本論文透過子空間學習法達到降低資料維度與擷取具鑑別性的特徵，將其運用於人臉偵測，辨識與角度估計的應用上。下面的文章中，我們會將檢視一些相關的子空間學習法，並針對在不同應用，提出相對應改良的方法。
第一個應用是使用在eigenspace下的流型特徵 (manifold learning)，而設計發展出一套多角度的人臉偵測與角度估計系統。在eigenspace 中，不僅可利用低維的投影量而發展一個簡易串接式的拒絕模組 (cascaded rejecter module)，且能達到濾除85%　非人臉影像的區域以加強整體系統的運算效能；同時透過人臉的流型分布，還可以得到粗略的角度資訊 （coarse pose estimation）。此外，為改良對於不同遮蔽，光線的影響，我們不同以往採用整張人臉做分析，而是採用一個具鑑別性的人臉子區域 (subregion) 做為分析對象。並透過eigenspace與獨立基底 （independent basis） 的建立，擷取每一個子區域其低頻與高頻的資訊。此外，因投影的特徵向量，包含在eigenspace下的投影權重向量 (projection weight vector) 與在獨立成份空間下的投影係數向量 (coefficient vectors) 有較分散的分布，因此我們採用混合高斯模型 (Gaussian mixture model GMM) 來描述這些投影向量，其中GMM模型參數和權重是藉由期望值最大化 (expectation-maximization, EM) 演算法來反覆估算)。人臉偵測是經由資訊的權重向量與係數向量，並考量相對應子區域位置的結合機率來執行一個可能性估算處理流程 (likelihood evaluation process)。而子區域的位置資訊可以提供錯誤偵測的風險。
在透過使用PCA+ICA 的子空間學習法來描述人臉的影像後，接著，在第二個應用中，我們透過延伸校準關係分析 (canonical correlation analysis (CCA)) 中，運用於人臉影像集(image sets) 的比對方式，進而提出一個核心鑑別轉換空間演算法 (kernel discriminant transformation (KDT) algorithm) 運用於人臉辨識應用中。使用人臉影像集的方式 (可包含受測影像任意的角度，臉部表情與光線條件等不同影像)，可提升辨識的效能。然而訓練資料中人臉的流型分布 (manifolds) 會因為這些不同變異性的影響，而使得人臉影像呈現非線性的分布 (non-linearly distributed)。因此，透過一個非線性的轉換，將原人臉影像投影至更高維的空間下，並在高維空間下，每一個人臉影像集將使用核心主成份分析 (kernel principal component analysis (KPCA)) 加以描述。為了擷取具鑑別性的人臉特徵，根據最大化同類別核心子空間 （within-kernel subspaces） 的相似度與最小化不同類別核心子空間 (between-kernel subspaces) 的相似度，提出一個KDT 轉換矩陣。然後，該矩陣並非真的需要在高維空間下做運算，而是透過一個所提出的迭代式核心鑑別性轉換演算法 (kernel discriminant transformation algorithm)，所求得的。透過KDT 轉換矩陣，所提出的人臉辨識系統，可以比現存的靜態影像(still-image-based) 辨識與集合影像 (set-based) 辨識的系統在 Yale face database B可提供較好的辨識效能。

This thesis concerns the subspace learning methods on performing dimensionality reduction and extracting discriminant features for face detection, recognition and pose estimation. We examine the subspace learning methods and then the novel subspace learning methods are derived according to the data distribution of each application.
The first application is based on analyzing the manifold in eigenspace to develop a statistic-based multi-view face detection and the pose estimation. In the eigenspace, not only the simple cascaded rejecter module can be developed to exclude 85% of the non-face images to enhance the overall system performance but also the manifold of face data can be applied to develop coarse pose estimation. In addition, to improve tolerance toward different partial occlusions and lighting conditions, the five-module detection system is based on significant local facial features (or subregions) rather than the entire face. In order to extract the low- and high-frequency feature information of each subregion of the facial image, the eigenspace and residual independent basis space are constructed. In addition, either projection weight vectors or coefficient vectors in the PCA (principal component analysis) or ICA (independent component analysis) space have divergent distributions and are therefore modeled by using the weighted Gaussian mixture model (GMM) with parameters estimated by Expectation-Maximization (EM) algorithm. Face detection is then performed by conducting a likelihood evaluation process based on the estimated joint probability of the weight and coefficient vectors and the corresponding geometric positions of the subregions. The use of subregion position information can reduce the risk of false acceptances.
Following the use of PCA+ICA to model the face images, in the second application the kernel discriminant transformation (KDT) algorithm is proposed by extending the idea of canonical correlation analysis (CCA) of comparing facial image sets for face recognition. The recognition performance is rendered more robust by utilizing a set of test facial images characterized by arbitrary head poses, facial expressions and lighting conditions. Since the manifolds of the image sets in the training database are highly-overlapped and non-linearly distributed, each facial image set is non-linearly mapped into a high-dimensional space and a corresponding kernel subspace is then constructed using kernel principal component analysis (KPCA). To extract the discriminant features for recognition, a KDT matrix is proposed that maximizes the similarities of within-kernel subspaces and simultaneously minimizes those of between-kernel subspaces. While the KDT matrix cannot be computed explicitly in the high-dimensional feature space, an iterative kernel discriminant transformation algorithm is developed to solve the matrix in an implicit way. The proposed face recognition system is demonstrated to outperform existing still-image-based as well as image set-based recognition systems using the Yale face database B.

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