A topological analysis on patches of optical flow

Abstract

The research of optical flow is vitally important topic in computer vision. In this paper we research a topological analysis of space of optical flow locally. We use the methods of computing topology to the spaces of 4 × 4 and 6 × 6 high contrast optical flow patches. We experimentally prove that in both cases there exist subspaces of the spaces of all high contrast optical flow patches that is topologically equivalent to a circle, which states that some results on the topological analysis of natural images and range images can be extended to the scope of image motion.

Non-linear Analysis on Optical Flow 9 x 9 HighContrast Patches

“…By using methods of the paper [5], it is shown that 9 9  optical flow patches with density subsets having topology of a circle. As increasing of the size of optical flow patches, the Klein bottle property of the optical flow spaces gradually weaken [3,4]. Here, we prove that the Klein bottle property of 9 9  optical flow patches may vanish.…”

“…As the optical flow database producing from the Brown range image database, optical flow patches may have similar topological properties as that of range image patches. The authors of papers [2,3,4] have found such similar topological features for optical flow n n  patches with range image patches when 3, 4, n  5, 6, 7 . In this short note, we enlarge the size of optical flow patches to 9. By using methods of the paper [5], it is shown that 9 9  optical flow patches with density subsets having topology of a circle.…”

An Analysis of Optical Flow 8x8 Patches

“…The authors of [3] shown a similar topological features for optical flow 3 3  patches with that of range image. The authors of papers [4,5] studied optical flow patches for 3, 4, 5, 6, 7 n  and got similar results as the case 3 n  . In this paper, we expand the size of optical flow patches to 8 and detect the topological features of spaces of 8 8  optical flow patches.…”

NEB in Analysis of Optical Flow 4 x 4 and 6 x 6-Patches

“…We use NEB to prove that there exist subsets of , and optical flow patches which has the homology of a circle, the results have been proven in [4] by using computational topology. For the same data sets, we used a completely different method from paper [4] to analyze them, and obtain the same result as in [4], therefore, these topological properties of the sets are their inherent nature.…”

“…We use NEB to prove that there exist subsets of , and optical flow patches which has the homology of a circle, the results have been proven in [4] by using computational topology. For the same data sets, we used a completely different method from paper [4] to analyze them, and obtain the same result as in [4], therefore, these topological properties of the sets are their inherent nature. The NEB method is simpler than the method of computational topology, in NEB, we only use several cell complexes to identify topological properties of spaces, but in the computational topology method, we need several thousands even tens of thousands of complexes to get same results.…”