Optic flow is a visual displacement flow field that can be used to explain changes in an image sequence. The underlying assumption that is used to find an equation is that the gray level is constant along the visual trajectories.
In other words, the particular derivative of the gray level It(x1,x2) along the optic flow (v1,v2) is zero, thus:
This equation alone is not sufficient to determine a unique vector field, since at any location, we only dispose of a single scalar constraint to find out a two dimensional vector (v1,v2). This is an ill-posed problem.
All approaches that have been introduced by various authours always rely on the same assumption: it is necessary to either find the smoothest optic flow field (according to an a priori chosen metric) or to make an additional assumption on the field class in which we want to find our optic flow.
Horn and Schunck[13,14] have suggested to replace this ill-posed problem with a convex minimization problem. The convex functional is then the sum of two functionals: a matching functional which is equal to 0 if the vector field v fulfills the optic flow equation, and a smoothing functional (like a Sobolev norm on v). The resulting functional is strictly convex and has a unique solution.
Spatiotemporal filtering methods have been suggested by Adelson and Bergen, Clifford et al., Fleet and Jepson, et Heeger. These methods also rely on an underlying assumption, that the optic flow is locally constant. Burns et al. have introduced a variation on these methods based on spatiotemporal wavelet analysis.
Later, Weber et Malik introduced a filtered differential method. The optic flow equation is filtered with a set of filters of various spectral contents. They obtain this way a set of equations they can solve to get a unique solution for the optic flow. This method also relies on the assumption that the optic flow has to be locally constant.
Whatever method is chosen, a universal problem in optic flow is that of time aliasing: fine scale measures cannot detect correctly large displacements.
An intuitive image of what is happening can be described to following way: if one looks trough a tiny hole at a moving pattern at two different times, it is only possible to estimate the real displacement between these if a same portion of pattern can be viewed through the hole at these two distinct times. Thus, the motion can be estimated only if the total displacement is smaller than the diameter of the hole.
``Why should we look at the picture sequence through a hole ?''. In practice, we look at portion of the picture, and do the assumption that the optic flow is constant over this portion. If this portion is large, the assumption is more restrictive.
Many authors have therefore suggested to use a multiresolution approach. Starting from coarse scale measurements (relying on coarse scale picture information and sampled on a coarse grid), we refine these measurements by using finer scale information, as far as possible.
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