## 2 Materials and Methods
### AR-Based Surgical Navigation
#### 2.2.1 Registration  
Our registration approach is based on our previous work on radiation-free surface digitization of the spinal anatomy [47] using a 3D printed pointing device (PD, Figure 5, step 1). The PD was designed such that a sterile marker (Clear Guide Medical, Baltimore, MD, USA) could be attached to it at a predefined position. The marker showed an AprilTag [52,63] pattern, which could be tracked by the two front-facing cameras of  
Figure 5: Workflow of the augmented reality (AR)-based surgical navigation. Please note that the pelvic bone in this image is shown opaque, as different rendering options could be chosen by the surgeon. The impression of visual misalignment is caused by the stereo-rendering technique which generates two images optimized to the left and right eye of the user. the HoloLens as described in the following. Please refer to Figure 6 for an overview of coordinate systems and transformations used in this subsection. Each camera detected the AprilTag using the detection functionality (pipeline of thresholding, contour filtering, bits extraction, marker identification and corner refinement) of the ArUco library [65; 66], which yields four 2D corner coordinates in a sorted manner for each image (left and right). The corner coordinates are then mapped to the respective image planes. For each corresponding corner in the left and right image, a ray from the camera origin through the image plane is formed and the closest point between the two rays is found by triangulation. The result consists of four 3D point estimates, one for each corner. However, these point estimates usually do not form a perfect square and are not coplanar. To solve this problem, the known 3D ground truth shape of the marker is mapped to the point estimates by applying [PERSON]'s absolute orientation method [67]. The found solution is then used as the 6 DoF marker pose. Due to the known geometry of the PD, fabricated with a printing tolerance of 0.1 mm, its tip position in world space can be inferred without the need of manual calibration by \(T_{\text{IntInt}}T_{\text{HP}}\) (Figure 6).In the first part of the registration procedure - the initialization-the surgeon manually captured the three preoperatively defined landmarks (Figure 2, step 4) at the lipophilic eminence (Figure 2, step 4: L1), the anterior inferior iliac spine (AIIS) (Figure 2, step 4: L2) and at the deepest point of convexity between anterior superior (ASIS) and inferior iliac spine (AIIS) (Figure 2, step 4: L3) using the PD (Figure 5, step 1). Applying [PERSON]'s absolute orientation method [67] to the intraoperatively captured landmarks and their preoperatively defined counterparts led to an initialization of \(T_{\text{IntPre}}\) (Figure 6) that was visually verified (Figure 5, step 2). If the result was not satisfactory, the initialization step could be repeated. Otherwise, the surgeon continued by sampling 10 to 15 additional points from accessible and well-suited areas on the pelvic bone, namely the area between superior and inferior iliac spine, the ala ossis liii, the quadrilateral space or the pelvic brim (Figure 5, step 3, green area). Starting from the initialization, these points were then registered to the preoperative 3D model by repeating the two ICP [68] steps being (1) building pairs of nearest neighbors using the KD-tree implementation of ALGLIB Project, Nizhny Novgorod, Russia) and (2) applying  
Figure 6: Overview of coordinate systems and transformations. A transformation from coordinate system A to coordinate system B is denoted as \(T_{\text{BA}}\). \(CS_{\text{Pre}}\) is the preoperative planning space defined by the computed tomography coordinate system and \(CS_{\text{Int}}\) is the intraoperative space defined by the HoloLens on application startup. \(CS_{\text{Plp}}\) is the pose of the planning in \(CS_{\text{Pre}}\). \(CS_{\text{TrgPre}}\) and \(CS_{\text{TrgPost}}\) are the pre- and planned postoperative fragment poses in \(CS_{\text{Pre}}\). \(T_{\text{IntPre}}\) is the final registration.