Mathematics, Computation and Geometry of Data

Volume 3 (2023)

Number 2

Geometric graph measures for textures classification

Pages: 59 – 99

DOI: https://dx.doi.org/10.4310/MCGD.2023.v3.n2.a1

Authors

Vladislav Barkanass (Intel Israel, Haifa, Israel)

Wei Chen (Dalian University of Technology, Dalian, China)

Na Lei (Dalian University of Technology, Dalian, China)

Emil Saucan (Braude College of Engineering, Karmiel, Israel)

Abstract

We propose a discrete curvature based approach to the intelligence and classification of textures in images, with special emphasis on natural ones. We make appeal to a number of discrete notions of curvature eminently suited for this task, namely both the graph and the full Forman–Ricci curvatures, the Ollivier–Ricci curvature and the Menger curvature measure. Furthermore, we consider a different type of geometric network measure, inspired by the early work of Duffin on electrical networks and stemming from Complex Function Theory, namely the so called network modulus. Combining these geometric measures with comparison methods developed originally for the study of networks, we are able to distinguish and classify various types of textures. In particular, we show that stochastic textures are not essentially distinguishable from other types of natural textures.

Keywords

stochastic textures, Forman–Ricci curvatures, Ollivier–Ricci curvatures, Menger curvature measure, graph modulus

2010 Mathematics Subject Classification

Primary 51K10, 68U10. Secondary 65D18.

Received 29 April 2023

Published 15 July 2024