## 4 Discussion and Conclusions  
Building a face recognition system that works robustly in various environments involves difficulties in securing the data needed to learn recognition algorithms. Moreover, large-scale face recognition applications typically use databases that contain SSPPs. A single image is not sufficiently representative for face recognition. The SSPP problem makes using the feature extraction method in a supervised manner quite difficult, because the interclass variations are unknown. To overcome this, several methods have been proposed. However, there have been limitations in that these methods did not reflect facial characteristics that could have various variations.  
We proposed an image generation method that uses a B-WIM that leverages the fact that the pixels of specific parts of the neutral face image vary significantly compared with other areas when there is an environmental variation in face recognition. The B-WIM statistically reflects the change in individual pixel values caused by the variation from the neutral and variational images included in the auxiliary set. For a given query image (neutral image), the proposed method creates a new variational image that reflects the characteristics of the variation while maintaining the unique characteristics of the face in the query image based on B-WIM. Through this, a training dataset containing only one sample per person can be made into a richer set that includes variational images for each person, further improving the performance of the face recognition system.  
The proposed method has the following advantages. The proposed method does not require a large amount of computation or a large dataset for creating new images. When the number of pixels in an image is \(n\), while SPCA+ has the complexity of \(O(n^{2})\), the complexity of the proposed method is \(O(n)\). Some methods, such as ICR, E(PC\({}^{2}\))A+, (ZD)\({}^{2}\)PCA, SLC, and MVI, require similar computations as the proposed method but do not address specific variations. In contrast, the proposed method generates high-quality variational images for query images in real-time, effectively improving the performance of existing face recognition systems at a low cost. Face recognition experiments using Bosphorus and RaFD databases showed that the proposed method outperformed the existing methods for solving the SSPP problem. In addition to general facial recognition algorithms, images generated using the proposed method can be utilized in the study of various facial images, including the fake image detection algorithms [49; 50].  
On the other hand, by comparing the recognition rates for two protocols of face recognition, \"_Closed Set_\" and \"_Open Set_,\" we found that the quality of the image created using the proposed method was affected by the images included in the auxiliary set. Although the proposed method can effectively generate new images for a specific variation, it does not control the degree of variation or handle more than two variations simultaneously. It is expected that the small sample-size problems, including the SSPP problem, can be solved more effectively by subdividing the degree of variation  
Figure 7: Face recognition results for a various number of DCV features in “_Open Set_” protocol.  
within the proposed method's algorithmic structure and applying the interpolation maps for two or more types of variations together. We leave these problems to future works.  
[PERSON] designed the experiments and drafted the manuscript. All authors have read and agreed to the published version of the manuscript.  
The present research was supported by a National Research Foundation of Korea(NRF) grant funded by the Korean government (MSIT) (No. 2018R1A2B6001400) and the MSIT(Ministry of Science and ICT), Korea, under the ICAN(ICT Challenge and Advanced Network of HRD) program(IITP-2020-2020-0-01824) supervised by the IITP(Institute of Information & Communications Technology Planning & Evaluation).  
The authors declare no conflict of interest.