Abstract
We used sampled data from a well-established database of natural images to construct a density estimator, which is designed to act as a Morse function. We constructed one-dimensional cell complexes through this Morse function by utilizing the nudged elastic band method. By making use of cell complexes, we detect some topological characterizations of natural image data, and we experimentally show that spaces of high-contrast n × n patches (n = 3, 4, 5, 6, 7) with different subsets have the same topology as a circle.
Highlights
A great deal of research on high-dimensional dada analysis has been carried out in recent years, and many insights have been gained by employing computational topology methods.1–4 To identify topological properties of sets of high-dimensional data, simplicial complexes, such as Rips complexes, and lazy witness complexes are usually applied to create simple combinatorial presentations of the data
We demonstrate that cell complex methods are effective for analyzing different sizes of patches of natural images
We have shown that the nudged elastic band (NEB) technique can be successfully applied to analyze topological spaces of m × m-patches of natural images
Summary
A great deal of research on high-dimensional dada analysis has been carried out in recent years, and many insights have been gained by employing computational topology methods. To identify topological properties of sets of high-dimensional data, simplicial complexes, such as Rips complexes, and lazy witness complexes are usually applied to create simple combinatorial presentations of the data. To identify topological properties of sets of high-dimensional data, simplicial complexes, such as Rips complexes, and lazy witness complexes are usually applied to create simple combinatorial presentations of the data. These simplicial complexes sometimes consist of too many simplices to allow for computing. Atanasov, and Carlsson applied the nudged elastic band (NEB) method to create cell complexes from Morse functions of the sampled data. Using this (NEB) method, we successfully established a more concise model for several datasets by using a few of the cell complexes. The data that have been examined in this paper were selected from the dataset used in Ref. 7
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