PCA Method Based on Singular Value Decomposition Theorem for Face Recognition on ORL Database

In face recognition tasks, the PCA method employing Singular Value Decomposition (SVD) serves as a classic dimensionality reduction technique. When applied to the ORL face database, this approach effectively extracts principal component features from facial images and combines them with a nearest neighbor classifier to achieve efficient recognition.

The core workflow comprises three key steps: Data Preprocessing The ORL database contains facial images of multiple subjects with varying poses. Each image must be flattened into a vector and centered to construct a sample matrix, thereby eliminating interference from lighting conditions and positional variations. In code implementation, this typically involves converting 2D image matrices into 1D vectors and subtracting the mean face from each sample.

Feature Extraction Perform SVD decomposition on the sample matrix to obtain eigenvectors (principal components) and singular values. Select the top k eigenvectors corresponding to the largest singular values to form a projection matrix. This matrix maps original high-dimensional data into a lower-dimensional space while preserving the most discriminative features. The SVD operation can be implemented using numpy.linalg.svd() or similar linear algebra functions, where the right singular vectors represent the principal components.

Classification and Recognition Apply a nearest neighbor classifier in the reduced feature space: calculate Euclidean distances between test samples and all training samples, then assign the class label of the closest match as the prediction result. This distance computation can be optimized using vectorized operations like numpy.linalg.norm() for efficient pairwise comparisons.

The method's key advantage lies in SVD's numerical stability for PCA implementation, while the moderate scale of the ORL database (e.g., 40 subjects × 10 images/subject) validates the algorithm's feasibility under controlled dimensionality. Future extensions could incorporate kernel PCA or integration with deep features to enhance recognition capabilities in complex scenarios. The algorithm demonstrates a balanced trade-off between computational efficiency and accuracy on small-scale datasets like ORL.