Now, we need to estimate the partial derivatives in the above equations with
*discretized formulas* starting from brightness values that are
*quantized* (say integers from 0 to **n**) and noisy. Given these
derivative estimation problems, the optimal step for the discretization grid
depends on local properties of the image. Use of a single discretization
step produces large errors on some images. Use of a *homogeneous*
multiscale approach, where a set of grids at different resolutions is
used, may in some cases produce a good estimation on an intermediate
grid and a bad one on the final and finest grid. Enkelmann and Glazer
[Enkelmann:88a], [Glazer:84a] encountered similar problems.

These difficulties can be illustrated with the following one-dimensional example. Let's suppose that the intensity pattern observed is a superposition of two sinusoids of different wavelengths:

where **R** is the ratio of short to long wavelength components. Using the
*brightness constancy* assumption ( or , see [Horn:81a]) the measured velocity is
given by:

where and are the three-point approximations of the spatial and temporal brightness derivatives.

Now, if we calculate the estimated velocity on two different grids, with
spatial step equal to one and two, as a function of the parameter,
**R**, we obtain the result illustrated in Figure 6.38.

**Figure 6.38:** Measured velocity for superposition of sinusoidal patterns as a
function of the ratio of short to long wavelength components. Dashed line:
, continuous line: .

While on the coarser grid, the correct velocity is obtained (in this case);
on the finer one, the measured velocity depends on the value of **R**. In
particular, if **R** is greater than , we obtain a velocity in the
opposite direction!

We propose a method for ``tuning'' the discretization grid to a measure of
the reliability of the optical flow derived at a given
scale. This measure
is based on a *local estimate of the errors* due to noise and
discretization, and is described in
[Battiti:89g;91b].

Wed Mar 1 10:19:35 EST 1995