PaperlayerView exact revision on arXiv

Source-linked AI summary

KASALv2: Fully Automatic 3D Rotational Symmetry Classification and Axis Localization

Mengxin Zhang, Yulin Wang, Chen Luo, Yongzhe Li, Yijun Zhou

arXiv:2610.09534v1cs.CVcs.AIcs.GR

TL;DR

Manual or predefined symmetry annotations limit scalable analysis of 3D objects, despite the role of rotational symmetry in 6D pose estimation. KASALv2 automatically classifies symmetry, infers rotational order, and localizes axes across all eight canonical types; it reports 94.75% accuracy on GSO and up to 0.9% higher FoundationPose accuracy across five BOP datasets.

  • Problem

    Existing 3D rotational-symmetry annotations remain largely manual or require predefined types or orders, limiting scalability for large object collections.

    Source: §1 · p. 1

  • Method

    KASALv2 uses the Geometric Degeneration Hierarchy to classify symmetry, infer rotational order, and reconstruct canonical axes without geometric references.

    Source: §1 · p. 2

  • Results

    94.75% accuracy on GSO and up to 0.9% higher FoundationPose accuracy across five BOP datasets are reported.

    Source: §5 · p. 8

  • Takeaways & Limitations

    KASALv2 provides annotation-free rotational-symmetry priors for unseen-object pose estimation.

    Source: §1 · p. 2

    eADI/d is reported only for samples correctly identified in both symmetry type and order.

    Source: §4.1.3 · p. 7

Abstract

from arXiv · show

Rotational symmetry is an important prior in 6D pose estimation, improving pose accuracy and supporting symmetry-aware evaluation. However, current symmetry annotations for 3D objects remain largely manual or semi-automatic, often requiring predefined types or orders, which limits scalability. This work introduces a fully automatic, reference-free framework for symmetry-type classification, rotational-order identification, and full-axis localization across all eight canonical 3D rotational symmetry types. The method localizes a dominant high-order axis, infers its rotational order through self-consistency analysis, and reconstructs the complete symmetry structure under a hierarchy-guided formulation. A texture-aware extension further models appearance-induced reductions in rotational order while preserving axis orientations. Experiments on idealized and real-world datasets demonstrate strong accuracy and generalization, achieving 94.75% accuracy on 438 symmetric objects in GSO. Training FoundationPose with these priors improves accuracy by up to 0.9% across five BOP datasets, showing that automatically estimated rotational priors improve downstream 6D pose estimation. Code is available at https://github.com/WangYuLin-SEU/KASAL.

1. Introduction

KASALv2 addresses the need for scalable rotational-symmetry annotation with a fully automatic, reference-free framework spanning all eight canonical types. It uses the Geometric Degeneration Hierarchy to infer symmetry structure and reports results on real-world objects and downstream pose estimation.

1. Introduction

  • Existing approaches cover only subsets of the eight canonical types, depend on geometric references, or require manually specified symmetry orders or categories.Large-scale datasets need per-object rotational symmetry, but manual annotation is infeasible at tens-of-thousands-of-model scale.Source: §1 · p. 1
  • KASALv2 jointly determines symmetry type, rotational order, and all canonical axes without prior input or predefined geometric references.It replaces exhaustive directional enumeration with the Geometric Degeneration Hierarchy (GDH), which encodes type-specific geometric relations.Source: §1 · p. 2
  • The method localizes a dominant high-order axis, estimates its order through periodic self-consistency, and reconstructs the symmetry structure with a texture-aware extension.The extension models appearance-induced reductions in rotational order while preserving axis orientations.Source: §1 · p. 2
  • 94.75% accuracy on real-world symmetric objects, with FoundationPose accuracy improving by up to 0.9% on BOP.The paper reports generalization across DSRSTO, BOP, and GSO, and the FoundationPose result uses symmetry priors.Source: §1 · p. 2

2. Related Work

Related work varies in its use of geometric references and its coverage of canonical rotational-symmetry types. Prior work addresses symmetry identification, axis localization, and pose estimation, but leaves gaps in automation and coverage.

2. Related Work

  • Existing symmetry-analysis methods differ in reliance on geometric priors and in how many of the eight canonical types they cover.The field includes both symmetry-type identification and axis localization, the latter often assuming the type is known.Source: §2 · p. 2

2.1. Symmetry Type Identification

  • Reference-free symmetry-type methods avoid predefined anchors but remain limited in coverage; examples include reflection-based methods that fail on continuously symmetric objects and approaches restricted to continuous and second-order discrete conical symmetries.Reference-based methods use anchors such as centroids, midpoints, or principal axes and are sensitive to surface noise, occlusion, and misalignment.Source: §2.1 · p. 2

2.2. Symmetry Axis Localization

  • Many axis-localization methods target specific scenarios and cannot handle texture-induced symmetries, while an all-type method still requires manual type and order specification.Manual specification can create annotation ambiguity for high-order or composite symmetries.Source: §2.2 · p. 2

2.3. Rotational Symmetry in Pose Estimation

  • Pose-estimation methods either implicitly capture symmetry cues or use known symmetry priors; the latter support pose disambiguation and symmetry-aware evaluation.KASALv2's depicted workflow localizes a dominant high-order axis, infers its order, verifies candidate types, and reconstructs canonical axes.Source: §2.3 · pp. 2–3, Figure 2 · p. 3

3. Methodology

The methodology combines a computable Geometric Degeneration Hierarchy (GDH) with an operational pipeline for reference-free rotational-symmetry analysis. It identifies a dominant axis and its order, classifies the symmetry, reconstructs canonical axes, and accounts for texture-induced order reductions.

3. Methodology

  • The GDH encodes canonical rotational-symmetry families and their degeneration conditions, providing the structural basis for reference-free analysis.The method uses this hierarchy to map an unknown object to its symmetry family.Source: §3 · p. 3
  • Guided by GDH, the pipeline localizes the dominant high-order axis, estimates its order, determines the symmetry type, and reconstructs the full canonical axis configuration.A texture-aware extension models appearance-induced order reductions while preserving geometric axis orientations.Source: §3 · p. 3
  • The section progresses from GDH's structural relations and degeneration pathways to axis localization, order inference, classification, axis recovery, and texture-aware analysis.The operational stages apply the hierarchy to appearance-aware settings using texture-induced order reductions.Source: §3 · p. 3

3.1. Geometric Degeneration Hierarchy (GDH) for 3D Rotational Symmetry

The Geometric Degeneration Hierarchy (GDH) organizes the eight canonical rotational symmetry types through geometric reductions of axis structure. Its computable relations constrain feasible types and support classification and axis inference.

3.1. Geometric Degeneration Hierarchy (GDH) for 3D Rotational Symmetry

  • GDH reorganizes the eight canonical rotational symmetry types into an operational hierarchy of axis multiplicities, orders, and inter-axis relations that constrain feasible symmetry families.Source: §3.1 · pp. 3–4
  • The hierarchy starts from the maximally symmetric sphere and separates into dihedral–cyclic and Platonic degeneration lineages.Source: §3.1 · p. 4
  • The dihedral–cyclic lineage models discretization from D∞ into finite rotational structures with or without orthogonal two-fold axes, while loss of orthogonal axes yields single-axis configurations.Source: §3.1 · p. 4
  • The Platonic lineage captures reductions among multi-axis structures A5, S4, and A4, with transitions simplifying the number and orientation of axes.Source: §3.1 · p. 4
  • Within GDH, a localized dominant high-order axis and its order narrow candidate types, while a geometrically consistent secondary axis verifies the type and its intrinsic relations specify the remaining axes.Source: §3.1 · p. 4

3.2. High-order Axis Localization and Order Identification

The method first localizes a dominant high-order axis through reference-free mixed-order alignment on the unit sphere, then estimates its rotational order from periodic self-alignment. The resulting axis and order support subsequent hierarchical classification and reconstruction.

3.2.1. Dominant Axis Localization

  • High-order axes tend to have lower aggregated mixed-order alignment loss, creating descent basins that bias reference-free search toward dominant directions.Source: §3.2.1 · pp. 4–5
  • Dominant-axis localization uses a global-to-local two-stage sampling strategy followed by Adam refinement to find and stabilize promising low-loss directions on the unit sphere.Source: §3.2.1 · pp. 4–5

3.2.2. Rotational Order Identification

  • Rotational order is inferred from the dominant frequency of Chamfer distances between the original shape and rotations sampled densely around the localized axis.Source: §3.2.2 · p. 5
  • The coarse frequency estimate is refined by testing neighboring candidate orders, with 180-degree and 45-degree rotation checks resolving two-fold, continuous, and asymmetric cases.Source: §3.2.2 · p. 5

3.3. Hierarchical Type Classification and Full-Axis Reconstruction

The GDH narrows candidate symmetry types using the dominant axis order and verifies the remaining ambiguity through secondary-axis geometry. Once the type is determined, its canonical axis arrangement is aligned to the detected axes to reconstruct the full set.

3.3. Hierarchical Type Classification and Full-Axis Reconstruction

  • The dominant axis and its estimated order restrict the GDH hypothesis space, and a secondary axis at a type-specific inclination resolves the remaining ambiguity.Source: §3.3 · p. 5

3.3.1. Secondary-Axis Identification and Hierarchical Type Classification

  • A secondary axis is searched on the prescribed inclination ring and refined by minimizing alignment loss; the order pair and GDH inclination identify the type.Source: §3.3.1 · p. 5
  • Continuous types are distinguished by additional perpendicular two-fold axes, discrete cyclic and dihedral types by orthogonal two-fold axes, and Platonic types by characteristic inter-axis inclinations.Source: §3.3.1 · p. 5

3.3.2. Canonical Axis Reconstruction

  • After classification, a rigid transformation aligns the type's canonical axis template with the localized dominant and secondary axes, transferring the canonical arrangement to reconstruct all axes.Source: §3.3.2 · p. 6

3.4. Texture-Symmetry Extension for Appearance-Aware Analysis

The texture extension models appearance-induced reductions in rotational order while preserving geometric axis orientations. It computes effective orders analytically from appearance consistency around the recovered axes.

3.4. Texture-Symmetry Extension for Appearance-Aware Analysis

  • Texture is treated as a secondary reduction of geometric rotational order, constrained to divisor orders while leaving each axis orientation unchanged.Source: §3.4 · p. 6
  • A geometrically six-fold object may appear three- or two-fold, and continuous cylindrical symmetry may appear as discrete Dn under repeated surface structure.Source: §3.4 · p. 6

4. Experiments

The experiments evaluate implementation and sampling choices, classification and localization, and performance across synthetic and real-world datasets. They include ablations on DSRSTO and cross-dataset tests on GSO and BOP benchmarks.

4.1.1. Implementation Details

  • Implementation uses 128 coarse upper-hemisphere directions, refines six within 10° caps, and optimizes the best three; the candidate order set is {3, 4, 5, 6}.The experiments were run in Python/PyTorch on an Intel i5-13400F / RTX 3050 GPU with 32 GB RAM.Source: §4.1.1 · p. 6

4.1.2. Adaptive Sampling Mechanism

  • The adaptive radius r controls coarse-sampling count, candidate directions, local samples, and Adam initializations, balancing narrow-axis coverage against broader exploration.Sampling covers the upper hemisphere to avoid antipodal redundancy, and candidate and initialization counts are capped at six.Source: §4.1.2 · p. 6

4.1.3. Metrics and Datasets

  • Axis localization uses ADI to measure discrepancy between the original model and reconstructed symmetric instances, with eADI/d reported only for objects correctly classified by type and order.ADI is the mean pointwise distance between the model and its rotated counterpart.Source: §4.1.3 · p. 7
  • The experiments use DSRSTO’s 38 CAD objects across seven categories and GSO’s 944 scanned models, including 438 with rotational symmetry, alongside six BOP pose benchmarks.DSRSTO includes 27 geometrically symmetric and 11 texture-symmetric models; GSO provides a large-scale testbed for generalization.Source: §4.1.3 · p. 7

4.2. Ablation Studies

The DSRSTO ablations examine the sampling radius and candidate rotational-order set, assessing classification accuracy, localization error, and runtime. They identify a radius near 10° and an order set of {3, 4, 5, 6} as effective choices.

4.2. Ablation Studies

  • The DSRSTO ablations assess how adaptive radius-driven sampling and candidate order-set composition affect accuracy, localization precision, and runtime.The radius study evaluates values from 5° to 30° and considers coverage, accuracy, and efficiency.Source: §4.2 · p. 7, §4.2.1 · p. 7

4.2.1. Sampling Configuration

  • At r = 10°, total accuracy reaches 81.48% and normalized alignment error is 4.18×10−3; performance is stable from 9° to 11° and worsens outside that range.Smaller radii add redundant refinements without accuracy gain, while larger radii reduce coverage and precision; 10° is adopted as default.Source: §4.2.1 · p. 7, Figure 4 · p. 7

4.2.2. Order-Set Composition

  • The order set {3, 4, 5, 6} is the minimal effective configuration: smaller sets reduce accuracy, while adding rare higher orders raises runtime without measurable benefit.Including order 2 causes instability and can misclassify high-order shapes; omitting order 4 impairs distinction between cubic and dihedral symmetries.Source: §4.2.2 · p. 7

4.3. Cross-Dataset Evaluation

Cross-dataset evaluation reports symmetry classification and axis localization on DSRSTO, BOP, and GSO. A separate FoundationPose evaluation finds improvements on all five BOP datasets when using KASALv2-derived symmetry priors.

4.3. Cross-Dataset Evaluation

  • Accuracy is 94.75% on GSO, 84.00% on re-annotated BOP symmetric instances, and 81.48% on DSRSTO; DSRSTOtex reaches 72.32%.On BOP, re-annotation raises accuracy from 80.00% under original labels and lowers eADI/d from 0.00290 to 0.00256; GSO eADI/d is 0.00262.Source: §4.3 · p. 8

4.4. Effect of Symmetry Priors on Pose Estimation

  • FoundationPose accuracy improves on all five BOP datasets with KASALv2-derived priors, with the largest mean gain of 0.9 points on IC-BIN.The comparison uses retrained models differing only in whether the symmetry priors are used.Source: §4.4 · p. 8, Table 3 · p. 8

5. Conclusion

KASALv2 provides fully automatic, reference-free rotational-symmetry analysis across all eight canonical types. Experiments show strong accuracy and generalization, alongside remaining order-estimation and geometry-robustness challenges.

  • 94.75% on GSO and up to 0.9% higher FoundationPose accuracy across five BOP datasets demonstrate strong results for the method and its symmetry priors.Source: §5 · p. 8
  • KASALv2 jointly classifies symmetry type, identifies rotational order, and localizes axes across all eight canonical types without reference input.The Geometric Degeneration Hierarchy and integrated inference pipeline support complete symmetry reasoning for real-world 3D objects.Source: §5 · p. 8
  • Underestimation of high rotational orders and sensitivity to local geometric irregularities remain failure modes; future work will explore frequency-sensitive estimation and greater robustness.Source: §5 · p. 8
Loading 2610.09534v1…