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

We apply the nudged elastic band technique to non-lineal high-dimensional datasets, we analyze spaces of 4 x 4 and 6 x 6 optical flow patches and detect their topological properties. We experimentally prove that subsets of 4 x 4 and 6 x 6 optical flow patches can be modeled a circle, which confirm some results of 4 x 4 and 6 x 6 optical flow patches by using a new method-NEB, and expend Adams et al's result to larger patches of optical flow.


Introduction
Optical flow is the pattern of clear motion of image objects between two continuous frames caused by the movement of objects [1]. Optical flow has many applications, for example, in object segmentation, motion estimation and video compression. Because the difficulty of collecting data of optical flow statistics, the spatial statistics of optical flow are comparatively unexploited. To analyze non-lineal high-dimensional data is a difficult problem, computational topology is an efficient method for analyzing non-lineal high-dimensional data, recently another tool (NEB) of analyzing nonlinear high-dimensional data appeared. In 2015, Adams, Atanasov, and Carlsson [2] used the nudged elastic band (NEB) technique to analyze data of optical flow, they found a new topological properties of an optical flow patches, that is, optical flow patches have a circle behavior, which initially demonstrate that the NEB method is an effective tool for high dimensional nonlinear data analysis.
In this paper, we use the nudged elastic band method [2] to high-contrast patches of optical flow, analyze , and optical flow patches, and identify topological structures in these data sets, the datasets used here comes from a database built in [3]. 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. This proves once again that NEB is an effective tool for analyzing high-dimensional nonlinear data sets and identifying their topological properties.

The spaces of optical flow patches
Our data are picked from the Roth and Black optical flow database [3], a sample is shown in Fig.1.
For each flow field sequence in the database, we utilize the second optical flow frame of a sequence, and randomly select data sets of high contrast 4 4 , and 6 6 patches from the optical flow database. The spaces M 4 , and M 6 are sets of 4 4 , and 6 6 patches of high contrast as created in [4], which is similar as [5], [6], [7]. And we use the set symbols of the paper [4]. Undefined concepts and symbols in this paper can be found in [2], [4~7].
We should note that one 4 4 patch is arranged as 1

Computing method
The outline of the computing method from the paper [2] is listed as following, for more details, please refer to [2].

Nudged elastic band
The nudged elastic band (NEB) comes from computational chemistry, that is an efficient method for finding a minimum energy path between a given two states. A NEB is computed by a optimization algorithm from an initial path according to the total force, the initial path is usually built from linear interpolation between the initial and final states. For more details about NEB, please refer to [8], [9], [10]. . The sequence { } n y converges to a local maxima of g [11]. In order to identify different 0-cells, we use single-linkage clustering to cluster the convergent points, and choose the densest member from each cluster as a 0-cell.

1-cells
For two 0-cells, there is a 1-cell between them if we find a convergent band between them by using NEB. For an initial band [U 0 , U 1 , ..., U N ], where U 0 and U N are 0-cells. The total force on each midterm node U i is computed by ( 2 ) n c e is the gradient constant. The smoothing force F sm is given by the similar formula as in [2].
Given a set n X from unknown probability density function : , we construct CW complex models Z to approximate the super-level sets X .
We construct only one-dimensional cell complexes. First step, we establish a differentiable density estimator to approach the unknown probability density function by using the given set. Then, we find local maxima of the density estimate to get 0-cells. Lastly, we randomly produce initial bands, then find the convergent bands by NEB, thus we get 1-cells. Hence we can detect the topology properties of the data set X by constructing cell complexes Z .

Experimental results
The author of the paper [4] use persistent homology to discuss the topological structure of spaces 4 M and 6 M of 4 4 and 6 6 Optical flow patches, and discover that the homology of the subsets changes from a circle to a 3-circle space. Specially, he experimentally show that there exist core subsets   (Fig.2).  (Fig.3).

Conclusion
In this short note we utilize the nudged elastic band method to discuss topological qualitative analysis of optical flow small patches. We experimentally prove that the spaces of 4 4 and 6 6 high contrast patches have subsets modeled as a circle. The most important parameter of the NEB method is standard deviation , we should choose its proper values to get a stable result. The advantage of cell complexes for analyzing high-dimensional dada is its simplicity. For example, we only use several cells to model 6 (200,30) MS as a circle, for the same result, we need several tens of thousands witness complexes to model 6 (200,30) MS as a circle. If we apply witness complexes to analyze subsets of 4 M and 6 M , we detect that the topology of the subsets changes from a circle to a 3-circle s space, but we can't find a 3-circle model for various subsets of 4 M and 6 M by cell complexes. Hence the NEB method may only identify coarse topology of a high-dimensional data set.