ON THE NONLINEAR ANALYSIS OF OPTICAL FLOW

We utilize the methods of computational topology to the database of optical flow created by Roth and Black from range images, and demonstrate a qualitative topological analysis of spaces of 3 x 3, 5 x 5 and 7 x 7 optical flow patches. We experimentally prove that there exist sub-spaces of the spaces of the three sizes high-contrast patches that are topologically equivalent to a circle and a three circles model, respectively. The Klein bottle is the quotient space described as the square [0, 1] x [0, 1] with sides identified by the relations (0, y) similar to (1, y) for y is an element of [0, 1] and (x, 0) similar to, (1- x, 1) for x is an element of [0, 1]. For the space of 3 x 3 optical flow patches we found a subspace having the same homology as that of the Klein bottle. As the size of patches increases, the Klein bottle feature of the spaces of 5 x 5 and 7 x 7 optical flow patches gradually disappears.