# Lattice Identification and Separation: Theory and Algorithm

## TL;DR

This paper introduces a comprehensive framework for representing, comparing, and separating lattice patterns in images, including new descriptors, a metric, and an algorithm capable of handling complex lattice mixtures.

## Contribution

It presents a novel lattice representation framework, a metric for lattice similarity, and the Lattice Identification and Separation Algorithm (LISA) for extracting multiple lattice patterns from images.

## Key findings

- LISA effectively identifies multiple lattice layers.
- The framework reduces the complexity of lattice equivalence classes.
- LISA demonstrates robustness against noise, moiré patterns, and missing particles.

## Abstract

Motivated by lattice mixture identification and grain boundary 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 helps to considerably reduce the size of equivalence classes of lattice bases. These finitely many equivalence relations are fully characterized by modular group theory. We construct the lattice space $\mathscr{L}$ based on the equivalent descriptors and define a metric $d_{\mathscr{L}}$ to accurately quantify the visual similarities and differences between lattices. Furthermore, we introduce the Lattice Identification and Separation Algorithm (LISA), which identifies each lattice patterns from superposed lattices. LISA finds lattice candidates from the high responses in the image spectrum, then sequentially extracts different layers of lattice patterns one by one. Analyzing the frequency components, we reveal the intricate dependency of LISA's performances on particle radius, lattice density, and relative translations. Various numerical experiments are designed to show LISA's robustness against a large number of lattice layers, moir\'{e} patterns and missing particles.