# Multi-Sensor Face Registration Based on Global and Local Structures
## 2 The Proposed Registration Method for Visible and Infrared Face Images
### Multi-Spectral Face Registration Using Global and Local Structures  
In order to estimate the mixture parameter \(\psi\) in the mixture probability density function (2), according to [22; 23], we first rewrite the function (2) as a complete data logarithm likelihood function \(\ln L_{\mathrm{C}}(\psi)=\sum\limits_{j=1}^{N}\ln\sum\limits_{i=1}^{M}w_{ij}f\! \!\left(x_{j}|y_{i},\Sigma_{i},v_{i}\right)\), and then use the Expectation Maximization (EM) to train the logarithm likelihood function. In general, the EM algorithm can be divided into the following two steps: Expectation step (E-step) and Maximization step (M-step), and the iterative process proceeds by alternating between the E- and M-steps until convergence. E-step: We first use the current parameter \(\psi^{(k)}\) to calculate the posterior probability \(\tau_{ij}{}^{(k)}\), and according to the Bayesian Theorem, it can be expressed in terms of the observation data \(x_{j}\) belonging to the \(i^{\mathrm{th}}\) component of the SMM  
\[\tau_{ij}{}^{(k)}=\frac{w_{ij}{}^{(k)}f\!\!\left(x_{j}|y_{i}{}^{(k)},\Sigma_{i }{}^{(k)},v_{i}{}^{(k)}\right)}{f\!\!\left(x_{j}|\psi^{(k)}\right)}, \tag{4}\]  
Next we need to calculate the \(k^{\mathrm{th}}\) iteration conditional expectation of the logarithm likelihood function \(\ln L_{\mathrm{C}}(\psi)\), which can be calculated as follows:  
\[\mathbf{Q}(\psi|\psi^{(k)})=\mathbf{Q}_{1}(\mathbf{w}|\psi^{(k)})+\mathbf{Q}_{2}(\mathbf{v}|\psi^{( k)})+\mathbf{Q}_{3}(\mathbf{y},\Sigma|\psi^{(k)}), \tag{5}\]where \(\mathbf{Q}_{1}\), \(\mathbf{Q}_{2}\) and \(\mathbf{Q}_{3}\) are respectively written as follows:  
\[ \begin{array}{l}\mathbf{Q}_{1}\Big{(}w|\psi^{(k)}\Big{)}=\sum\limits_{j=1}^{N} \sum\limits_{i=1}^{M}\tau_{ij}{}^{(k)}\ln w_{ij}\\ \mathbf{Q}_{2}\Big{(}v|\psi^{(k)}\Big{)}=\sum\limits_{j=1}^{N}\sum\limits_{i= 1}^{M}\tau_{ij}{}^{(k)}\Bigg{(}-\ln\Gamma\Big{(}\frac{v_{i}}{2}\Big{)}+\frac{ v_{i}}{2}\ln\Big{(}\frac{v_{i}}{2}\Big{)}+\frac{v_{i}}{2}\Bigg{\{}\sum\limits_{j=1}^{N} \left(\ln u_{ij}{}^{(k)}-u_{ij}{}^{(k)}\right)+\gamma\left(\frac{v_{i}^{(k)}+D }{2}\right)-\ln\left(\frac{v_{i}^{(k)}+D}{2}\right)\Bigg{\}}\Bigg{)}\Bigg{)} \Bigg{)}\,\\ \mathbf{Q}_{3}\Big{(}y,\Sigma_{i}|\psi^{(k)}\Big{)}=\sum\limits_{j=1}^{N}\sum \limits_{i=1}^{M}\tau_{ij}{}^{(k)}\Bigg{(}-\frac{D}{2}\ln(2\pi)-\frac{1}{2}\ln [\Sigma_{i}]+\frac{D}{2}\ln u_{ij}{}^{(k)}-\frac{u_{ij}{}^{(k)}|y_{ij}-y_{i}| ^{2}}{2\Sigma_{i}}\Bigg{)}\end{array} \]  
with the gamma in \(\mathbf{Q}_{2}\) that denotes the Digamma function, and the \(u_{ij}{}^{(k)}\) in \(\mathbf{Q}_{3}\) that represents the conditional expectation about additional missing data in the complete-data; it can be written as:  
\[u_{ij}{}^{(k)}=\frac{v_{i}{}^{(k)}+D}{v_{i}{}^{(k)}+D\Big{(}x_{j},y_{i}{}^{(k) };\Sigma_{i}{}^{(k)}\Big{)}}, \tag{6}\]  
In addition, we use the displacement function \(\rho\) to denote the non-rigid transformation in the point set: \(\boldsymbol{X}=\boldsymbol{T}(\boldsymbol{Y},\rho)=\boldsymbol{Y}+\rho( \boldsymbol{Y})\), and add the regularization term \(\boldsymbol{\varphi}(\boldsymbol{\rho})\) to the \(\mathbf{Q}_{3}\), which can enforce the smoothness of the displacement function in the alignment of two point sets. So this \(\mathbf{Q}_{3}\) term can be rewritten as:  
\[\widetilde{\mathbf{Q}}_{3}=\mathbf{Q}_{3}(\rho,\Sigma_{i}{}^{(k+1)}|\psi^{(k) })+\frac{\lambda}{2}\boldsymbol{\varphi}(\boldsymbol{\rho}), \tag{7}\]  
where \(\lambda\) is the regularization parameter and controlling the trade-off of two terms in Equation (7).