# Measurement-Driven Synthesis of Female Digital Mannequin Using Convex Sub-Volumes
## 3 Materials and Methods
### Cleaning of Reference Mannequin Geometries
Such processes intend to minimize said variation by maximizing the projection between analogous axes by: (1) **relabeling** the orientation axes (e.g., making \(\mathbf{u}=\mathbf{v}\) and \(\mathbf{v}=\mathbf{u}\)), and (2) **inverting** the directions of the newly labeled axes when needed. For the correction of each set of corresponding ellipsoids, one ellipsoid is selected as the reference, and the processes are executed over all the other four corresponding ellipsoids in the set. Note that the dimensions \(D\) associated with each axis are considered and also re-labeled accordingly. The orientation axes of each ellipsoid are obtained by performing principal component analysis (PCA) of the point-cloud. The resulting eigenvalues have a \(\pm\) sign, which does not allow the definition of a concrete tendency of direction for the eigenvectors  
Figure 11: World Coordinate System definition, registration process, and definition of pivot point (origin) of WCS. (**a**) Orthogonal views. (**b**) Isometric view. Figure 10: Anatomical planes defined for the Reference Mannequins. (orientation axes). Hence, when **re-labeling** is performed, the resulting axes may possess the maximum projections but with inverted direction (See Figure 12b). **Symmetry Correction**  
For the symmetry correction, both the central and lateral ellipsoids (ellipsoids with right or left distinction) are considered but approached differently. For instance, the central ellipsoids present asymmetry with respect to themselves through the sagittal plane. In comparison, lateral ellipsoids posses asymmetry in terms of the analogous ellipsoid in the right hemisphere of the sagittal plane possessing a deviation in the orientation, position, and/or dimensions. As a consequence, the following steps are performed in order to fix the asymmetry of the models:  
1. For lateral ellipsoids, the right ellipsoids are omitted, and the analogous left ellipsoids are mirrored with respect to the sagittal plane. 2. Due to the mirroring effect, the resulting coordinate systems are left-handed. Hence, the Z-axes have their direction inverted to preserve the SO(3) property. See Figure 13.
3. For central ellipsoids, the X-axes are defined orthogonal to the sagittal plane. 4. The Y-axes are projected to the sagittal plane, and the Z-axes are computed as \(\overrightarrow{Z}=\overrightarrow{X}\times\overrightarrow{Y}\). See Figure 14.  
Note that for Figures 12-14 some parts of the ellipsoids are rendered in front of others despite really being occluded. Consider this as simply a rendering order issue with transparent objects in the rendering engine. Figure 12: Minimization of the variation between orientations of corresponding ellipsoids with **re-labeling** and **direction inversion**. Yellow ellipsoids, 13.a Rectus Abdominalis from Tall Chubby (reference \([\mathbf{u}_{\text{c}},\mathbf{v}_{\text{c}},\mathbf{w}_{\text{c}}]\)) and Tall Slim (modified orientation \([\mathbf{u},\mathbf{v},\mathbf{w}]\)). (**a**) Incorrect status for axes \(\mathbf{u},\mathbf{v},\mathbf{w}\). (**b**) **Re-labeled** axes \(\mathbf{u},\mathbf{v},\mathbf{w}\) in (**a**). Correct labeling, incorrect direction for axes \(\mathbf{u},\mathbf{v},\mathbf{w}\). (**c**) **Direction inversion** of axes \(\mathbf{u},\mathbf{v},\mathbf{w}\) in (**b**). Correct status for axes \(\mathbf{u},\mathbf{v},\mathbf{w}\). Figure 13: Correction of lateral ellipsoid symmetry of ellipsoids from Tall Chubby with only legs displayed for the example. (**a**) Asymmetric model. (**b**) Symmetric model.