EIGENFACES AND DIGITAL IMAGE RECOGNITION

David Mumford, Alan Yuille, and Peter Hallinan at Harvard have taken a different tack in the computer matching of faces. One problem in analyzing a picture of a person's face is that the image depends largely on the presence or absence of shadows, which depend, in turn, on lighting conditions. Mumford and Hallinan address this by computing what they call "eigenfaces".
To obtain these, they photograph a subject under as many as a hundred different lighting conditions. Once digitized, these images can be interpreted as points in a very high-dimensional space (the dimension is equal to the number of pixels in the image). The resulting cloud of points is shaped something like an ellipse. The axes of the ellipse and their lengths can be determined by a computation in linear algebra (to be precise, by computing the eigenvectors and eigenvalues of the Matrix IIt where I is a column-by-column listing of the images and It is the transpose of I). The eigenfaces lie along these axes. The idea is to reconstruct a good approximation to a face under general lighting conditions by combining just a few eigenfaces and then "warping" the result.
The theory of linear agebra carries over into the high-dimensional world of digital images with the computation of "eigenfaces" in an application of control theory to an important problem in pattern recognition. (Figure courtesy of Harvard Robotics lab.)
(From: The Gentle Art of Control , in "What's Happening in the Mathematical Sciences", Vol. 3, by Barry Cipra, and published and available from the American Mathematical Society .)