alone, in particular, significantly brings down the initial error by 76.8%, thereby assisting the subsequent steps in the algorithm to achieve global registration more efficiently. Furthermore, to estimate the importance of including the distance transform step, a cost function analysis is performed. The final cost function, which determines the accuracy of the registration, LC\({}^{2}\), is plotted against any two parameters from the initial transform. We selected the rotation x, rotation z parameters from the output transforms of both pre-in and DT stages, varied their values within the range \(\pm 10^{\circ}\) and plotted against the LC\({}^{2}\) metric for a given subject. The resulting curves (Figure 5) revealed that the inclusion of distance transform produces a more peaked objective surface (Figure 5b), which would enable better convergence, whereas, without the distance transform (i.e., using LC\({}^{2}\) directly on volumes), the objective surface is undulated with no clear optimum (Figure 5a). In Figure 5b, a global peak is seen with maximum LC\({}^{2}\) value of 0.2, whereas, in Figure 5a, the maximum LC\({}^{2}\) value is observed to be a little lesser with minor local optimum.