Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching

TL;DR

This work tackles the challenge of crease-aware non-isometric shape matching by integrating intrinsic Laplace-Beltrami (LBO) eigenfunctions with extrinsic elastic thin-shell Hessian eigenfunctions into a unified hybrid functional map framework. By formulating functional maps in a non-orthogonal Hilbert space and employing the Hilbert-Schmidt norm, the method accommodates both low-frequency isometric structure and high-frequency extrinsic details. The authors derive a separable, block-diagonal optimization that combines two independent maps for the LBO and elastic bases, and they propose learning strategies with linear annealing to stabilize training in deep pipelines. Extensive experiments across near-isometric, non-isometric, and topologically noisy datasets demonstrate consistent improvements over state-of-the-art methods, including up to 15% mean geodesic error reduction and up to 45% gains under topological noise, across supervised, unsupervised, and axiomatic frameworks. This hybrid basis approach thus enables robust, crease-aware shape correspondences while remaining compatible with existing functional map pipelines, with potential impact on 3D shape analysis, animation, and transfer tasks.

Abstract

Non-isometric shape correspondence remains a fundamental challenge in computer vision. Traditional methods using Laplace-Beltrami operator (LBO) eigenmodes face limitations in characterizing high-frequency extrinsic shape changes like bending and creases. We propose a novel approach of combining the non-orthogonal extrinsic basis of eigenfunctions of the elastic thin-shell hessian with the intrinsic ones of the LBO, creating a hybrid spectral space in which we construct functional maps. To this end, we present a theoretical framework to effectively integrate non-orthogonal basis functions into descriptor- and learning-based functional map methods. Our approach can be incorporated easily into existing functional map pipelines across varying applications and is able to handle complex deformations beyond isometries. We show extensive evaluations across various supervised and unsupervised settings and demonstrate significant improvements. Notably, our approach achieves up to 15% better mean geodesic error for non-isometric correspondence settings and up to 45% improvement in scenarios with topological noise.

Figures (17)

• Figure 1: We propose a novel approach of hybridizing the eigenbases originating from different operators for mapping between function spaces in deformable shape correspondence. While the Laplace-Beltrami operator (LBO) eigenbasis is robust to coarse isometric deformations, it fails to encapsulate extrinsic characteristics between shapes. In contrast, elastic basis functions hartwig_elastic_2023 align with high curvature details but lack the robustness for coarse isometric matching. The proposed hybrid basis can be used as a drop-in replacement for the LBO basis functions in modern functional map pipelines, improving performance in near-isometric, non-isometric, and topologically noisy settings.
• Figure 2: A Percentage-Correct-Keypoint ablation between the pure LB basis, pure elastic basis (orthogonalized), and our hybrid approach at the same spectral resolution ($k = 200$). The elastic basis attains better detail alignment than the LB basis but yields inferior overall global correspondences. The proposed hybrid approach achieves the best of both worlds. Experiments are conducted with the ULRSSM cao_unsupervised_2023 framework on SMAL.
• Figure 3: Correspondence quality visualized through the transfer of normals (left) and vertex positions (right) from the source to the target shape. We compare the results from LB, Elastic, and Hybrid basis functions by encoding and recovering ground truth correspondences through functional map representations at a spectral resolution of $k = 60$. Additional results at a spectral resolution of $k = 200$ are provided in the supplementary.
• Figure 4: Hybrid Functional Maps in a typical pipeline. Features are first extracted from a pair of shapes with a Siamese network (a). They are then projected onto eigenbasis sets from different linear operators (b). We then solve for a block diagonal functional map spanning the constructed hybrid function space (b). Additional regularization can be used to impose structure on parts of the hybrid functional map (c).
• Figure 5: Qualitative Results on SMAL, DT4D-H, and TOPKIDS. Comparison of ULRSSM in the LBO basis and in the proposed hybrid basis. Hybrid functional maps yield higher-quality correspondences, particularly under topological noise. ULRSSM in the LB basis frequently creates coarse mismatches such as incorrectly assigning appendages, whereas the elastic basis better represents these details. The first six columns show texture transfer. The last columns transfer normals making the less accurate alignment of creases in ULRSSM visible.

Theorems & Definitions (7)

• Lemma 4.1
• Proposition 4.2
• proof
• proof : Proof of \ref{['lemma:edata_induced_norm']}
• proof : Proof of \ref{['thm:ereg_hs_norm']}
• Theorem C.1
• proof : Proof of \ref{['thm:seperable_block_matrix']}