Symmetry-aware super-resolution of crystal orientation maps via invariant latent-space learning

Umang Garg; Warren Zamudio; McLean P. Echlin; Samantha H. Daly; Tresa M. Pollock; B. S. Manjunath

Source-linked AI summary

Symmetry-aware super-resolution of crystal orientation maps via invariant latent-space learning

Umang Garg, Warren Zamudio, McLean P. Echlin, Samantha H. Daly, Tresa M. Pollock, B. S. Manjunath

arXiv:2609.10898v1Sep 9, 2026cs.CVcond-mat.mtrl-scics.LG

TL;DR

EBSD super-resolution must handle orientations defined on SO(3)/G while preserving discontinuous grain boundaries. SG-SRAN uses a frozen symmetry-invariant, locally isometric latent representation with routed local synthesis, achieving strong cross-material accuracy and zero-shot transfer with only 27–49k trainable parameters. Its main limitations are difficult extreme orientation choices, costly dictionary decoding, and limited validation breadth.

Problem
EBSD orientations are defined only up to crystal symmetry, so Euclidean interpolation can mix equivalent representations and blur physically distinct grain boundaries.
Method
SG-SRAN performs super-resolution in a frozen symmetry-invariant, locally isometric latent space and routes each high-resolution token to a feature-consistent local orientation branch before dictionary decoding.
Results
Across FCC and HCP benchmarks, the 27–49k-parameter SG-SRAN configurations perform strongly on tail-error, boundary-sensitive, and fidelity metrics and transfer zero-shot to unseen alloys.
Takeaways & Limitations
The results support evaluating orientation-field super-resolution jointly through angular accuracy, boundary localization, local orientation-branch recovery, and compactness.
Takeaways & Limitations
Extreme p95–p99 errors remain comparable to other learned methods, decoding dominates inference time at approximately 4.63 s per patch, and broader cross-specimen validation remains necessary.

Abstract

from arXiv · show

Crystal-orientation maps are physical fields defined only up to crystal symmetry; electron backscatter diffraction (EBSD) resolves them experimentally, but acquisition-time constraints limit spatial resolution. Unlike conventional images, EBSD data lie on the quotient space $\mathrm{SO}(3)/G$, where $G$ is the crystal-symmetry group. Standard Euclidean interpolation can therefore mix symmetry-equivalent representations and blur grain boundaries. We introduce the Symmetry-Group-Aware Super-Resolution Attention Network (SG-SRAN), which incorporates crystal symmetry and boundary preservation by design. A frozen, locally isometric encoder maps equivalent orientations to a common latent representation in which Euclidean distance approximates misorientation. Super-resolution is performed in this space, with each high-resolution token restricted to a feature-consistent local support to prevent cross-boundary mixing. A dictionary-based decoder then recovers valid orientations. Across FCC and HCP benchmarks, SG-SRAN matches 15-16 million parameter backbones using only 27-49k trainable parameters, while achieving the lowest p68 errors, highest inverse-pole-figure fidelity, and zero-shot transfer to unseen alloys.

1 Results

SG-SRAN performs symmetry-aware super-resolution in a locally isometric latent space, using routed local supports to preserve distinct orientation branches near grain boundaries. Across FCC and HCP benchmarks, it combines competitive or leading accuracy with very small trainable backbones, strong boundary diagnostics, and zero-shot transfer.

Pipeline overview: Latent-space routing partitions local support into orientation slots and assigns each high-resolution token to one feature-consistent branch before refinement.The pipeline performs encoding, latent refinement, routed upsampling, HR refinement, and dictionary decoding; routing is designed to avoid merging distinct grains across boundaries.
Encoder fidelity: The frozen encoder and dictionary decoder reconstruct held-out orientations with milliradian-level mean symmetry-aware errors of 0.0076 rad for FCC IN718 and 0.0079 rad for HCP Ti-6Al-4V.This round-trip test isolates the fidelity of the latent interface before spatial upsampling.
Benchmark on held-out IN718 and Ti-6Al-4V at 4 × 4: SG-SRAN leads key accuracy and fidelity measures with 49k parameters on IN718 and 27k on Ti-6Al-4V, while competing methods lead selected individual metrics.On IN718 it leads p68/p95 errors and IPF fidelity; on Ti-6Al-4V it leads pooled mean, median, boundary-band mean, Boundary F1, PSNR, and SSIM.
Grain Boundary analysis: SG-SRAN recovers local orientation branches at boundaries with Composition F1 scores of 0.891 on IN718 and 0.782 on Ti-6Al-4V, close to SAN’s leading values.The metric evaluates whether reconstructed boundary windows recover the distinct ground-truth orientation groups without spurious observations.
Grain Boundary analysis: Routing accounts for nearly all decoded-error reduction, with 98.0% on IN718 and 99.7% on Ti-6Al-4V; replacing it with latent interpolation lowers IN718 Boundary F1 from 0.656 to 0.445 or 0.425.The stagewise analysis shows little additional mean-error change from subsequent HR refinement.
Zero-shot transfer: Without target retraining, SG-SRAN transfers across symmetry classes to CoNi and Ti-Al, remaining strongest on several angular and fidelity metrics while competitors lead selected composition or boundary measures.It leads the reported Ti-Al metrics listed in the passage and remains strongest on CoNi mean, median, p68, p95, and IPF fidelity.

2 Discussion

SG-SRAN treats orientation super-resolution as reconstruction on SO(3)/G and combines geometry-aware representations with local routing to preserve interfaces. Across crystal systems, it performs strongly on tail-error and boundary-sensitive metrics, while decoding cost and validation breadth remain limitations.

2 Discussion: SG-SRAN combines quotient-space geometry with local routing to produce accurate, physically valid outputs while preserving distinct orientation branches at grain boundaries.The encoder gives nearby latent distances a crystallographic interpretation, while routing assigns among local branches instead of averaging across interfaces.
2 Discussion: Across five seeds, the compact 27–49k-parameter model performs strongly on tail-error and boundary-sensitive metrics across both crystal systems, although competitors lead individual measures.
2 Discussion: Dictionary decoding can make valid but incorrect orientation selections, with extreme Ti-6Al-4V p95–p99 errors comparable to other learned methods.Dictionary search and refinement require approximately 4.63 s per patch, compared with 28.5 ms for encoding and latent super-resolution.
2 Discussion: Evidence from CoNi and Ti–Al transfers is limited to FCC and HCP symmetry classes, so broader cross-specimen validation remains necessary.
2 Discussion: The findings motivate router calibration, learnable decoding, and broader cross-specimen validation as directions for improving the representation-first strategy.

3 Methods

The methods define notation for quaternion fields, the frozen encoder and dictionary decoder, the symmetry-aware metric, and latent feature fields across backbone stages. The evaluated spatial upsampling factor is r = (4, 4).

3 Methods: The method denotes low- and high-resolution quaternion fields by QLR and QHR, the frozen encoder and decoder by E and D, and the symmetry-aware metric by dG.
3 Methods: The studied task uses spatial upsampling factor r = (4, 4), with successive latent backbone fields denoted Z0, . . . , Z4.

3.1 Crystal orientations and quotient geometry

Crystal orientations are represented as active crystal-to-specimen rotations and scalar-first unit quaternions, with crystal symmetry acting on the right. The physical orientation space is the quotient SO(3)/G, and symmetry-aware errors identify equivalent orientations as identical.

3.1 Crystal orientations and quotient geometry: An orientation is represented by an active rotation R satisfying vs = Rvc and by a scalar-first Hamilton unit quaternion q.
3.1 Crystal orientations and quotient geometry: The antipodal quaternions q and −q represent the same rotation, so quaternion space double-covers SO(3).
3.1 Crystal orientations and quotient geometry: Passive specimen-to-crystal inputs are inverted once at import, after which R and q consistently denote active crystal-to-specimen orientations.
3.1 Crystal orientations and quotient geometry: Crystal symmetry acts from the right as R ∼ Rg, making the physically meaningful orientation space the quotient manifold SO(3)/G.For FCC metals, the proper rotational crystal point group is the cubic rotation group with 24 elements.
3.1 Crystal orientations and quotient geometry: The fundamental zone selects a representative from each symmetry orbit, while the symmetry-aware angular error is zero for symmetry-equivalent orientations.

3.2 Locally isometric orientation encoder

The encoder maps crystal-symmetry-equivalent orientations to invariant latent features while retaining SO(3)-equivariant structure. Degree selection and metric normalization make local latent distances track crystallographic misorientation, with accuracy degrading beyond approximately 0.2 rad.

Symmetry-invariant orientation encoding: Crystal-symmetry-equivalent orientations map to identical latent representations, while global rotations act equivariantly on the feature blocks.The encoder uses Reynolds-projected Wigner-D features and distinguishes right crystal-symmetry invariance from left global-rotation equivariance.
Finite harmonic descriptors: The encoder retains non-scalar invariant blocks, using l ≤4 for FCC and l ≤6 for HCP; the HCP descriptor has 40 dimensions.For HCP, the retained blocks are 1×2e + 1×4e + 2×6e, giving 5 + 9 + 2 × 13 = 40 dimensions.
Local-isometry normalization: Metric normalization makes the combined local feature metric equal to I3 up to numerical precision.Before normalization, the HCP equal-weight metric has only small anisotropy, λmax/λmin = 37/36 ≈ 1.03.
Local-isometry normalization: Mean-squared error in normalized latent space tracks squared geodesic orientation error to leading order.The approximation remains tight for misorientations up to approximately 0.2 rad, or 11°, before higher-order Taylor terms become non-negligible.
Representation scope: Finite degree sets produce computationally tractable representations whose local-isometry analysis characterizes infinitesimal geometry, while round-trip error measures implemented orientation recovery.The selected degrees determine harmonic content, latent dimension, and computational cost.

3.3 Orientation-cluster routed-patch super-resolution backbone

The routed-patch backbone performs super-resolution entirely in the symmetry-invariant latent space. It clusters local orientation supports, synthesizes token-conditioned proposals, and assigns each high-resolution token to a selected local orientation branch.

Latent-space upsampling: SG-SRAN predicts one 4×4 high-resolution latent patch per low-resolution pixel from an orientation-conditioned local support bank.The FCC and HCP backbones operate on their respective frozen invariant descriptors rather than directly on quaternions.
Feature-consistent local support: Latent-direction masking restricts convolutional context to aligned neighbors, suppressing crystallographically dissimilar contributions.The cosine-similarity gate uses τcos = 0.97, followed by equivariant tensor products and residual connections.
Orientation clustering: Local orientation regions are formed by connected components in an 8-neighbor clustering graph and packed into up to K = 6 deterministic slots.Adjacency corresponds under local isometry to 2° for IN718 and 5° for Ti-6Al-4V.
Token proposal synthesis: Cross-attention synthesizes token-conditioned proposals from slot members, allowing different high-resolution tokens to attend to different members of one local orientation cluster.The attention weights are invariant scalars and the values are equivariant features, so the emitted proposals remain equivariant.
Geometric routing: A router uses complete slot-composition masks and subpixel position to select each token’s owner slot with a straight-through argmax.Empty slots are masked, and a center-prior weight of 5.0 biases selection toward the slot containing the window center when occupancy evidence is weak.

3.4 Dictionary-based decoder and fundamental-zone canonicalization

The decoder converts predicted latent features into valid canonical quaternions through dictionary retrieval, local refinement, and fundamental-zone canonicalization. Its frozen lookup-and-refinement procedure also determines the current inference bottleneck.

Dictionary retrieval and refinement: The decoder retrieves the 2 nearest entries from a 857,973-quaternion fundamental-zone dictionary, then refines candidates by minimizing latent reconstruction error.Candidates are renormalized after each Adam step, and the lowest-residual candidate is selected.
Fundamental-zone canonicalization: The selected quaternion is canonicalized to the fundamental zone, ensuring that decoding returns a valid unit-quaternion orientation representation.The dense cubochoric dictionary establishes a small frozen-interface fidelity floor below downstream super-resolution errors.
Decoding cost: 4.63 s of the 4.65 s end-to-end IN718 inference time is spent in dictionary decoding and refinement.The encoder takes 0.70 ms and the super-resolution backbone 27.8 ms per 256 × 256 HR patch on an NVIDIA A100-PCIE-40GB.

3.5 Training objective and implementation

Training minimizes feature-space error between predicted and frozen target latent maps, using the local-isometry relation as a surrogate for squared geodesic error. The implementation combines e3nn equivariant operations with precomputed invariant bases and fixed training schedules.

Training objective: The backbone is trained with feature-space mean-squared error between predicted high-resolution latents and detached frozen encoder targets.The feature loss is optimized alone, while the geometric router and straight-through owner assignment learn end to end from its gradient.
Geometric interpretation: The feature loss is a first-order surrogate for squared geodesic orientation error in the locally isometric small-angle regime.The correspondence remains empirically effective at larger angles.
Implementation: Runtime invariant features use precomputed Reynolds projector bases and GPU spherical-harmonic synthesis, while eigensolve machinery is offline.This separates basis construction from runtime feature evaluation.
Optimization: Training uses AdamW with cosine annealing, two warmup epochs, gradient clipping, mixed precision, and 150 epochs.The peak learning rate is 3 × 10^-4, the minimum is 10^-6, and weight decay is 10^-4.

3.6 Datasets and evaluation protocol

The evaluation uses FCC IN718 and HCP Ti-6Al-4V datasets, aligned 4 × 4 LR/HR patches, classical and learned baselines, and symmetry-aware angular, image-space, and boundary metrics.

Datasets: The study uses IN718 FCC and Ti-6Al-4V HCP datasets, with 1174/147/147 and 581/73/72 train/validation/test paired patches, respectively.HR patch sizes are 256 × 256 for IN718 and 128 × 128 for Ti-6Al-4V.
Evaluation scope: The frozen encoder–dictionary-based decoder round trip is reported as a representation analysis, whereas SG-SRAN is trained and evaluated end to end on both materials.Datasets and encoded lookup tables are planned for public deposition, and code, checkpoints, basis files, and dictionaries are planned for release.
Dataset construction: LR patches are formed by regular stride-4 subsampling of aligned HR patches, preserving spatial registration without an additional interpolation step.The construction models the spatial-resolution component of the LR/HR trade-off.
Baselines: SG-SRAN is compared with four classical interpolants and seven learned baselines trained independently on the same evaluation framework.The classical methods include nearest, bicubic, SLERP, and symmetry-aware SLERP; learned baselines include quaternion, real-valued, attention, and inpainting models.
Metrics: Angular accuracy is evaluated with symmetry-aware misorientation dG, using the 24 proper rotations of O for FCC and 12 proper rotations of D6 for HCP.Super-resolution errors are pooled over held-out HR pixels and summarized by mean, median, p68, p95, and p99.
Metrics: Boundary recovery is assessed with Boundary F1 and boundary-window composition metrics, while IPF-X, IPF-Y, and IPF-Z renderings provide averaged PSNR and SSIM measures.Boundary masks use a 5° crystallographic misorientation threshold, and composition analysis evaluates spurious rate, recall, and Composition F1.

Extended Data

The extended data document the EBSD/IPF representation, latent encoder–decoder interface, visual benchmark comparisons, stagewise and boundary-focused behavior, scale dependence, and zero-shot transfer diagnostics.

EBSD and IPF representation: IPF maps encode the crystal direction parallel to a selected sample axis, making contiguous grains and sharp orientation changes visually legible.The IPF-Z representation uses the crystal direction aligned with the sample Z axis and a colored fundamental triangle.
Frozen orientation interface: The frozen encoder maps orientations to symmetry-invariant latent features, which are matched against a cubochoric-sampled dictionary and decoded to unit quaternions.The 4 × 4 annotation refers to the surrounding super-resolution task rather than the encoder–decoder interface itself.
Held-out comparisons: SG-SRAN visual comparisons include LR inputs, multiple interpolants or learned baselines, SG-SRAN outputs, and HR targets for IN718 and Ti-6Al-4V.The extended figures show common IPF views and residual displays across held-out examples.
Stagewise analysis: The routed upsampler accounts for 98.0% of the encoded-LR-to-final decoded-error decrease on IN718 and 99.7% on Ti-6Al-4V.Stagewise visualizations include decoded LR embeddings, context aggregation, routed patch synthesis, final outputs, HR targets, and boundary maps.
Boundary routing: In a boundary-rich IN718 crop, route changes and lower router confidence align with grain-boundary neighborhoods.The walkthrough compares owner-slot assignments, router confidence, the HR boundary mask, and the owner-boundary overlay.
Scale dependence: Scale-dependent comparisons report normalized median misorientation, IPF-space PSNR and SSIM, Boundary F1, and boundary-window Composition F1 at 2 × 2, 4 × 4, and 8 × 8.The plots cover IN718 FCC with O symmetry and Ti-6Al-4V HCP with proper D6 symmetry.
Zero-shot transfer: Zero-shot figures evaluate IN718-trained models on CoNi and Ti-6Al-4V-trained models on Ti-Al without target retraining.Extended Data Table 2 applies the same boundary-window composition diagnostic to the headline out-of-distribution targets.