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Can One Hear Geometry? Gauge-Invariant Spectral Descriptors for Text-to-B-rep Retrieval

Hongzhu Yi, Hao Liu, Tao Yu, Tiankun Yang, Yujia Yang, Zhenghao Zhang, Tianyu Zong, Zhenyu Guan, Bingkang Shi, Haopeng Jin, Yuanxiang Wang, Zhongtian Luo, Jungang Xu

arXiv:2610.33484v2cs.CV

TL;DR

Text-to-B-rep retrieval must bridge language about intrinsic shape and B-rep encodings that vary with pose, scale, and boundary conventions. The paper uses group-action equivalence classes and spectral descriptors for invariant geometry-text alignment. It reports state-of-the-art retrieval performance, including 0.886 R@5 on Text2CAD.

  • Problem

    B-rep encodings entangle precise geometry with pose, scale, and boundary conventions, while discretization loses analytic precision and graph encoders lack explicit invariance guarantees.

    Source: §1 · p. 3

  • Method

    The method formalizes geometric equivalences with group actions and combines regional, extrusion, and boundary spectral features in a descriptor used for multi-positive cross-modal alignment.

    Source: §1 · pp. 3–4, §1.1 · p. 4, §4.1 · p. 6

  • Results

    The method achieves state-of-the-art retrieval performance, including 0.886 R@5 on Text2CAD.

    Source: §6 · pp. 11–12

  • Takeaways & Limitations

    Geometrically equivalent solids can share textual supervision through descriptor-consistent multi-positive sets.

    Source: §1 · p. 4

Abstract

from arXiv · show

Text-to-3D solid retrieval faces a fundamental representation gap: while natural language describes intrinsic shapes, Boundary Representation (B-rep) entangles precise geometry with arbitrary variations in pose, scale, and boundary encoding conventions. Existing methods either discretize B-reps at the cost of analytic precision, or learn features from topological graphs without explicit invariance guarantees. We propose a text-to-B-rep retrieval framework based on geometric equivalence classes and spectral invariants, mathematically eliminating representation ambiguities prior to cross-modal alignment. Formalizing translation, rotation, reflection, uniform scaling, and contour re-indexing via group actions, we exploit the planar-profile-and-extrusion structure of engineering solids to construct a unified spectral descriptor across cross-sectional regions, contour boundaries, and extrusion depths. On public benchmarks, our method achieves state-of-the-art recall performance across all evaluation metrics.

1 Introduction

Text-to-B-rep retrieval must match language about intrinsic shape despite B-rep encodings that vary with pose, scale, and boundary conventions. The paper formalizes shape-preserving equivalences and combines invariant spectral descriptors with multi-positive supervision.

1 Introduction

  • B-rep encodings can change with coordinate frame, contour start, or traversal direction even when the solid and its description remain unchanged.Discretization loses analytic precision, while graph encoders must learn to separate geometry from encoding conventions.Source: §1 · p. 3, Figure 1 · p. 3
  • The method models translation, rotation, reflection, uniform scaling, and contour re-indexing as group actions on geometric equivalence classes.Shape-preserving instances share a description, whereas removing the bore changes the geometry.Source: §1 · p. 3, Figure 1 · p. 3
  • A unified descriptor combines regional spectra, boundary turning Fourier magnitudes, and extrusion depth for planar-profile-and-extrusion solids.The components describe complementary geometry without a volumetric eigensolve.Source: §1 · p. 4
  • Descriptor-consistent multi-positive sets let geometrically equivalent solids share textual supervision instead of becoming false negatives in single-positive contrastive learning.The paper evaluates retrieval on Text2CAD construction records and native STEP models from MMCAD-B.Source: §1 · p. 4

1.1 Contributions

  • The contributions frame group-action equivalence classes as mathematically guaranteed invariants and combine Dirichlet eigenvalues with boundary Fourier magnitudes without discretization losses.The descriptor targets regional, boundary, and extrusion geometry directly from B-reps.Source: §1.1 · p. 4

2 Related Work

Prior text-to-3D retrieval largely targets point clouds, meshes, or rendered images, while CAD representation learning addresses construction sequences and graphs. The cited methods do not resolve the paper’s focus on invariant retrieval over parametric B-reps.

Text-to-3D retrieval

  • Text-to-3D retrieval methods align language with point clouds, meshes, or rendered images, including ShapeGlot, Text2Shape, ULIP, and OpenShape.These methods target mesh or point-cloud representations rather than parametric CAD models with construction-sequence structure.Source: §2 · p. 4

CAD representation learning

  • CAD representation work includes construction-sequence generation and learning, while Text2CAD generates parametric CAD sequences from text and CADIR [30] supports retrieval of construction graphs.DeepCAD, SkexGen, and CAD-SIGNet address sequences, sketch/extrusion generation, and construction-graph signatures, respectively.Source: §2 · p. 4

CAD retrieval methods

  • Published results for BRepCLIP [31], Duoduo CLIP, CLIP-GS, CADGCL, and BRT are summarized in Appendix I.The passage points to those results but does not report their values.Source: §2 · p. 4

3 Preliminaries

The problem formulation reconstructs B-rep geometry from text queries and construction sequences, then removes representation variations before cross-modal alignment. Straight-extrusion structure decomposes each unit into a planar region and depth, enabling tractable geometric subproblems.

3.1 Problem Formulation

  • Given a text query and construction sequence, the formulation reconstructs the corresponding B-rep geometry before retrieval.R denotes the reconstruction mapping.Source: §3.1 · p. 4
  • A deterministic quotient map removes pose and recording variations before cross-modal alignment, yielding an invariant descriptor under geometric equivalences.The encoding illustration contrasts rendered views, face-adjacency graphs, and the group-theoretic descriptor.Source: §3.1 · p. 5, Figure 2 · p. 5

3.2 Structural Priors and Equivalence Relations

  • The equivalence relation factors out extrinsic similarity transformations and contour recording conventions while remaining sensitive to substantive geometric changes.The cited conventions include contour starts, traversal directions, and unit indices.Source: §3.2 · p. 5
  • Each straight-extrusion unit has a Cartesian product structure consisting of planar cross-section Ω and depth h, decomposing 3D invariance into regional and boundary subproblems.For models with multiple units, canonical ordering and permutation-invariant aggregation produce a fixed model-level representation.Source: §3.2 · p. 5, Figure 3 · p. 6

4 Method

The method builds orthogonal invariant features for cross-sectional shape, extrusion depth, and contour structure, then maps descriptor and text representations into a shared space. Its spectral construction uses extrusion structure to avoid direct volumetric eigensolving.

4.1 Constructing Orthogonal Spectral Invariants

  • The unit descriptor assembles regional, extrusion, and boundary features to represent the planar region, extrusion depth, and contour structure.The method then uses a dual encoder to map the invariant geometry descriptor and text into a shared D-dimensional space.Source: §4.1 · pp. 6–7, §4.2 · p. 8
  • The regional Dirichlet eigenvalues encode cross-sectional shape independently of planar translation, rotation, and reflection.The eigenvalue problem is solved on the planar cross-section, including internal holes.Source: §4.1 · p. 6
  • Cartesian-product separation of variables combines cross-sectional eigenvalues with extrusion depth to obtain exact 3D Dirichlet eigenvalues without volumetric meshing.Normalizing by 3D volume removes absolute scale, and the resulting spectrum captures the cross-section-to-depth aspect ratio.Source: §4.1 · pp. 6–7
  • Boundary turning-angle Fourier magnitudes remove contour start-point, direction, and reflection effects while retaining local corners and arcs.The descriptor combines these magnitudes with the normalized regional and extrusion spectra.Source: §4.1 · p. 7

4.2 Equivalence-Aware Cross-Modal Alignment

  • A residual MLP encodes each invariant geometry descriptor, while a text encoder processes its paired text for shared-space alignment.Cross-modal similarity is measured with cosine inner product.Source: §4.2 · p. 8

5 Experiments

Experiments evaluate text-to-CAD retrieval on Text2CAD and MMCAD-B, then test representation stability and the effects of language-model size, descriptor capacity, descriptor components, and geometric supervision.

5.1 Setup

  • Evaluation uses Text2CAD and MMCAD-B, with each query ranked against its target and 999 randomly sampled distractors across 10 seeds.The method is compared with retrieval baselines and zero-shot CLIP and BLIP-2 on rendered views; retrieval reports R@1, R@5, R@10, and median rank.Source: §5.1 · p. 8

5.2 Main Retrieval Results

  • Expert R@5=0.886 on Text2CAD exceeds reproduced CADIR (0.641) by 24.5 percentage points; retrieval declines for less detailed descriptions.R@5 is 0.502 for Intermediate, 0.200 for Beginner, and 0.151 for Abstract descriptions; repeated Abstract descriptions further limit individual-model identification.Source: Table 1 · p. 9, §5.2 · p. 9

5.3 Feature Alignment under Reordering

  • Reordering 6,514 B-rep serializations yields 99.961% overlap in projected geometry distributions, while text–B-rep matching scores correlate at r = 0.982 across 746 text variants.For 980 parts, all seven variant distances remain below 10−6; the median absolute score change is 0.011.Source: §5.3 · pp. 9–10, Figure 6 · p. 10

5.4.1 Effect of Language Model Size

  • With LoRA, the 0.6B text model achieves Expert R@5=0.859, 2.7 percentage points below the 4B model; freezing the 0.6B encoder lowers R@5 to 0.596.The descriptor is evaluated unchanged across both backbone settings.Source: §5.4.1 · p. 10

5.4.2 Effect of Unit Capacity

  • R@5 rises from 0.467 at U = 1 to 0.721 at U = 4, then falls slightly to 0.710 at U = 8, supporting U = 4 as the default capacity.R@10 is similar for U = 4 and U = 8: 0.803 and 0.805, respectively.Source: Table 2 · p. 10, §5.4.2 · p. 11

5.4.3 Contributions of Descriptor Components

  • All four descriptor-block removals lower R@5 for both backbones; local topology and face statistics produce the largest drops, while the full descriptor has the highest R@1, R@5, and R@10.Turning, regional, and extrusion spectra also improve retrieval; the effects of equally sized regional and extrusion blocks differ, so retained dimensionality alone does not describe component contributions.Source: Table 3 · p. 11, §5.4.3 · p. 11

5.4.4 Geometric Supervision and Positive Groups

  • Removing either auxiliary geometric heads or spectral equivalence classes leaves the 4B contrastive loss near ln B.The complete configurations and results are reported in Appendix G.3.Source: §5.4.4 · p. 11

6 Conclusion

The conclusion reports 0.886 R@5 on Text2CAD with less supervisory information than coordinate-based counterparts. The appendices explain how the descriptor uses extrusion structure and spectral features while formalizing invariance to geometric and contour-recording changes.

6 Conclusion

  • 0.886 R@5 on Text2CAD; the conclusion also reports state-of-the-art retrieval performance using less supervisory information than coordinate-based counterparts.The conclusion frames the approach around spectral geometry, group theory, and spectral equivalence classes.Source: §6 · pp. 11–12

A.1.1 Product Geometry and Boundary Decomposition; A.1.2 Geometric Roles of the Descriptor Components

  • The descriptor combines regional, boundary-turning, and extrusion spectra to represent cross-sectional structure, contour detail, and extension depth.The product structure separates the planar region from its extrusion direction; the boundary exposes contour features including corners, arcs, and holes.Source: §A.1.2 · p. 16

A.2 Reconstruction of Explicit B-rep Geometry; A.2.3 Boolean Composition

  • Construction records are reconstructed into explicit B-rep geometry through planar-profile reconstruction, solid generation, and Boolean composition.The scalar-field formulation is used during reconstruction; retrieval features are built from recovered B-rep regions, boundaries, extrusion units, and relations.Source: §A.2 · p. 16, §A.2.3 · p. 17

A.3.1 Similarity Transformations and Discretization; A.3.2 Operation-Level Equivariance and Recording Conventions

  • Translation, rotation, reflection, uniform scaling, contour re-indexing, and changes in unit ordering can preserve reconstructed geometry under the stated implementation conditions.Scale-independent thresholds and dimensionless sampling are required for uniform-scale compatibility.Source: §A.3.1 · p. 18, §A.3.2 · p. 18

B.1.2 Scale Normalization and Contour Conventions

  • Area-normalized Dirichlet eigenvalues are invariant to translation, rotation, reflection, and uniform scaling, and the result also applies to cross sections with holes.Changing a closed contour's starting point or traversal direction leaves its region and eigenvalue problem unchanged.Source: §B.1.2 · p. 19

B.2.1 Product Modes and Eigenvalues; B.2.3 Relative Extrusion Depth

  • The normalized extrusion spectrum combines cross-sectional shape with relative extrusion depth while eliminating absolute size.The lowest 12 extrusion eigenvalues are retained for each straight-extrusion unit.Source: §B.2.1 · p. 19, §B.2.3 · p. 20

B.3.1 Continuous Turning Measure; B.3.2 Discrete Coefficients and Sampling Conditions; B.3.3 Change of Contour Origin; B.3.4 Reversal of Traversal; B.3.5 Spatial Transformations

  • The turning representation accounts for smooth-segment curvature and corner turns; its Fourier magnitudes are insensitive to contour-origin changes and transform consistently under the stated spatial transformations.For curved segments, the invariance identities require sampling locations and turning increments to correspond under transformation.Source: §B.3.1 · p. 20, §B.3.3 · p. 20, §B.3.4 · p. 20

B.4.1 Harmonic Structure of Rotational Symmetry

  • For an ideally q-fold symmetric contour, continuous turning-spectrum coefficients vanish whenever q does not divide the harmonic index.The discrete result requires sampling that preserves the repeated structure.Source: §B.4.1 · p. 21

B.4.2 Finite-Band Symmetry Estimation

Finite-band concentration estimates contour symmetry order, but its output is bounded by available positive-frequency energy and the selected band. The model descriptor then orders extrusion units geometrically and assembles invariant local and global features into a fixed representation.

B.4.2 Finite-Band Symmetry Estimation

  • A finite band and ρ(q) ≥ 0.98 threshold select the largest qualifying symmetry candidate, with q = 1 ensuring a candidate when energy is positive.The DC term is excluded because it gives only total turn ±2π; the estimate is not a certificate of exact symmetry.Source: §B.4.2 · p. 21

B.4.3 Degenerate Cases and Geometric Consistency

  • When positive-frequency energy is zero or unresolved, symmetry order cannot be determined; analytic geometry or a wider band may be needed.For sections with holes, symmetry also requires preserving all boundary components and their relative positions under the same rotation.Source: §B.4.3 · p. 21

C.1.1 Spectral Features

  • Each extrusion unit’s 40-dimensional spectral representation combines 12 normalized Dirichlet eigenvalues, 12 extrusion eigenvalues, and 16 boundary turning Fourier magnitudes.Source: §C.1.1 · p. 22

C.1.2 Local Topology and Geometric Statistics

  • The spectral block is augmented with 24 dimensions of contour-topology, rotational-symmetry, face-level, and geometric statistics.Source: §C.1.2 · p. 22

C.2.1 Geometric Ordering of Extrusion Units

  • Extrusion units are ordered by normalized cross-sectional area, with isoperimetric ratio and aspect ratio as secondary keys rather than construction indices.Source: §C.2.1 · p. 22

C.2.2 Global Aggregation and Descriptor Dimension

  • The 280-dimensional model descriptor combines four retained 64-dimensional units with a 24-dimensional global block; extra units are summarized by permutation-insensitive statistics.The global block includes model-level quantities such as extrusion-unit count, global topology counts, rotational symmetry, and Boolean-operation statistics.Source: Table 4 · p. 22, §C.2.2 · p. 22

C.3.1 Retained Units, Padding, and Overflow

  • The fixed layout retains four local units and uses input-independent padding when fewer than four are present; overflow summaries must ignore unit order.Source: §C.3.1 · pp. 22–23

C.3.2 Invariant Attributes and Deterministic Ordering

  • Complete-model invariance requires invariant attributes and sorting keys; tied keys need identical local descriptors or an index-independent tie-breaker.Under these conditions, transformations leave local vectors, their ordering, and the aggregate block unchanged.Source: §C.3.2 · p. 23

D Descriptor Validation

Validation reports numerical invariance checks and measures how well the descriptor distinguishes models and supports self-retrieval. The paper also formalizes construction-record reindexing and commuting operations in its gauge-group structure.

D.1 Numerical Validation Details

  • Cyclic shifts and traversal reversal give zero error, turning-spectrum errors stay below 3 × 10^-15, and finite-resolution planar rotation-and-scaling checks have maximum relative error 0.015.Source: D Descriptor Validation · p. 23, Table 5 · p. 23

D Descriptor Validation

  • The descriptor distinguishes 3,315 of 4,000 sampled models versus 3,601 distinct raw sequences, retaining 92% of distinguishing capacity; self-retrieval reaches R@1=0.913, R@5=0.951, and R@10=0.989.These self-retrieval scores provide an empirical reference for additional error from text alignment.Source: D Descriptor Validation · p. 23

D.2 Descriptor Distinctness and Pairwise Distances

  • Among 4,000 sampled test models, 3,315 descriptor representations correspond to 3,601 distinct raw sequences; within-duplicate distances are small relative to between-model distances.Within-duplicate distances have mean 0.004 and 95th percentile 0.000; between-model distances have mean 20.18 and 5th percentile 6.24, supporting descriptor-based positive grouping while retaining most distinctions.Source: §D.2 · p. 23

E Gauge Group Structure

  • The construction-record gauge group combines contour reindexing by Cn or Dn and extrusion-unit reindexing by Sm when geometric and operation relations are preserved.The group description also identifies signed axis permutations and a trace monoid over commuting Boolean operations, with Foata normal form as canonical representatives.Source: E Gauge Group Structure · p. 24

F Training and Supervision Details

The training setup adapts a 4B text backbone with LoRA and attaches geometry-prediction heads to shared text features. The appendices describe a reduced-capacity backbone comparison and visualize descriptor-component ablations, without stating their plotted values in these passages.

F.1 Optimization and Model Configuration

  • The main retrieval model uses Qwen3-Embedding-4B with LoRA for 30 epochs, 18,210 steps, and global batch size 1,024 across 8 GPUs.It has 77.4M trainable parameters: 70.3M in text LoRA and 7.1M in the CAD tower.Source: §F.1 · p. 24

F.2 Auxiliary Geometric Targets

  • Auxiliary heads predict five geometric classifications and five geometric regressions computed automatically from B-rep geometry without manual annotation.Classification targets include rotational symmetry, hole topology, circular-contour count, extrusion-unit count, and face count; regression targets include shape ratios and depth dispersion.Source: §F.2 · p. 24

G.1 Language-Model Adaptation Across Description Levels

  • The 0.6B text-backbone comparison holds the Full descriptor fixed and evaluates frozen versus LoRA-adapted encoding across four description levels under the main protocol.The supplied table caption describes recall proportions over 1,000 trials per level and notes that bold marks the better score; it does not provide the cell values.Source: §G.1 · p. 24, Table 6 · p. 24

G.2 Descriptor-Component Visualization

  • Figure 7 compares Expert R@5 changes after component removals for frozen 4B and 0.6B text backbones, with both panels on the same scale.Arrows run from Full to each removal; orange labels give R@5 and percentage-point changes, while positive groups are recomputed per variant.Source: §G.2 · p. 25, Figure 7 · p. 25

G.3 Geometric Supervision and Positive Groups

The alignment ablation reports that removing either geometric supervision or equivalence-class positive groups leaves training near the random-matching loss baseline. The extended related-work section situates the paper among representation-learning, retrieval, multimodal, and agent research.

G.3 Geometric Supervision and Positive Groups

  • Removing either auxiliary heads or spectral equivalence classes leaves contrastive loss near ln B, while their joint use with LoRA reaches Expert R@5=0.886.Auxiliary heads provide per-sample geometric targets, while positive groups avoid treating models with equivalent descriptors as negatives.Source: §G.3 · p. 25, Table 7 · p. 25

H Extended Related Work

  • Extended related work spans contrastive and multimodal embeddings, retrieval and evidence gathering, multimodal applications, and foundation-model systems research.Source: H Extended Related Work · p. 26

I Published Results of Additional Methods

The section compiles published retrieval, classification, and face-labeling results from different evaluation settings. Tables 8–10 retain their original metrics and settings, so their percentages are not measurements on the Text2CAD split.

I Published Results of Additional Methods

  • Tables 8–10 reproduce published results in their original settings; their percentages are not measurements on the Text2CAD split.Table 8 reports text-to-shape retrieval with different metrics and ranks across methods.Source: I Published Results of Additional… · p. 26, Table 8 · p. 26
  • BRepCLIP uses full galleries of 91K ABC, 40K CADParser, and 65K Automate models; Duoduo CLIP and CLIP-GS use different training and evaluation collections.Duoduo CLIP uses approximately 1.4K Text2Shape shapes and 7.4K captions; CLIP-GS uses 20K Objaverse-GS shapes and excludes Objaverse-LVIS from training.Source: I Published Results of Additional… · p. 26
  • CADGCL reports CAD-to-CAD retrieval with an 80:20 split, while BRT reports classification and face labeling with 70:15:15 splits; neither reports text-to-CAD retrieval.Using either encoder for text retrieval requires text alignment.Source: I Published Results of Additional… · p. 26
  • Table 9 covers CADGCL’s published CAD-to-CAD retrieval, while Table 10 covers BRT’s classification and face-labeling results.Table 10 uses a dash where IoU is not a metric for classification.Source: Table 9 · p. 26, Table 10 · p. 26

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