# Multi-Sensor Face Registration Based on Global and Local Structures
## 2 The Proposed Registration Method for Visible and Infrared Face Images
### Student's \(t\) Mixtures for Feature Point Set Registration  
In order to fully exploit the global structures in the feature point sets, we treat the alignment of two point sets as a probability density estimation of an SMM distribution with the IDSC local feature descriptors. Compared to the GMM distribution, the main advantage of SMM is that Student's t distribution has a heavier tail than the exponentially decaying tail of Gaussian distribution. As is shown in the Figure 3, each component of the SMM has an additional parameter \(\upsilon\) called the degrees of freedom, which can change the shape of a Student's t distribution curve, and hence SMM provides a more robust model than the classical GMM.  
Figure 3: The Student’s t distribution for various degrees of freedom.  
Figure 2: Schematic illustration of face silhouette feature descriptors based on Inner-Distance Shape Context (IDSC). (**a**) Bellman-Ford shortest path graph built using face silhouette landmark points; (**b**,**c**) Four marked points and their corresponding IDSC feature histograms.  
Given a point set \(\mathbf{X}_{N\times D}=(x_{1},\,\ldots,\,x_{N})^{\mathrm{T}}\), which can be considered as an observation datum, the other point set \(\mathbf{Y}_{M\times D}=(y_{1},\,\ldots,\,y_{M})^{\mathrm{T}}\) is treated as the SMM centroids, where \(N\) and \(M\) represent the number of points in \(\mathbf{X}\) and \(\mathbf{Y}\), respectively, and \(D\) represents the dimension of each point. Then the goal of registration is to align the centroids point set \(\mathbf{Y}\) to the observation data point set \(\mathbf{X}\). Typically, the point sets contain noise and outliers, which can be supposed to be a uniform distribution \(1/N\), and the weight of the uniform distribution is denoted as \(\eta\in[0,\,1]\). Let \(w_{ij}\in[0,\,1]\) denote a mixing proportion corresponding to the \(i^{\mathrm{th}}\) component of the SMM for point \(x_{j}\,(\sum\limits_{i=1}^{M}w_{ij}{=}\,1)\), then the mixture probability density function can take the form:  
\[f(\mathbf{x}|\psi)=\eta\frac{1}{N}+(1-\eta)\sum\limits_{i=1}^{M}\omega_{ij}f(x_{j }|y_{i},\Sigma_{i},v_{i}), \tag{2}\]  
where \(\psi=(\psi_{1},\,\psi_{2},\,\ldots,\,\psi_{M})\) is mixture parameter set with \(\psi_{i}=(w_{ij},\,y_{i},\,\sum_{i},\,v_{i})\), and \(\mathbf{f}\) (\(x_{j}|y_{i},\,\sum_{i},\,v_{i}\)) is the Student's t distribution probability density function for the \(i^{\mathrm{th}}\) component of SMM; we express that:  
\[f\!\!\left(x_{j}|y_{i},\Sigma_{i},v_{i}\right)=\frac{\Gamma\!\left(\frac{v_{ij }+D}{2}\right)\!\left|\Sigma_{i}\right|^{-\frac{1}{2}}}{(\pi v_{ij})^{\frac{1} {2}D}\Gamma\!\left(\frac{v_{i}}{2}\right)\!\left[1+\frac{d\left(x_{j},y_{i} \Sigma_{i}\right)}{v_{i}}\right]^{\frac{v_{i}+D}{2}}}, \tag{3}\]  
where \(d(x_{j},\,y_{i};\,\sum_{i})\) is the Mahalanobis squared distance between observation point \(x_{j}\) and centroid point \(y_{i}\), while \(\sum_{i}\), \(\Gamma\) and \(v\) represent the covariance matrix, Gamma function and degrees of freedom, respectively.