     6.7.1 Errors in Computing the Motion Field

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].     