PCA+Fisher: Applying Kernel Functions to Face Recognition Research

PCA combined with Fisher discriminant analysis represents a classical methodology for applying kernel functions to face recognition research. This approach enhances the classification performance of traditional linear algorithms through nonlinear mapping techniques.

In conventional PCA+Fisher implementations, the process begins with Principal Component Analysis (PCA) for dimensionality reduction of facial images, extracting principal feature components. This is typically implemented using eigenvalue decomposition of the covariance matrix. Subsequently, Fisher Linear Discriminant Analysis (LDA) further optimizes these features by maximizing between-class scatter while minimizing within-class scatter through computing scatter matrices. However, this linear approach demonstrates limitations when handling complex data distributions.

The introduction of kernel functions employs nonlinear transformations to map original data into higher-dimensional feature spaces. In this transformed space, originally linearly non-separable data may become linearly separable. Common kernel implementations include polynomial kernels (K(x,y) = (x·y + c)^d) and Gaussian radial basis function kernels (K(x,y) = exp(-γ||x-y||²)). The kernel trick avoids explicit computation of high-dimensional mappings by only requiring kernel function calculations in the original space.

This kernel-based methodology offers significant advantages for face recognition: improved handling of nonlinear variations like lighting conditions and pose changes; extraction of more discriminative features through kernel mapping; and superior pattern differentiation capabilities compared to traditional linear methods. Experimental validation on standard databases like ORL demonstrates that kernel-based PCA+Fisher approaches significantly enhance recognition rates, often achieving improvements of 5-10% over linear counterparts through proper kernel parameter optimization.