Lattice Identification and Separation: Theory and Algorithm

TL;DR

This work develops a geometric and algorithmic framework for identifying and separating multiple lattices in images. It introduces a lattice space $\mathscr{L}$ built from scale and shape descriptors $\beta$ and $\rho$, with a metric $d_{\mathscr{L}}$ grounded in modular-group actions and the Poincaré metric, enabling quantitative comparison of lattice patterns. The main algorithm, LISA, formulates lattice separation as a variational problem and extracts lattices sequentially via Fourier-domain peak analysis, optional refinement, and remainder updates, performing robustly under moiré effects, missing particles, and Gaussian perturbations. The paper provides analytical insights into how spectrum, translations, particle size, and density affect performance, supports the approach with extensive numerical experiments, and highlights practical applications to materials science and grain-boundary analysis. Overall, it delivers a principled, unsupervised method for decomposing complex lattice mixtures with strong theoretical underpinning and empirical validation.

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, moiré patterns and missing particles.

Figures (24)

• Figure 1: [Challenges of pattern separation] Each images have two lattices superposed. (a) Each red squares are units of one lattice, yet they have different interior patterns which can confuse the texton approach. (b) Using non-superposed lattice identification method, such as MSBP, wrong local feature L-shapes (the red arrows) can be identified. These red arrows do not correspond to true underlying lattices. (c) The pink and the yellow L-shapes on the top-left corner denote the true lattice components. There are three different types of Moiré patterns present (red, blue and green regions).
• Figure 2: [Equivalent lattice and minimal bases] (a) $\Lambda(3,4i)$, (b) $\Lambda(4i,-3+4i)$, and (c) $\Lambda(-3-4i,-6-4i)$ are all equivalent. (a) is a minimal basis: $|\text{Re}(\frac{4i}{3})|=0<\frac{1}{2}$. (b) is not minimal: $|\text{Re}(\frac{-3+4i}{4i})|=1>\frac{1}{2}$, and (c) is not positive: $\text{Im}(\frac{-6-4i}{-3-4i})=-\frac{12}{25}<0$.
• Figure 3: [Descriptors $\beta$ and $\rho$] (a) $\Lambda\langle 1,i\rangle$, (b) $\Lambda\langle 2,i\rangle$, (c) $\Lambda\langle e^{i\pi/6},i\rangle$, (d) $\Lambda\langle 1,2i\rangle$, (e) $\Lambda\langle 1,e^{2\pi i/3}\rangle$, and (f) $\Lambda\langle 2,e^{2\pi i/3}\rangle$. From (a) to (b), only $\beta$ changed from 1 to 2. From (a) to (c), $\beta$ is rotated. From (a) to (d), $\rho$ changed from $i$ to $2i$. From (a) to (e), $\rho$ is rotated,. From (a) to (f), both $\beta$ changed and $\rho$ rotated.
• Figure 4: The shaded region in (a) with the boundary is $\mathcal{P}$. Vectors $\rho$, $\rho'$ and $\rho"$ are the shape descriptors for the bases $\Lambda(3,4i)$, $\Lambda(4i,-3+4i)$, and $\Lambda(-3-4i,-6-4i)$ in Figure \ref{['F:basisdemo']} (a)-(c) in order. All represents the same lattice, while $\rho$ for $\Lambda(3,4i)$ is a minimal basis. (b) A fundamental set of the modular group $\Gamma$ acting on the upper half plane $\mathcal{H}$. If $\text{Re}(\rho)=-1/2$, $\rho$ and $\rho+1$ are in the same orbit.
• Figure 5: [Examples of subspaces of $\mathscr{L}$] (a) A square lattice $\Lambda\langle \beta, i \rangle$. The red and blue arrows indicate two directions. Stretching along them represents two different families of lattices. They form a subspace of $\mathscr{L}$ shown in (b), and it is homeomorphic to $\mathbb{R}$ as in (c). The second row (d) shows a lattice $\Lambda\langle \beta, e^{i\pi/3} \rangle$. Stretching along the three marked directions generates three distinct families of lattices. (e) is the subspace they form in $\mathscr{L}$, which is homeomorphic to (f).

Theorems & Definitions (21)

• Definition 2.1: 2D Lattice, Basis
• Definition 2.2: Equivalent Bases
• Theorem 2.3: Fourier Slice Theorem
• Definition 2.4: Quotient pseudometric
• Definition 2.5: Product Metric
• Remark 2.6
• Remark 2.7
• Definition 3.1: Scale and Shape Descriptor
• Proposition 3.2: Necessary condition
• proof