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

cs.CVOct 24, 2026

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

Authors: Hongzhu Yi, Hao Liu, Tao Yu, Tiankun Yang, Yujia Yang, Zhenghao Zhang, Tianyu Zong, Zhenyu Guan,

Organizations: University of Chinese Academy of Sciences · Institute of Automation, Chinese Academy of Sciences · Institute of Information Engineering, Chinese Academy of Sciences · Tencent

Abstract

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.

Figures & tables

Figure 1: Geometry is shared across B-rep encodings. (a) Changing the coordinate frame, contour starting point, or traversal direction changes the numerical encoding while preserving the solid. (b) The equivalent instances $B_{1}\sim B_{2}\sim B_{3}$ share the same textual description and form a multi-positive set. Removing the bore changes the geometry and no longer matches the description.

Figure 2: Three ways to encode B-reps for text-to-CAD retrieval. (a) Rendered-view encoding converts B-reps into multi-view images for an image encoder. (b) Graph encoding represents faces and their adjacency as an attributed graph processed by a GNN. (c) Our group-theoretic encoding factors out geometric equivalence transformations and combines regional, extrusion, and boundary spectra into an invariant descriptor before geometry encoding, satisfying $\Phi(g\cdot\mathcal{B})=\Phi(\mathcal{B})$ for $g\in G$ .

Figure 3: Product decomposition of a straight-extrusion unit. (a) A rounded B-rep solid with a through-hole, showing faces, edges, vertices, and oriented boundary loops (dashed arrows). (b) Its planar region

\Omega

, boundary

\partial\Omega

, and extrusion depth

h

;

\partial\Omega

sweeps along thickness

h

to form the lateral and inner-hole walls.

Figure 4: Spectral components on real B-reps. Four MMCAD-B parts are shown with regional Dirichlet modes on their top faces, solid modes $u_{m}(x,y)\sin(\pi z/h)$ on oblique interior cuts, and turning harmonics along their boundaries. The two rows use $m=2,4$ and $k=2,4$ ; the triangular frame uses $k=3,6$ . Colors indicate signed amplitudes normalized per field or boundary loop; zero harmonics remain neutral. Boundary panels include outer and hole contours, while the descriptor retains outer-contour Fourier magnitudes.

Figure 5: Invariance of the boundary spectrum to recording conventions. Changing the contour starting vertex from

A

C

or reversing traversal direction only changes phase or conjugates coefficients in the frequency domain; the Fourier magnitudes

|F_{k}|

remain identical.

Figure 6: Feature alignment under reordering on MMCAD-B. (a) Original and reordered B-rep features in a fixed PCA space; contours enclose 50%, 80%, and 95% probability mass, with overlap computed on a shared density grid. (b) Full-dimensional $\ell_{2}$ drift of normalized features; Coordinates denotes the sampled input-sequence diagnostic, and values below $10^{-6}$ share a display bin. (c) Original versus reordered text–B-rep matching scores; hexagon color gives pair counts and the dashed line indicates unchanged scores.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: Full versus ablated descriptors. Arrows show the change in Expert R@5 from Full (blue) to each component removal (orange) with frozen text encoders. Orange point labels report R@5 (%); negative labels report changes in percentage points. Both panels use the same scale. Positive groups are recomputed for each variant.

Table 8

3D Shape Retrieval

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BrepLLM: Enabling Large Language Models to Understand Boundary Representations

Liyuan Deng, Hao Guo, Yongkang Dai, Yunpeng Bai+4

Northwestern Polytechnical University · National University of Singapore · Hong Kong University of Science and Technology+1

Multimodal Large Language Models 3D Representation Learning 3D VLMs

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Current token-sequence-based Large Language Models (LLMs) struggle to directly process 3D Boundary Representation (B-rep) models that contain complex geometric and topological information. To this end, we propose BrepLLM, the first multimodal framework that enables LLMs to directly parse and reason over raw B-rep data. BrepLLM adopts a two-stage training pipeline: cross-modal alignment pre-training and two-stage LLM fine-tuning. In the first stage, we design an adaptive UV sampling strategy to convert B-reps into graph representations that integrate geometric and topological information. Subsequently, we construct a hierarchical BrepEncoder to extract features from geometric elements (faces and edges) and topology, generating a global token and a sequence of node tokens. Then, via contrastive learning, we conduct an initial alignment between this global token and the text embeddings of a frozen CLIP text encoder (ViT-L/14). In the second stage, we integrate the pre-trained BrepEncoder into the LLM and employ a two-stage progressive strategy to align the sequence of node tokens: (1) training an MLP-based semantic mapping network that utilizes the prior knowledge of a 2D-VLM to align the B-rep representation to the 2D visual semantic space; (2) utilizing LoRA for parameter-efficient fine-tuning of the Q-Former and the LLM backbone network to achieve the final 3D-language generation capability. Furthermore, we construct the Brep2Text dataset, which contains 269,444 B-rep and text question-answer pairs. Experiments demonstrate that BrepLLM achieves SOTA performance on 3D object classification and captioning tasks. The project page is available at https://user-deng.github.io/BrepLLM/.

Abstract

3D Shape Retrieval

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