eigenfaces

Eigenfaces are a set of eigenvectors derived from the covariance matrix of facial images, used to represent and recognize human faces in a compact, low-dimensional subspace. The idea is that although face images are high-dimensional, the variation among well-aligned faces lies in a much smaller subspace.

The typical process starts with a training set of grayscale, aligned face images. Each image is converted into a vector, and a mean face is computed. All vectors are centered by subtracting the mean, and principal component analysis (PCA) is applied to obtain the principal components, called eigenfaces, ordered by the amount of variance they explain. Because the number of pixels is large, the computation is usually done via a singular value decomposition of the data matrix or by solving a smaller eigenproblem, then reshaping the eigenvectors into image form to visualize the features they capture.

To recognize a new face, the image is projected onto the eigenface subspace, producing a set of weights that describe its coordinates in that space. The weights are compared to those of known faces using a distance measure; the closest match (or a thresholded decision) yields identification or rejection. A reconstruction can be produced by combining the mean face with a weighted sum of eigenfaces, illustrating how much of each eigenface is present in the input.

Eigenfaces, introduced by Turk and Pentland in 1991, popularized PCA-based face recognition and demonstrated the value of dimensionality reduction for this task. They are efficient and intuitive but can be sensitive to lighting, pose, and expression, leading to later approaches that add discriminative features or use nonlinear methods.