The optic flow constraint was introduced by Horn and Schunck in 1981. It gives a simple linear relation between the optic flow and the instantaneous change in the illumination at a point. This linear relation forms the basis for many "gradient-based" motion estimation techniques. The general approach is to compute motion by imposing the constraint on all points of a large region, assumed to be a single rigid object moving in a simple motion. The result is an overdetermined system of linear equations that can be solved by standard least-squares techniques.

Now consider these steps in reverse order. Given images related by known motion, compute a linear relation between the optic flow and the instantaneous change in the illumination that satisfies the following condition. If this relation is used instead of the Horn and Schunck constraint to estimate the motion as explained above, the result is a good approximation to the known motion. It turns out that the solution can be written in a simple closed form. This can be viewed as a technique for learning the optic flow constraint from a provided training data of images with known motion.

As described above this learning procedure is only of limited interest because its output (the learned optic flow constraint) depends on the training data. A key observation is that this training data can be obtained from a single image. For example, by shifting the image to the right by one pixel one obtains a new image that can be used for the training with the motion parameters taken as the known shift value. Additional training data for translational motion can be obtained by using different shift values in various directions. One can also compute linear constraints for additional motion parameters (e.g, rotation or general affine) using the same technique.

It is interesting to investigate the relation between the constraints computed as explained above and the Horn and Schunck constraint. It can be shown that if the training data is composed of symmetric shifts of the image, the result can always be viewed as a variation of the Horn and Schunck constraint which differ only in the way in which the derivatives are computed.