# Learning Kernel-Based Robust Disturbance Dictionary for Face Recognition
## 3 Fundamental Techniques
### Concise Steps of KRDD  
The steps of KRDD are concisely generalized below:  
1. KDA projection was performed on the training sample \(X\) to get \(p^{\phi}=[p_{1}^{\phi},p_{2}^{\phi},\ldots,p_{C-1}^{\phi}]\), where \(C\) is the quantity of subjects.
2. The gallery samples can be outside the training samples. The gallery samples \(\bar{X}=[\bar{x_{1}},\bar{x_{2}},\ldots,\bar{x_{n}}]\) were projected to obtain the basic dictionary \(D\) by KDA through Formula (9): \[D=(p^{\phi})^{T}\phi(\bar{X})\] (15)  
Figure 4: Original Training Sample (**Upper**) & Samples after Random Square Block Processing (**Lower**).  
Figure 3: Original Training Sample (**left**) & Samples after Mirror Processing (**right**).  
KPCA projection was performed on the training sample \(X\) to obtain \(q^{\phi}=[q_{1}^{\phi},q_{2}^{\phi},\ldots,q_{r}^{\phi}]\), then the \(r\) principal component of disturbance variations in the kernel space was used as the kernel real disturbance model.
4. KDA was used to project the kernel principal component \(q^{\phi}=\sum_{i=1}^{m}\tilde{a}_{i}\Phi(i)\) with the following equation without having to explicitly know the nonlinear mapping by (9). \[\begin{split}(p^{\phi})^{T}q^{\phi}&=\langle p^{ \phi},q^{\phi}\rangle=\langle\sum_{i=1}^{m}a_{i}\phi(X_{i}),\sum_{j=1}^{m} \tilde{a}_{j}\Phi(j))\rangle\\ &=\langle\sum_{i=1}^{m}a_{i}\phi(X_{i}),\sum_{j=1}^{m}\tilde{a}_{ j}(\phi(X_{j})-\phi(\tilde{\gamma}_{j}))\rangle\\ &=a^{T}K\tilde{a}-a^{T}K^{\gamma}\tilde{a}\end{split}\] (16) where \(a\) and \(\tilde{a}\) are the coefficient vectors of KDA and the real disturbance model, respectively. \(K_{i,j}=\langle\phi(X_{i}),\phi(X_{j})\rangle\), \(K_{i,j}^{\gamma}=\langle\phi(\tilde{z}_{i}),\phi(\tilde{\gamma}_{j})\rangle\). \(p^{\phi}\) was projected by KDA to obtain the kernel real disturbance dictionary: \[D_{r}=(p^{\phi})^{T}q^{\phi}\] (17)
5. Three special schemes were used on \(Z\) to obtain alternative training samples \(\tilde{Z}\), the sample in \(Z\) can be different from the sample in \(X\).
6. The \(r\) principal components of disturbance changes were trained in kernel space as the simulated disturbance model, i.e., \(l^{\phi}=[l_{1}^{\phi},l_{2}^{\phi},\ldots,l_{r}^{\phi}]\), from the set of \(\tilde{Z}\), as the kernel simulated disturbance model.
7. \(l^{\phi}\) was projected by KDA to obtain the kernel simulated disturbance dictionary via (17) as follows: \[D_{s}=(p^{\phi})^{T}l^{\phi}\] (18)
8. A testing sample \(y\) was projected by KDA: \[y_{KDA}=(p^{\phi})^{T}\phi(y)\] (19)
9. The sparse representation of \(y_{KDA}\) was found according to the basic dictionary \(D\), the real disturbance dictionary \(D_{r}\) and the simulated disturbance dictionary \(D_{s}\): \[\min_{\beta,\beta_{r},\beta_{s}}\|y_{KDA}-[D,D_{r},D_{s}]\left[\begin{array}[] {c}\beta\\ \beta_{r}\\ \beta_{s}\end{array}\right]\|_{2}^{2}+\lambda\|\left[\begin{array}{c}\beta \\ \beta_{r}\\ \beta_{s}\end{array}\right]\|_{1}.\] (20)
10. \(y\) was classified to the subject that minimizes the residual as follows: \[r_{c}(y_{KDA})=\|y_{KDA}-[D,D_{r},D_{s}]\left[\begin{array}{c}\delta_{c}( \beta)\\ \beta_{r}\\ \beta_{s}\end{array}\right]\|_{2}.\] (21)