Abstract
In this paper we use the nudged elastic band tech-nique from computational chemistry to investigate sampled high-dimensional data from a natural image database. We randomly sample 8 × 8 and 9 × 9 high-contrast patches of natural images and create a density estimator believed as a Morse function. By the Morse function we build one-dimensional cell complexes from the sampled data. Using one-dimensional cell complexes, we identify topological properties of 8 × 8 and 9 × 9 high-contrast natural image patches, we show that there exist two kinds of subsets of high-contrast 8 × 8 and 9 × 9 patches modeled as a circle, by the new method we confirm some results obtained through the method of computational topology.
Highlights
Computational topology becomes a very important and efficiently method to analyse high-dimensional dada in the recent years [1], [2], [3], [4]
Atanasov, and Carlsson [5] used the nudged elastic band method to construct cell complexes through density functions of sampling data, they built more low-priced reasonable models for some nonlinear data sets by a few of cell complexes, and effectively detect the homology of the nonlinear data sets, it initially shows that cell complex models are efficient ways for analysing high-dimensional nonlinear data
In the paper [7], Xia shown that there exist some core subsets of 8 × 8 and 9 × 9 natural image patches that are topologically equivalent to a circle and the three circle model respectively
Summary
Computational topology becomes a very important and efficiently method to analyse high-dimensional dada in the recent years [1], [2], [3], [4]. To analyse high-dimensional dada, we usually construct a sequence of simplicial complexes from the finite sampled data set to produce a simple combinatorial presentations of the data, the most commonly used complexes include Cech complexes, Rips complexes and lazy witness complexes. Atanasov, and Carlsson [5] used the nudged elastic band method to construct cell complexes through density functions of sampling data, they built more low-priced reasonable models for some nonlinear data sets (such as, sets generated from social networks, from range image analysis, and from microarray analysis) by a few of cell complexes, and effectively detect the homology of the nonlinear data sets, it initially shows that cell complex models are efficient ways for analysing high-dimensional nonlinear data. In the paper [7], Xia shown that there exist some core subsets of 8 × 8 and 9 × 9 natural image patches that are topologically equivalent to a circle and the three circle model respectively
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