# Measurement-Driven Synthesis of Female Digital Mannequin Using Convex Sub-Volumes
## 3 Materials and Methods
### Rationale for the Mathematical Model  
In order to obtain a parametric mannequin, a set of pre-processing and processing steps are carried out. The procedural enumerations and descriptions of these steps are addressed in Sections 3.2-3.4 for pre-processing and Sections 3.5 and 3.6 for processing. The current subsection intends to highlight the main mathematical and scientific reasoning that provide support for such steps.  
**Geometrical and Topological Pre-Processing**  
Pre-processing seeks to facilitate the bijective mapping ellipsoid set \(\leftrightarrow\) tailor measurements by reducing (a) mathematical complexity and (b) numerical unstability. Pre-processing (1) manually constructs an ellipsoid set (\(BM\)) that approximates a mesh-based model, (2) reduces the degrees of freedom of the ellipsoid set, and (3) lowers the noise level within the ellipsoid set. Pre-processing consists of:  
1. Selection of ellipsoids due to their low geometrical complexity, low data-storage, and the benefits of convex volumes present in physics engines;
2. Manual approximation of the 3D reference model by fitting a set of ellipsoids;
3. Symmetrical location of ellipsoids with respect to the sagittal plane;
4. Construction of a set of Reference Mannequins through rigid transformations of the set of ellipsoids;
5. Compliance of Reference Mannequins to sets of tailor measurements;
6. Data processing is performed over the set of Reference Mannequins in order to achieve: 1. Enforcement of symmetry with respect to the sagittal plane; 2. Enforcement of right-handedness in the ellipsoid coordinate systems; 3. Congruence of the coordinate system (definition of World Coordinate System); 4. Re-labeling of SO(3) coordinate systems (i.e., ellipsoid axes) to ensure minimal variation of the interpolation functions in set \(F\).  
Figure 1: Flow diagram of the convex-volume-based measurement-driven approximation. Input: Reference 3D model. Output: Measurement-driven mannequin \(B\). Steps 1 and 2 pre-processing. Steps 3 and 4 are processing.  
By definition, our approach seeks large-scale (ellipsoid approximation) geometrical information of the mannequin. This scope filters out small details and noise. In this particular manuscript, such filtering was manually executed by fitting the ellipsoid set to the Base Mannequin. For this process to be executed in a semi-automated or automated manner, we suggest these topics: (a) alpha shape fitting [37] and (b) signal denoising [38].  
**Processing; Bijective Mapping; Tailor Measurements \(\leftrightarrow\) Ellipsoid Sets**  
Many possible sets of ellipsoids could satisfy the measurements of a particular \(x\in A\), generating infinite \(m_{T_{i}}^{-1}\). Therefore, a particular \(F\subset m_{T}^{-1}\) has been selected, such that each \(B=F(x)\) satisfies the measurements of an arbitrary \(x\). \(F\) is then considered as the set of parametric functions that synthesizes a particular mannequin \(B\) given a desired set of measurements \(x\) and a set of known tuples of \(RM\) (set of Reference Mannequins) containing geometry and measurements.  
In addition to the previous precision made about the mapping from \(x\longrightarrow B\), a set of reasonings is presented for the processing steps (interpolation method). These reasonings seek to avoid instability (erroneous sizing and unnatural demeanor) of the synthesized mannequins (see Section 3.6).  
1. Selection of a multivariate weighted average (interpolation) method for the definition of \(F\);
2. Selection of inverse distance weighting (IDW) as the interpolation method due to its minimal tuning and boundary condition characteristics;
3. Limitation of the dimension of the measurement space to reduce the complexity of the interpolation scheme;
4. Segmentation of the interpolation functions dictated by mannequin neighborhoods;
5. Synthesis of neighborhoods by a subset of functions \(f_{i}\in F\) given a subset of measurements (e.g., shoulder width and height);
6. Definition of neighborhoods based on a measurement dependency rationale.