The determination of distances between images of objects based on persistent spectra of eigenvalues of Laplace matrices

Abstract

The work uses the method of filtering simplicial complexes, similar to the method used in the formation of persistent homology. The filtering process creates a number of nested simplicial complexes encoded with topological information. In papers [1-6] algorithms for the formation of persistent barcodes are used to compare images of objects. The use of persistent homology in relation to the methods of traditional algebraic topology provides additional information about the image of an object. To increase the diversity of information and the number of machine learning features, this work proposes algorithms for the formation of persistent spectra of eigenvalues of Laplace matrices for comparing images of objects. When comparing the shapes of objects, it is proposed to construct a modified Wasserstein distance based on the determination of the spectra of the eigenvalues of the Laplace matrix of the compared shapes.