Interpretable liquid crystal phase classification via two-by-two ordinal patterns

Abstract

Liquid crystal textures encode rich structural information, yet mapping these images to mesophase identity remains challenging because visually similar patterns can arise from distinct structures. Here we present a simple, interpretable representation that maps textures to a 75-dimensional frequency vector of two-by-two ordinal patterns, grouped into eleven symmetry-based types to characterize a large-scale dataset spanning seven mesophases. Combined with a simple machine learning classifier, this lightweight representation yields near-perfect phase recognition, including the difficult distinction between smectic A and smectic B mesophases. Our approach generalizes to unseen compounds and accurately distinguishes between phase identity and material origin. Unlike deep learning methods, each ordinal pattern is readily interpretable, and model explanations augmented with network visualizations of pattern interactions reveal the specific types and pairwise dependencies that drive each mesophase decision, providing compact, physically meaningful summaries of texture determinants. These results establish two-by-two ordinal patterns as an interpretable and scalable tool for liquid crystal image analysis, with potential applications to other complex patterned systems in materials science.

Figures (8)

• Figure 1: Visualization of the $75$ possible two-by-two ordinal patterns. Patterns are aggregated into 11 groups (labeled with capital letters from A to K) according to their number of unique symbols (distinct intensity levels) and smoothness degree tarozo2025two. Each two-by-two matrix represents one ordinal pattern, where the entries are ranks of pixel intensities within an image partition. For example, the single type A pattern, $[0000]$, corresponds to a uniform partition where all pixels have the same intensity, whereas the first type B pattern, $[0123]$, indicates that pixel intensities increase in rank from top-left to bottom-right within an image partition. Dashed outlines highlight the primary patterns whose rotations by quarter, half, and three-quarter turns yield all remaining patterns of the corresponding group.
• Figure 2: Mapping the ordinal-pattern space of liquid crystal textures. The main panel depicts a UMAP projection of the ordinal-pattern space, where each dimension in the original space is the normalized probability of one of the 75 possible two-by-two ordinal patterns. Each data point represents a texture in our dataset, with mesophases encoded by distinct markers and colors. The surrounding inset panels show the ordinal distributions (circular bars on a logarithmic scale) and the corresponding texture for one randomly selected example per mesophase; their locations in the UMAP plane are indicated by numbers from 1 to 7. An interactive version is provided in Ref. voltarelli2026umaplc.
• Figure 3: Overall prevalence of ordinal patterns across liquid crystal textures. Bars show the (a) average, (b) the standard deviation, and (c) the coefficient of variation (standard deviation divided by the mean) of the probability of each two-by-two ordinal pattern, calculated across all images. Patterns are arranged according to their 11 groups, indicated by colors and separated by vertical dashed lines labeled at the top. Numbers above bars denote the mean within each pattern group, and the insets in panels (a) and (b) magnify probabilities for groups D to K.
• Figure 4: Ordinal-pattern fingerprints of each liquid crystal mesophase. (a) Matrix plot depicts the $z$-score probability of each two-by-two ordinal pattern (columns) across all mesophases (rows). Patterns are grouped by type and separated by vertical dashed lines. (b) Matrix plot of the $z$-score probability calculated for each pattern type (columns) and mesophase (rows). For both panels, $z$-scores are calculated by subtracting the average probability of a pattern or pattern type within a particular phase from the overall average probability and dividing the result by the overall standard deviation across all images. Positive values (green shades) indicate patterns or types that occur more frequently than the overall average, whereas negative values (purple shades) denote those that occur less frequently. Bar plots show (c) the distance to the average pattern, determined by the absolute sum of the $z$-score probabilities for each phase, and (d) the mean absolute coefficient of variation of the ordinal probabilities (in $z$-score units) calculated across all images within each mesophase.
• Figure 5: Classifying mesophases with two-by-two ordinal patterns. (a, b) Training (circles) and cross-validation (triangles) accuracy as a function of the number of neighbors $k$ in the nearest-neighbor classifier. (c, d) Training (circles) and cross-validation (triangles) accuracy as a function of the fraction of data used for training the nearest-neighbor algorithm with $k=1$. Shaded regions in panels (a-d) represent one standard deviation band from a five-fold cross-validation procedure. (e, f) Confusion matrices showing the fraction of classifications on the test set for each true (rows) and predicted (columns) phase. Upper panels (a, c, and e) use the prevalences of the 75 two-by-two ordinal patterns as features; lower panels (b, d, and f) use the prevalences of the 11 pattern types. The inset in (d) compares the average accuracy obtained with the 75 patterns (red) and the 11 pattern types (blue) to a dummy classifier that predicts mesophases uniformly at random (green). Results in (e), (f), and the inset are averages over 50 independent train–test splits with 80% of the data used for training and $k=1$; error bars in the inset denote one standard deviation.