Lattice identification and separation: Theory and algorithm

Motivated by the problems of lattice mixture identification and grain irregularity detection, we present a framework for lattice pattern representation and comparison and propose an efficient algorithm for lattice separation. We define new scale and shape descriptors, which considerably reduce the size of equivalence classes of lattice bases. These finite number of equivalence relations are fully characterized by the modular group theory. We construct the lattice space L based on the equivalent descriptors and define a metric d_L to accurately quantify the visual similarities and differences between lattices. We introduce the lattice identification and separation algorithm (LISA), which identifies individual lattice patterns from superposed lattices. LISA finds lattice candidates from the high responses in the image spectrum, then extracts different layers of lattice patterns one by one. By analyzing the frequency components, we explore the intricate dependency of LISA's performances on particle radius, lattice density, and relative translations. Various numerical experiments are presented to show LISA's robustness against a large number of lattice layers, moiré patterns, and missing particles. © 2019 Society for Industrial and Applied Mathematics.