## 2 Materials and Methods
### Multi-Shape Free-Form Deformation (MSFFD)  
This paper uses free-form deformation (FFD) [15; 16] to transmit the deformation of the vulva and cervix to the HMD. FFD has been widely used in the field of computer graphics to manipulate high-resolution meshes with a small number of control points. The target mesh is embedded in a lattice of control points, as shown in Figure 2. A mapping is defined between the control points and the surface. Surface vertices \(S\) can be represented by a mapping matrix \(M\) and a control points vector \(P\) as shown in Equation (1). \[S=MP \tag{1}\]Due to the compact nature of FFD, it is possible to substantially reduce the amount of data transmission to the HMD by delivering control points of the surface mesh instead of the surface mesh itself. The surface mesh is then reconstructed in the HMD from the received control points and a mapping matrix. The deformation of the surface mesh and corresponding haptic feedback are both computed in a server PC. Forces resulting from the interaction between the virtual instrument and the mesh are sent to a haptic device for producing haptic feedback. Control points are updated by an inverse mapping matrix and the deformed surface mesh. The inverse mapping matrix can be obtained in advance by the pseudo inverse, as shown in Equation (2). \[P=\left(M^{\mathrm{T}}M\right)^{-1}M^{\mathrm{T}}S \tag{2}\]  
The surface mesh can be updated and rendered on the HMD for visual feedback by using the received control points and a prestored mapping matrix. In the case of medical training simulators, organ models may be deformed by interaction with virtual instruments or virtual hands or as part of the training scenario. We categorise deformations into local deformation that occurs as the result of user interaction (Figure 3 bottom) and global deformation that changes the complete shape according to the simulation scenario (Figure 3 top). In the case of VE, the posture of the baby and the cervix change, with the cervix opening dilating as labour proceeds. A simulation state is defined by the diameter of the cervix and the posture of the cervix and baby's head. Figure 1: Interaction between a user avatar and vulva model (**top**) and internal view (**bottom**) of a pregnant woman model in AR-based training simulation of vaginal examination. Figure 2: Vulva mesh model (**left**) embedded in a lattice of control points (grey). Cervix mesh model (**middle**, red) embedded in a lattice of control points (yellow). Cervix mesh deformed by moving the control points (**right**). Local deformation can be updated on the HMD by transmitting the position of the control points computed from the deformed surface mesh. Global deformation, however, is difficult to represent by updating the relatively small number of control points because there are not enough degrees of freedom (DoF) to fully represent the deformation. Figure 4 shows the difference between the dilated cervix model and the model reconstructed from the control points, which are computed by an inverse mapping matrix in the vaginal examination training simulation. The red surface indicates the resulting deformed surface mesh in the server PC. Blue points indicate surface vertices reconstructed in the HMD using 32 control points computed from the deformed surface mesh and the inverse mapping matrix. The root mean square error between two surfaces is 4.70 mm. The error is higher at the cervix entrance. The diameter of the cervix entrance (red) is 30 mm, but the diameter of the reconstructed cervix is 10 mm. One of the essential skills of the vaginal examination is assessing cervix dilation in cm by using two fingers. Incorrect visual feedback can confuse trainees as a result of the discrepancy between the visual and haptic feedback, which in turn  
Figure 4: The difference (**c**) between the reconstructed surface (**a**) and the original surface (**b**). Figure 3: Global deformation—dilation of the cervix (**top**). Local deformation—user manipulation (**bottom**).