# 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
By reproducing Kernel Hilbert Space and Fourier transformation, the displacement function \(\rho\) can be denoted as:  
\[\rho\Big{(}y_{j}\Big{)}=\sum\limits_{i=1}^{M}h_{i}\boldsymbol{G}\Big{(}y_{i}, y_{j}\Big{)},\ \boldsymbol{G}\Big{(}y_{i},y_{j}\Big{)}=e^{-\frac{1}{2}\|\frac{\eta^{i}-y_{i}}{ \beta}\|^{2}}, \tag{8}\]  
where \(\boldsymbol{G}\) is a Gaussian kernel matrix, \(\beta\) determines the width of the smoothing Gaussian filter, and the Equation (8) can be conveniently denoted in matrix form as: \(\rho(\boldsymbol{Y})=\boldsymbol{G}\boldsymbol{H}\), where \(\boldsymbol{H}_{M\times D}=(h_{1},\ldots,h_{M})\) is a matrix of coefficients. Thus, the Equation (7) can be expanded in the following matrix form:  
\[\widetilde{\mathbf{Q}}_{3}=\sum\limits_{j=1}^{N}\sum\limits_{i=1}^{M}\tau_{ij }{}^{(k)}\Bigg{(}-\frac{D}{2}\ln(2\pi)-\frac{1}{2}\ln\lvert\Sigma_{i}\rvert+ \frac{D}{2}\ln u_{ij}{}^{(k)}-\frac{u_{ij}{}^{(k)}\|x_{j}-y_{i}\|^{2}}{2\Sigma_ {i}}\Bigg{)}+\frac{\lambda}{2}tr\Big{(}\big{(}\boldsymbol{H}^{T}\big{)}^{(k)} \boldsymbol{G}\boldsymbol{H}^{(k)}\Big{)}, \tag{9}\]  
M-step: On the M-step at the \((k+1)^{\text{th}}\) iteration of the EM algorithm, considering that \(\mathbf{Q}_{1}\), \(\mathbf{Q}_{2}\) and \(\mathbf{Q}_{3}\) in Equation (5) can be computed independently of each other, thus the maximization of the objective function \(\mathbf{Q}_{1}\), \(\mathbf{Q}_{2}\) and \(\mathbf{Q}_{3}\) with respect to the parameters \(w\), \(v\), \(\sum\), \(\rho\) can be operated separately. The mixing proportion for SMM is updated by our consideration of the first term \(\mathbf{Q}_{1}\). Given the feature descriptors in Section 2.1, it can obtain the coarse correspondences \(\boldsymbol{\Omega}\) between the point sets \(\boldsymbol{X}\) and \(\boldsymbol{Y}\). For an observation data \(x_{j}\), we define \(\iota\) (\(0\leq t\leq 1\)) as a confidence by IDSC feature matching, and then the mixing proportion \(w_{ij}\) by incorporating the local structures among neighboring points can be updated by the following rule:  
\[w_{ij}{}^{(k+1)}=\left\{\begin{array}{ll}\sum\limits_{j=1}^{N}\frac{\tau_{ij }{}^{(k)}}{N}\,,\ \exists\Big{(}x_{j}\to y_{i}\Big{)}\in\boldsymbol{\Omega}\\ \iota\,,\Note that in Equation (10) the mixing proportion \(w_{ij}\) does not only depend on prior assignment by local structures, but also depends upon the posterior probability of observation data \(x_{j}\) belonging to the \(i^{\text{th}}\) component of the SMM by global structures. In addition, we obtain degree of freedom \(\mathbf{v}\) by taking the corresponding derivative of \(\mathbf{Q}_{2}\) to zero, and the updated value \(\mathbf{v}^{(k+1)}\) is a solution of the following equation:  
\[1-\gamma\left(\frac{v_{i}^{(k+1)}}{2}\right)+\ln\left(\frac{v_{i}^{(k+1)}}{2} \right)+\frac{\sum\limits_{j=1}^{N}\tau_{ij}{}^{(k)}\left(\ln u_{ij}{}^{(k)}- u_{ij}{}^{(k)}\right)}{\sum\limits_{j=1}^{N}\tau_{ij}{}^{(k)}}+\gamma\left( \frac{v_{i}^{(k)}+D}{2}\right)-\ln\left(\frac{v_{i}^{(k)}+D}{2}\right)=0, \tag{11}\]  
Then, to update the estimates of covariance matrices \(\sum\) and displacement function \(\mathbf{\rho}\) in Equation (7), we also need to take the derivative of \(\overline{\mathbf{Q}}_{3}\), and the update values can be denoted as:  
\[ \begin{array}{l}\mathbf{H}^{(k+1)}=\left(diag\big{(}\mathbf{P}^{(k)}\mathbf{1}\big{)} \mathbf{G}+\lambda\Sigma_{i}^{(k)}\mathbf{I}\right)^{-1}\big{(}\mathbf{P}^{(k)}\mathbf{X}-diag \big{(}\mathbf{P}^{(k)}\mathbf{1}\big{)}\mathbf{Y}\big{)},\mathbf{P}_{ij}{}^{(k)}=\tau_{ij}{}^ {(k)}u_{ij}{}^{(k)}\\ \Sigma_{i}{}^{(k+1)}=\frac{\sum\limits_{j=1}^{N}\tau_{ij}{}^{(k)}u_{ij}{}^{(k )}\left(\right)\left(\right)}{\sum\limits_{j=1}^{N}\tau_{ij}{}^{(k+1)}}\,,\;(i =1,\ldots,N)\end{array}  \tag{12}\]  
where \(\mathbf{1}\) is a column vector of all ones, and \(\mathbf{I}\) is an identity matrix, \(diag(\bullet)\) denotes a diagonal matrix and \(\mathbf{G}(i,\bullet)\) denotes the column vector in the kernel matrix \(\mathbf{G}\).