## 2 Proposed Method  
### Main Ideas  
Sparse representation is the existing method for super-resolution, which is not appropriate for global face images, as we need to explore non-linear information hidden in the data. Moreover, the available Eigenface model and Markov network model have very low accuracy. The processing step starts when the low-resolution image is interpolated to the size of the target high-resolution image [11]. The low-resolution interpolated image is a blurred image that is deficient in high-frequency information, and it is used as a preliminary estimation of the target HR image. Then, the image will be divided further into non-overlapping image patches that are not separate. The framework uses the generated patches to identify the image, which is most matched by searching a training data set of low-resolution and high-resolution image patches. The selected high-resolution image is generated to study high-resolution information. Finally, the trained high-resolution image and the interpolated images are combined to estimate the high-resolution target image. The embedded space is learned for producing solutions with representation, distinct solutions for the same functionality, and picking the group of embedding vectors. Therefore, mapping within the embedding spaces is restricted. Figure 1 demonstrates the block diagram of SR based face hallucination.  
The proposed algorithm was employed to discover the non-linear structure in the data. The algorithm was developed under the assumption that any object is \"nearly\" flat on small scales. The initial object was to map data from one space globally to another space by embedding the locations of the neighbors for each point. For face hallucination, it was about LR and HR face image spaces. The main idea of the proposed methodology was minimizing the reconstruction error of the set of all local neighborhoods in the data set. The reconstruction weights were computed by reducing the reconstruction error using the cost function in Equation (1).  
\[\delta_{w}=\sum\
olimits_{i=1}^{n}\left(x_{i}-\sum\
olimits_{j=1}^{k}w_{i,j}x _{N(j)}\right)^{2} \tag{1}\]  
For face hallucination, such reconstruction weights are computed in the LR face image space and then applied to the HR space to reconstruct the HR face image that corresponds to the LR input. The proposed method normally contains the weight matrix \(W\) computed from the neighborhood in the same space as the training samples. Similar to LLE, the ALLE (Adaptive locally linear embedding) computes the weights from the space of training samples, which is built adaptively from only the neighborhood of each input, not all training samples. Since some information contained in all training  
Figure 1: Block diagram of super-resolution (SR) based face hallucination.  
samples can make the optimizer misleading to another optimal value, the threshold of similarity \(\theta\) is defined for building each subspace in Equation (2).  
\[\frac{\sum_{i=1}^{k}d_{i}}{\sum_{i=1}^{n}d_{i}}\leq\theta \tag{2}\]  
Face image is represented as a column vector of all pixel values. Based on structural similarity, the face image can be synthesized using the linear combination of training objects. In other words, a face image at an unfixed view can also be reconstructed using a linear combination of other objects in the same view aligned as computed using Equation (3).  
\[I_{p}=W_{p}L_{p} \tag{3}\]  
There is no information to determine the construction coefficients at other views. After obtaining the LR in all views, these LR images are hallucinated to the HR face images of all views using the position-patch methodology. The proposed framework for multi-view face hallucination is to replace the linear combinations in all processes to improve the performance of hallucinating face images in multiple views. The framework means the number of the neighbors can be adapted for each input patches, and this would relieve the deviation from optimal values. The symbols used to construct the proposed framework are demonstrated in Table 1.