An Adaptive Finite-Element Method For Image Representation

by Pierre Moulin

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 solve numerically partial differential equations
while preserving important 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.
Our particular representation is based on hierarchical finite elements.
A multiresolution algorithm computes the solution to the approximation problem
in $O(N)$ time on a sequential machine and in $O( log N)$ time
on a single-instruction, multiple-data, fine-grain parallel architecture,
where $N$ is the number of pixels in the image.
Applications to the problems of image compression and restoration are given.