Comparison of Objects’ Images based on Computational Topology Methods

Abstract

The paper considers methods for comparison of objects’ images represented by sets of points using computational topology methods. The algorithms for construction of sets of real barcodes for comparison of objects’ images are proposed. The determination of barcodes of object forms allows us to study continuous and discrete structures, making it useful in computational topology. A distinctive feature of the use of the proposed comparison methods versus the methods of algebraic topology is obtaining more information about objects’ form. An important area of application of real-valued barcodes is studying invariants of big data. Proposed method combines the technology of barcodes construction with embedded non-geometrical information (color, time of formation, pen pressure), represented as functions of simplicial complexes. To do this, barcodes are expanded with functions from simplexes to represent heterogeneous information. The proposed structure of extended barcodes increases the effectiveness of persistent homology methods when comparing images and pattern recognition. A modification of the Wasserstein method is proposed for finding the distance between images by introducing non-geometric information about the distances between images, due to inequalities of the functions of the source and terminal images of the corresponding simplexes. The geometric characteristics of an object can change with diffeomorphic deformations; the proposed algorithms for the formation of expanded image barcodes are invariant to rotation and translation transformations. We considered a method for determining the distance between sets of points representing the curves, taking into account an orientation of curves’ segments. The article is intended for a reader who is familiar with basic concepts of algebraic and computational topology, the theory of Lie groups, and diffeomorphic transformations.