# Measurement-Driven Synthesis of Female Digital Mannequin Using Convex Sub-Volumes
## 3 Materials and Methods
### Synthesis of Mannequins from Tailor Measurements  
Now that a set of Reference Mannequins has been manually (see Section 3.3) created and made consistent (see Section 3.4), the synthesis of a 3D mannequin \(B_{f}\) (set of convex volumes) from input tailor measurements \(Me_{f}\) can be achieved. Such synthesis is accomplished by computing a set of functions \(F(Me_{f})\). The synthesized mannequin (\(B_{f}\)) is called a measurement-driven mannequin, and the construction of \(F\) corresponds to the parameterization of the convex volume approximation of the mannequins. The synthesized mannequins are displayed in gray in order to differentiate them from \(RM\) and \(BM\). **Parameterization Scheme--Creation of the Set \(F\). **  
The parameterization scheme consists of building the set of functions \(F(Me)\) that synthesizes the geometry (position, orientation, and dimensions of each ellipsoid) of model \(B_{f}\) from input tailor measurements \(Me_{f}\), such that \(B_{f}\) satisfies \(Me_{f}\). Each function \(f_{i}\in F(Me)\) synthesizes a geometrical property (\(S\) or \(D\)) of a particular ellipsoid \(E_{i}\) of \(B_{f}\). As a consequence, there are \(2k\) functions in \(F\), with \(k\) the number of ellipsoids in \(BM\). Take into account that each \(f_{i}\in F(Me)\) is an interpolating function (weighted average) with the known tuples (geometry and measurements) of \(RM\) as parameters and the input measurements (\(Me_{f}\)) as the variable. Note that each ellipsoid \(E_{i}\in B_{f}\) is synthesized by interpolating only the geometrical information (\(S\) and \(D\)) of its corresponding ellipsoids in \(RM\) (see Section 3.4 for a definition of corresponding ellipsoids). To understand the definition of the interpolation method, take into consideration the following definitions:  
1. \(RM\): set of \(n\) Reference Mannequins as a set of tuples \(RM=\{(Me_{1},B_{1}),(Me_{2},B_{2}),\ldots,(Me_{n},B_{n})\}\). 2. \(Me_{j}\): point in \(\mathbb{R}^{5}\) containing the measurements of the \(j\)th Reference Mannequin \(B_{j}\), \(Me_{j}=[He_{j},Sh_{j},Br_{j},Wa_{j},H_{j}]\). 3. \(B_{j}\): Set of ellipsoids \(B_{j}=\{E_{1},E_{2},\ldots,E_{k}\}\). \(B_{j}\in RM\), with 1. \(k\): the number of ellipsoids defined in the \(BM\); 2. \(E_{i}=[D_{i},S_{i}]\) : the lilsoid defined by dimension and coordinate system (position and orientation), respectively. 4. \(Me_{f}=[He,Sh,Br,Wa,Hi]\) : measurements of target model \(B_{f}\) describing the shape of the desired female body. 5. \(B_{f}\): Target model as a set of ellipsoids \(B_{f}=F(Me_{f},h)\). **Measurement Dependency. **  
Considering that each ellipsoid approximates a particular part of the body, it is rational to think that not every ellipsoid in \(B_{f}\) will influence every single measurement in \(Me_{f}\) and vice versa. Thereby, each ellipsoid should not be synthesized as \(f(Me_{f})\) but as a function of a subset of measurements \(me_{f}\subset Me_{f}\) that influences it. In other words, each neighborhood of the model (associated subset of ellipsoids) has a unique subset of interpolating functions given by the measurements that influence the specific neighborhood (see Figure 16). For example, the measurement \(Sh\) has no influence on ellipsoids \(E\) from the waist neighborhood (e.g., 11.a_Abdominals). As a consequence, \(Sh\) is not an input nor a parameter for the interpolation function \(f\) of the waist ellipsoids, but \(Wa\) and \(He\) are. The construction of specific interpolating functions that depend on subsets of measurements (\(me\)) rather than full sets (\(Me\)) is called the segmented interpolation scheme. **Interpolation Method**  
As stated previously, the set of functions \(F\) consists of interpolating functions (\(f\)). The functions \(f\) perform a weighted average of the known geometric data of \(RM\) to produce the geometry of a new mannequin \(B_{f}\). The weights are calculated as a function of the distance between points in space representing the measurements \(Me_{j}\) of each mannequin in \(RM\) and the input measurements \(Me_{f}\).