cs.CVMar 29, 2026
SaveRINO: Rotation-Invariant Non-Rigid Correspondences
Organizations: TUM · MCML · Stanford University · MIT
Abstract
Dense 3D shape correspondence remains a central challenge in computer vision and graphics as many deep learning approaches still rely on intermediate geometric features or handcrafted descriptors, limiting their effectiveness under non-isometric deformations, partial data, and non-manifold inputs. To overcome these issues, we introduce RINO, an unsupervised, rotation-invariant dense correspondence framework that effectively unifies rigid and non-rigid shape matching. The core of our method is the novel RINONet, a feature extractor that integrates vector-based SO(3)-invariant learning with orientation-aware complex functional maps to extract robust features directly from raw geometry. This allows for a fully end-to-end, data-driven approach that bypasses the need for shape pre-alignment or handcrafted features. Extensive experiments show unprecedented performance of RINO across challenging non-rigid matching tasks, including arbitrary poses, non-isometry, partiality, non-manifoldness, and noise.
Figures & tables
Figure 2: Our learned SO(3)-invariant features. We visualize the Euclidean distance to the blue surface point in the learned feature space (darker red means higher similarity). Semantic correspondences have a similar feature similarity pattern (left & mid.). The learned feature is invariant to rotations (mid. & right). Shape orientations are depicted by RGB frames.
Figure 3: SO(3)-invariant correspondences. Our method is unaffected by shape orientations. In contrast, baselines (represented by DUOFM) perform well only when train and test shapes are pre-aligned (I/I), failing dramatically on unseen rotations (I/SO(3)).
Figure 4: We propose RINONet , which learns SO(3)-invariant features from input shape . Our novel network inherits all nice properties from DiffusionNet [ 54 ] , and learns smooth, high-quality per-point features, while additionally retaining invariant to SO(3) actions applied to input shapes by employing vector neurons as the feature representation in our hidden layers. The RINONet has a simple structure and contains four consecutive RINONet Blocks at its core, which is combined with VN-EdgeConv and VN-linear layers to achieve the desired I/O dimensionality. The VN-invariant layer is employed to convert SO(3)-equivariant features to invariant ones.
Figure 5: Our RINONet block is the core of our RINONet, and it consists of three main modules: a VN-Diffusion layer, a VN-Gradient layer, and a VN-MLP. In contrast to the original DiffusionNet block [ 54 ] , ours is SO(3)-equivariant by design.
Figure 6: Intrinsic symmetry. Ours fully resolves symmetry, while baselines suffer. Top: partial left-right (legs) and front-back (belly-back) flips. Bottom: full left-right flips (arms and legs).
Figure 7: Overview of RINO. Our novel RINONet is used to extract SO(3)-invariant features and from input shapes and . The features are further processed in three non-learnable but differentiable blocks: the FMaps solver to compute the functional map , the CFMaps solver to compute the complex functional map (based on feature gradients), and the pointwise map block to compute the soft pointwise map based on feature similarity. Finally, our unsupervised loss prompts structural properties of and consistency between different map representations.
| Train/Test | I/I | I/SO(3) | SO(3)/SO(3) | Y/Y |
|---|---|---|---|---|
| CnsFM | 5.4 | 58.7 | 9.1 | 5.4 |
| DUOFM | 11.7 | 41.4 | 25.2 | 16.5 |
| URSSM | 4.8 | 62.1 | 24.8 | 7.9 |
| SMS | 4.6 | 58.5 | 57.6 | 39.3 |
| SmpFM | 5.6 | 62.6 | 9.9 | 7.3 |
| HbrFM | 5.6 | 61.8 | 26.4 | 5.3 |
Table 1: SO(3)-invariance on SMAL. The rotation type is indicated by: I (aligned), SO(3) (fully random rotation), and Y (y-axis rotation). e.g. I/SO(3) means aligned shapes for training and fully randomly rotated shapes for testing. We report mGeoErr ( ) and ours consistently outperforms baselines, especially on unseen rotations. Best results are bold , and highlighted cells indicate the experiment settings used in baseline papers.
| wks | xyz | |||||
|---|---|---|---|---|---|---|
| SMAL | DT4D | FSCAN | SMAL | DT4D | FSCAN | |
| CnsFM | 23.4 | 9.3 | 3.6 | 9.1 | 7.4 | 50.5 |
| DUOFM | 6.7 | 15.8 | - | 25.2 | 55.8 | - |
| URSSM | 29.6 | 8.4 | 3.0 | 24.8 | 59.3 | 24.8 |
| SMS | 30.6 | 58.6 | 12.1 | 57.6 | 64.9 | 54.0 |
| SmpFM | 29.9 | 15.3 | 5.7 | 9.9 | 5.5 | 24.3 |
Table 2: Non-isometric and raw scan matching. We report mGeoErr ( ) and highlight the best and 2nd best results. Ours outperforms all baselines regardless of the input types. Highlighted cells indicate the experiment settings used in baseline papers.
Figure 8: Non-isometric matching on SMAL and DT4D. Our method outperforms all baselines; it is invariant to extrinsic rigid poses (no pre-alignment) and robust to intrinsic shape symmetry by directly processing raw shape geometry.
| FAUST | SCAPE | |||||
|---|---|---|---|---|---|---|
| E | ES | #Flips | E | ES | #Flips | |
| CnsFM | 4.4 | 2.8 | 25 | 27.2 | 7.4 | 249 |
| DUOFM | 28.0 | 7.2 | 204 | 28.3 | 8.9 | 232 |
| URSSM | 30.0 | 7.8 | 240 | 26.9 | 7.7 | 245 |
| SmpFM | 3.0 | 2.7 | 1 | 29.9 | 7.9 | 63 |
| HbrFM | 29.9 | 7.2 | 227 | 27.2 | 8.1 | 236 |
Table 3: Symmetry analysis . We compute mGeoErr ( ) by allowing only symmetry-free correspondences (col. E ) and additionally the symmetric ones (col. ES ), and report the number of symmetrically flipped estimations out of 400 test pairs (col. #Flips ).
Figure 9: Qualitative results on SHREC16-Partiality.
| URSSM | EchoMatch | Wormhole | Ours | |
|---|---|---|---|---|
| CUTS | 42.1 | 12.7 | 48.2 | 6.7 |
| HOLES | 29.8 | 57.1 | 26.9 | 12.09 |
Table 4: mGeoErr ( ) on SHREC16-Partiality. Ours outperforms all baselines, including the supervised method EchoMatch.
Figure 10: Human Segmentation. We show the segmentation results of DiffusionNet and ours on the composite dataset [ 40 ] . Our RINONet produces sharper segments.
Figure 11: Robustness under noise perturbations.
| Ours | URSSM | DUOFM | CnsFM | SMS | SmpFM | HbrFM | |
|---|---|---|---|---|---|---|---|
| #params(k, ) | 327.7 | 510.3 | 477.8 | 477.8 | 1,266.2 | 1,848.4 | 494.3 |
| runtime(s, ) | 0.065 | 0.023 | 0.071 | 0.055 | 15.311 | 0.171 | 0.043 |
| mGeoErr ( ) | 4.6 | 24.8 | 25.2 | 9.1 | 57.6 | 9.9 | 26.4 |
Table 5: Comparison of runtime, #params and mGeoErr on SMAL.
Figure 12: MGeoErr( ) on SCAPE, FAUST, SCHREC19. (Left) trained on SCAPE; (Right) trained on FAUST. Y-axis dataset labels indicate the test dataset. MGeoErr is annotated on top of each bar. All baselines significantly deteriorate when changing from wks to xyz, whereas ours is more robust to these changes. To our best knowledge, RINO is the first method, which learns competitive features and correspondences directly from raw 3D geometry in a purely data-driven manner.
Figure 13: Statistics of wks vs. xyz. The box plot illustrates the statistical distribution of the mean feature channel values at semantically corresponding points across the entire shape collections of the SMAL and SCAPE datasets. The methodology involved aggregating the channel-wise mean of features across corresponding points for every shape, yielding a feature matrix (with dimensions #correspondences #feature channels), where each column’s distribution is plotted as a box. The statistics shown for the 5th, 10th, 15th, and 20th wks channels demonstrate a pronounced difference in behavior between the datasets. Specifically, the wks statistics for SMAL display a substantially larger variance than those for SCAPE . This higher dispersion is a direct consequence of the non-isometries present in the SMAL dataset, which contrasts sharply with the stability observed in the xyz descriptor statistics.
Figure 14: Non-manifold shape matching. Methods employing the wks as input exhibit a significant performance deterioration, even for SoTA methods such as URSSM and HbrFM, an observation that also applies to our proposed method (Ours+wks). Fortunately, our SO(3)-invariance network allows for effective learning directly from the raw xyz coordinates (Ours). This approach offers robustness against the difficulties posed by non-manifoldness, bypassing the fragility observed when using wks.
| FAUST | FAUST non-manifold | |
|---|---|---|
| URSSM(wks) | 1.6 | 17.6 |
| HbrFM(wks) | 1.4 | 16.6 |
| Ours(wks) | 1.6 | 15.6 |
| Ours(xyz) | 1.6 | 2.7 |
Table 6: MGeoErr( ) on FAUST non-manifold shapes.
Figure 15: Human Segmentation. We show the segmentation results of DiffusionNet and our RINONet on the composite dataset [ 40 ] . Our RINONet produces sharper and more semantically meaningful segments.
| Method | Human dataset [ 40 ] |
|---|---|
| DiffusionNet(hks, w/o aug) | 91.3% |
| Ours(hks, w/o aug) | 91.0% |
| DiffusionNet(xyz, w/ aug) | 90.3% |
| DiffusionNet(xyz, w/o aug) | 83.6% |
| Ours(xyz, w/ aug) | 90.6% |
| Ours(xyz, w/o aug) | 90.1% |
Table 7: Segmentation accuracy ( ) on human dataset [ 40 ] .
Figure 16: RNA segmentation
Figure 17: 3D rigid registration on Stanford dataset (bunny)
| No. | Ablation Setting | SMAL |
|---|---|---|
| 1 | w\o | 7.3 |
| 2 | w\o | 7.6 |
| 3 | w\o | 5.7 |
| 4 | w\o the Qmaps branch | 5.1 |
| 5 | w\o | 5.0 |
| 6 | Ours + coupling | 26.9 |
Table 8: Ablation study on SMAL.
Figure 18: Backbone comparison. Spider emoji denotes DiffusionNet and rhinoceros emoji denotes RINONet.
| mGeoErr ( ) | DUOFM | CnsFM | URSSM | HbrFM | |
|---|---|---|---|---|---|
| (i) | DiffusionNet | 25.2 | 9.1 | 24.8 | 26.4 |
| RINONet | 5.7 | 7.3 | 5.1 | 7.4 | |
| (ii) | SCAPE | FAUST | SMAL | DT4D | |
| DiffusionNet | 26.9 | 30.0 | 24.8 | 59.3 | |
| RINONet | 2.0 | 1.6 | 5.1 | 5.9 | |
| RINONet + CFMaps | 2.0 | 1.5 | 4.6 | 5.3 |
Table 9: Ablation: SO(3)-invariant features & CFMaps
Figure 19: Matching shapes with Gaussian perturbation.
Figure 20: Failure cases.
Figure 21: Qualitative results on the real-world FAUSTSCAN.
Figure 22: SHREC16-Partiality CUTS cat
Figure 23: SHREC16-Partiality CUTS david
Figure 24: SHREC16-Partiality CUTS centaur
Figure 25: SHREC16-Partiality CUTS dog
Figure 26: SHREC16-Partiality CUTS horse
Figure 27: SHREC16-Partiality CUTS victoria
Figure 28: SHREC16-Partiality CUTS michael
Figure 29: SHREC16-Partiality HOLES cat
Figure 30: SHREC16-Partiality HOLES david
Figure 31: SHREC16-Partiality HOLES centaur
Figure 32: SHREC16-Partiality HOLES dog
Figure 33: SHREC16-Partiality HOLES horse
Figure 34: SHREC16-Partiality HOLES victoria
Figure 35: SHREC16-Partiality HOLES michael