eigenfaces

Eigenfaces is the foundational 1991 face-recognition method (Turk and Pentland) that treats a face image as a single point in a very high-dimensional space and then dramatically compresses it. A 100×100 grayscale face is a vector of 10,000 numbers, but real faces do not scatter randomly through that 10,000-dimensional space — they cluster on a thin sheet, because faces share enormous structure (two eyes, a nose, smooth skin). Eigenfaces finds the few dozen directions along which faces actually vary the most and describes any face by how far it moves along each of those directions, collapsing 10,000 numbers into perhaps 50.

The machinery is principal component analysis (PCA). You collect many aligned, same-size face images, compute the average face, and subtract it from each so you are left with deviations from the mean. PCA then finds the orthogonal directions of maximum variance in these deviations — the eigenvectors of the data's covariance matrix, ordered by how much variance they capture. Reshaped back into images, these eigenvectors look like ghostly, generic face-like patterns and are called the eigenfaces. Any face is then approximated as the average face plus a weighted sum of the top eigenfaces; the list of weights (the projection coefficients) is the face's compact signature. Recognition is just nearest-neighbor in this low-dimensional weight space: project a new face, compare its weights to the gallery's, and pick the closest identity. The distance of a face from the eigenface subspace itself also flags whether the input is even a face at all.

Eigenfaces matters because it introduced the subspace view of appearance — the idea that a class of images lives on a low-dimensional manifold you can learn from data — which seeded a whole generation of appearance-based recognition. Its weaknesses are instructive: PCA keeps the directions of greatest total variation, but the biggest variations in face images are usually lighting and pose, not identity, so eigenfaces can be more sensitive to illumination than to who the person is. It also demands tight alignment and consistent scale, and it is a purely linear, unsupervised compression. Fisherfaces fixed the identity-versus-nuisance problem by using class labels, and modern deep face embeddings (DeepFace, FaceNet, ArcFace) replaced the hand-built linear subspace with learned nonlinear ones, but eigenfaces remains the canonical first lesson in representing faces.