Hilbert-space Karhunen-Loève transform with application to image analysis

A generalization of the Karhunen-Loève (KL) transform to Hilbert spaces is developed. It allows one to find the best low-dimensional approximation of an ensemble of images with respect to a variety of distance functions other than the traditional mean square error (L2 norm). A simple and intuitive characterization of the family of Hilbert norms in finite-dimensional spaces leads to an algorithm for calculating the Hilbert-KL expansion. KL approximations of ensembles of objects and faces optimized with respect to a norm based on the modulation transfer function of the human visual system are compared with the standard L2 approximations.