SIGMA: Scale-Invariant Global Sparse Shape Matching

TL;DR

SIGMA tackles sparse non-rigid shape matching by formulating a global mixed-integer program that is invariant to rigid motions and global scaling. It introduces the projected Laplace-Beltrami operator ($\Delta^{(\mathcal{X})}_{\mathrm{proj}}$) that blends intrinsic geometry with extrinsic coordinates, enabling a deformation prior that avoids oversmoothing. The final objective combines a reconstruction term, a PLBO-based deformation term, and an orientation-aware term over a permutation matrix $\mathbf{P}$, yielding provable invariances and often global optimality within a 1-hour budget. Empirically, SIGMA achieves state-of-the-art performance on isometric and non-isometric benchmarks, including mesh-to-point-cloud matching, and scales favorably with mesh resolution, demonstrating robust applicability to challenging 3D geometry problems.

Abstract

We propose a novel mixed-integer programming (MIP) formulation for generating precise sparse correspondences for highly non-rigid shapes. To this end, we introduce a projected Laplace-Beltrami operator (PLBO) which combines intrinsic and extrinsic geometric information to measure the deformation quality induced by predicted correspondences. We integrate the PLBO, together with an orientation-aware regulariser, into a novel MIP formulation that can be solved to global optimality for many practical problems. In contrast to previous methods, our approach is provably invariant to rigid transformations and global scaling, initialisation-free, has optimality guarantees, and scales to high resolution meshes with (empirically observed) linear time. We show state-of-the-art results for sparse non-rigid matching on several challenging 3D datasets, including data with inconsistent meshing, as well as applications in mesh-to-point-cloud matching.

Figures (12)

• Figure 1: We propose a sparse non-rigid shape matching approach that is provably invariant to rigid transformations and global scaling, can (often) be solved to optimality, and scales linearly with mesh resolution. (a, b) Our matchings for non-rigid shapes with drastically different scale and partiality. (c, d) Our method is the only one that is able to both solve the majority of pairs to global optimality within a time budget of $1$h, as well as scaling up to high mesh resolutions.
• Figure 2: We utilise synergies between shape matching and shape reconstruction. To this end, rigid motion-invariant geometric information of shape $\mathcal{X}$, encoded in the projected LBO $\Delta^{(\mathcal{X})}_{\mathrm{proj}}$, and the keypoint coordinates $\mathbf{Y}_\mathcal{J}$ of shape $\mathcal{Y}$, are combined to reconstruct shape $\mathcal{X}$ in the pose of shape $\mathcal{Y}$. Our motivation is that high quality correspondences will lead to better reconstruction and vice versa.
• Figure 3: Comparison of reconstructed shapes using the standard LBO and our proposed projected LBO. From the definition in Eqn. \ref{['eq:proj-lbo']} we can show that $\Delta^{(\mathcal{X})}_{\mathrm{proj}}\mathbf{X}=0$, i.e. this yields a deformation prior that incurs no cost on exact reconstructions of the initial geometry $\mathbf{X}$. Thus, the projected LBO is able to better preserve local geometry and leads to more realistic reconstructions. See the supplementary material for details.
• Figure 4: PCK curves on datasets TOSCA Bronstein:2008:NGN:1462123, SMAL Zuffi:CVPR:2017, SHREC20 Dyke:2020:track.b and DT4D-M magnet2022smooth. The values in the legends are Area-Under-the-Curves (AUC $\uparrow$). Across all datasets our method consistently outperforms all other approaches. For SMAL and SHREC20 our method produces nearly perfect results. The orientation-aware term improves our performance in most cases.
• Figure 5: (Left) Relative optimality gap statistics on the TOSCA. The Inf gap reflects cases where SM-comb roetzer2022scalable fails to find a solution. Our proposed method produces the lowest and most concentrated relative optimality gap. (Right) Mean geodesic error for shape pairs with different shape scales. PMSDP maron2016point and our method are the only ones which perform consistently across different shape scales, with ours being more accurate.

Theorems & Definitions (7)

• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof