Determination of features based on the formation of persistent spectra of eigenvalues of Laplace matrices

An algorithm for determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes has been developed in the paper. The spectrum of eigenvalues of the Laplace matrix is used as features in the data structure for image analysis. Similarly to the method of persistent homology, the filtering of embedded simplicial complexes is formed, approximating the image of the object, but the topological features at each stage of filtration is the spectrum of eigenvalues of the Laplace matrix of simplicial complexes. The spectrum of eigenvalues of the Laplace matrix allows to determine the Betti numbers and Euler characteristics of the image. Based on the method of finding the spectrum of eigenvalues of the Laplace matrix, an algorithm is formed that allows you to obtain topological features of images of objects and quantitative estimates of the results of image comparison. Software has been developed that implements this algorithm on computer hardware. The method of determining the spectrum of eigenvalues of the Laplace matrix has the following advantages: the method does not require a bijective correspondence between the elements of the structures of objects; the method is invariant with respect to the Euclidean transformations of the forms of objects. Determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes allows you to expand the number of features for machine learning, which allows you to increase the diversity of information obtained by the methods of computational topology, while maintaining topological invariants. When comparing the shapes of objects, a modified Wasserstein distance can be constructed based on the eigenvalues of the Laplace matrix of the compared shapes. Using the definition of the spectrum of eigenvalues of the Laplace matrix to compare the shapes of objects can improve the accuracy of determining the distance between images.