Institute of Computer Science X, Universitaet Erlangen-Nuernberg, Cauerstr. 6, D-91058 Erlangen, Germany

El Mostafa Kalmoun

Abstract

It is currently recognized that optical flow computation has many
applications
in image processing, pattern recognition, data compression, and biomedical
technology. Differential approaches, which estimate velocity vectors from the
spatial and temporal intensity derivatives are the most common visited
techniques in the literature. The basic assumption behind that is the
intensity
variations are weak and only due to a movement in the image plan.

This constant brightness assumption leads to an ill-posed problem that can
only be solved by imposing additional assumptions. A standard technique,
which
is due to to Horn and Schunck, is to require the flow
field to be smooth by means of a standard regularization approach.
This results in a system of elliptic PDEs of reaction diffusion type.
The second order terms are induced by the regularization and become
straightforward Laplace operators. The zero order terms are linear
(but variable) and are computed from the derivatives of the
image data. Since the image data (and even more so its derivatives)
are usually nonsmooth, this poses some nontrivial problems.
The PDEs are usually discretized by standard finite differences.

The Horn-Schuck algorithm is the standard way to discretize and solve
the PDEs. In multigrid terminology, it is simply a coupled pointwise
relaxation. As expected, it has acceptable convergence only if the
zero-order terms dominate but the performance is poor, when the diffusion
character dominates.
In this case, a multigrid strategy can be expected to provide a significant
accelleration by allowing a quick propagation of information from non-zero
flow
field regions into homogeneous or untextured image areas. The Horn-Schunck
algorithm can still be used as a smoother for such a multigrid method. The
most
difficult aspect is to deal with the strongly discontinuous coefficients in
an efficient way.
This will be discussed in our presentation.