# Measurement-Driven Synthesis of Female Digital Mannequin Using Convex Sub-Volumes
## 3 Materials and Methods  
The proposed methodology consists of approximating a 3D model with convex volumes (ellipsoids and cushions), which we call a 3D mannequin. This mannequin is parameterized (synthesized as a function of tailor measurements) in order to obtain measurement-driven 3D mannequins. Such mannequins satisfy the desired measurements with a minimum margin of error. The main steps performed to produce the measurement-driven mannequins are as follows (see also Figure 1):  
1. A reference 3D model is manually approximated/traced with ellipsoids to produce a Base Mannequin (\(BM\)). Furthermore, an ellipsoid-to-cushion post-process is presented to improve the convex volume approximation after it has been parameterized. See Section 3.2.
2. The \(BM\) (set of ellipsoids) is then used to instance a set of Reference Mannequins (\(RM\)). Each one of these Reference Mannequins (\(RM_{i}\)) satisfies a particular set of five tailor measurements. See Section 3.3.
3. Based on the known data of \(RM\) (geometry and measurements), a set of functions \(F(Me)\) is created. This set of functions synthesizes the geometry of a measurement-driven mannequin that in return satisfies the given measurements in \(Me\). See Section 3.6.
4. Finally, the measurement-driven mannequin can be synthesized by computing \(F(Me)\).  
**Problem Specification**  
These steps are explained in more detail in the following subsections (see Sections 3.2, 3.3 and 3.6). The problem of finding the parameterization for the 3D mannequins can be specified per the following **given** and **goal**:  
**Given:**  
1. \(A\): A set of points in \(\mathbb{R}^{q}\). Each component \(j\) of a point \(x\) (\(x\in A\)), with \(j=1,2,\ldots,q\), represents a measurement of the human body in \(Me\) (e.g., height, shoulder width, etc.). The following conditions must be considered: 1. \(x_{j}>0\), \(\forall\,j=1,2,\ldots,q\). 2. \(L\): A set of \(q\) ordered pairs corresponding to the lower and upper limits of the components \(j\) of \(x\). \(L:\{l\,|\,l_{1,2}>0\), \(l_{1}<l_{2}\,\}\)
2. \(H\): A set of sets of ellipsoids in \(\mathbb{R}^{3}\). Each set of ellipsoids \(B\), with \(B\in H\), approximates the shape and size of a human body described by a point \(x\in A\). 1. Each \(B\) has the same number \(m\) of ellipsoids and the same topology. 2. Each \(B\) satisfies the measurements of a specific point \(x\in A\). 3. Each ellipsoid of \(B\) is defined by a homogeneous SO(3) coordinate system \(S\) and a triad \(D\) describing the dimensions of its semi-axes.  
**Goal:**  
1. \(m_{T}\): Function \(m_{T}:H\longrightarrow A\); \(m_{T}\) obtains the tailor measurements \(Me\) of any model \(B\in H\).
2. \(F\): A set of functions \(F:A\longrightarrow H\), \(F=m_{T}^{-1}\). More specifically, \(F:x\longrightarrow B\), such that \(m_{T}\) applied to \(B=F(x)\) produces the measurements in \(x\). \(F\) is a set of functions that, given a set of measurements \(x\), synthesizes a model (geometry) \(B\) that satisfies \(x\).  
From Figure 1 it must be taken into consideration that Steps 1 through 3 are performed only once so as to produce \(F\). The set of functions \(F\) is then computed each time a new mannequin is synthesized from defined measurements \(Me\). It is also worth noting that the subject approximated by convex volumes in this manuscript is the female body. The same process could be applied to approximate the male body.