A multiresolution image representation is proposed as a basis for constructing an approximation to an original image. The method is based on adaptive finite-elements, a technique used in applied mathematics to approximate the solution of partial differential equations while providing an adequate description of the features of the solution at different scales. Theory and experiments suggest that adaptive finite-elements is a natural and computationally-powerful approach to image approximation problems. We propose a fast multiresolution algorithm to compute the solution to the approximation problem. Applications to problems of image compression and restoration are given. We demonstrate compression of the image "Lena" by a 17:1 ratio with virtually unnoticeable degradation.