在人臉辨識的相關研究中，線性鑑別式分析（Linear Discriminant Analysis，LDA）是一種常見的線性轉換技術，可以求出一組具鑑別性的特徵參數。然而這個方法在訓練樣本數不足（small sample size）時，會因為矩陣的奇異性問題，而導致無法實現。針對這個問題，本篇論文提出一種改良式的線性鑑別式分析方法：先將原始的特徵參數經由線性轉換投射至一個保留相同散佈程度的特徵空間，在此空間內的類別內散佈矩陣（Within-class scatter matrix）沒有奇異性（Singularity）的問題。若原始類別內散佈矩陣的秩（Rank）為r，則該轉換矩陣是由原始類別內散佈矩陣前r大特徵值所對應之特徵向量所組成。最後再對新的特徵參數做線性鑑別式分析。經由五種不同人臉資料庫實驗的結果，我們可以發現在樣本數目足夠的情形下，辨識率明顯地比原始的LDA提昇許多，而在訓練樣本數不足時也在較少的訓練成本下能維持不錯的辨識效果。

In the literature of face recognition, LDA (Linear Discriminant Analysis) is a popular linear transformation technique to extract the discriminant feature vector. However, this technique may occur the singularity problem on calculating the inverse of the within-class scatter matrix when the training sample size is small. To overcome the problem, an improved linear discriminant analysis is proposed in this thesis. The proposed method aims to transform the original feature vector to a new feature space with the same degree of scattering and without the singularity problem. Then, We use the LDA to extract the new feature vector. In the experiments on five face databases, the results show that the recognition rates of the proposed method is significantly better than other LDA-based techniques when training data are sufficient. In case of very limited training data, the proposed method achieves desirable recognition performance with moderate training cost.

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