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BridgeMatch: Conditional Transport Bridges in Matching Matrix Space for 3D Deformable Registration
Qianliang Wu, Haobo Jiang, Guangwei Gao, Shuo Chen, Jin Xie, Jian Yang, Yaqing Ding
TL;DR
Non-rigid registration needs reliable correspondences, but Top-K coarse-to-fine pruning can discard weak yet correct hypotheses. BridgeMatch preserves complete matching matrices through coarse diffusion, hierarchy-preserving lifting, and deterministic or stochastic high-resolution transport refinement. Both variants improve correspondence estimation, downstream registration, and zero-shot generalization, with larger gains under low overlap.
Problem
Top-K pruning in coarse-to-fine matching may remove weak but correct correspondence hypotheses and restrict fine matching to an incomplete search space.
Method
BridgeMatch uses coarse-resolution diffusion, hierarchy-preserving lifting of the complete matrix, and a high-resolution conditional bridge implemented as endpoint-CFM ODE or Brownian-bridge SDE.
Results
BridgeMatch improves correspondence estimation, downstream registration, and zero-shot generalization on 4DMatch, 4DLoMatch, CAPE, and DeepDeform, with the largest gains under low overlap.
Takeaways & Limitations
Maintaining all soft matching hypotheses enables high-resolution refinement without target-domain adaptation while using the same deformation solver.
Takeaways & Limitations
Zero-shot comparisons on CAPE do not imply identical training or input settings, and compared methods may use different pretraining or target-domain supervision.
Abstract
from arXiv · showhide
Reliable non-rigid point cloud correspondences are important for deformable anatomical registration, embodied perception and manipulation, and dynamic 3D reconstruction. Coarse-to-fine methods reduce computational cost by selecting the top-\(K\) coarse regions. However, this pruning may remove weak but correct hypotheses and restrict fine matching to an incomplete search space. We present \paper, a two-stage generative solver that maintains the complete soft matching matrix at both coarse and high resolutions. Stage~I uses denoising diffusion to estimate a global matching matrix in the compact coarse-resolution space. We then lift this matrix to high resolution while preserving its hierarchy. The lifted matrix is rank-bounded and block-constant. Stage~II refines it through a conditional transport bridge. We implement the bridge with two types of dynamics: a deterministic endpoint-parameterized conditional Flow Matching (CFM) ODE and a stochastic Brownian-bridge SDE inspired by Schrödinger bridges. Both variants share the lifted source, a time-conditioned transformer, and a matching-matrix endpoint predictor. Experiments on 4DMatch and 4DLoMatch show that both variants produce more accurate correspondences than the compared methods and improve downstream registration, with larger gains in low-overlap cases. They also improve cross-dataset generalization on CAPE and DeepDeform without target-domain adaptation while using the same deformation solver.
1 INTRODUCTION
BridgeMatch addresses incomplete fine-matching search caused by Top-K pruning by lifting complete coarse matching estimates and refining them in matching-matrix space. Its deterministic and stochastic bridge variants improve correspondence and registration results, especially under low overlap.
- Method: BridgeMatch connects compact coarse-resolution diffusion with a high-resolution conditional transport bridge using the complete coarse matching matrix as its source.The high-resolution ground-truth matching matrix defines the target, enabling Stage II to correct coarse correspondence estimates.
- Method: Stage II evolves the complete high-resolution matching matrix so multiple target-candidate scores can change while coarse errors and within-block details are corrected.The method studies deterministic and stochastic transport for refining ambiguous correspondence estimates.
- Results: 5.26 and 11.79 points: on 4DLoMatch, the stochastic bridge improves NFMR and inlier ratio over Diff-Reg, respectively.Experiments show consistent gains on 4DMatch and 4DLoMatch, particularly under low overlap.
- Motivation: Top-K pruning can remove weak but correct hypotheses, whereas BridgeMatch retains all candidate correspondences for subsequent refinement.The bridge updates every candidate score without pruning, although retaining candidates alone does not determine the final matches.
- Results: Both bridge variants improve correspondence estimation, downstream registration, and zero-shot generalization, with ablations testing high-resolution refinement and solver-step budgets.The contributions include deterministic endpoint-CFM and stochastic paired Brownian-bridge implementations.
2 RELATED WORK
BridgeMatch extends generative transport for 3D registration from pose, coordinate, or latent spaces to complete matching-matrix space. Its two-stage overview combines coarse diffusion, hierarchy-preserving lifting, and high-resolution bridge refinement.
- BridgeMatch Overview: Stage I performs reverse diffusion on a compact coarse matching matrix, then lifts the solution to initialize Stage II at high resolution.Both bridge variants share the time-conditioned transformer and matching-matrix endpoint predictor.
- BridgeMatch Overview: Stage II refines the lifted high-resolution matrix through either a deterministic endpoint-CFM ODE or a stochastic Brownian-bridge SDE.The stochastic branch additionally predicts bridge noise.
- Generative Transport in 3D Geometry: Unlike pose-, coordinate-, or latent-space models, BridgeMatch keeps both stages in matching-matrix space.The method connects compact matching-matrix diffusion with deterministic or stochastic high-resolution transport.
3 METHOD
BridgeMatch keeps complete soft matching matrices through a two-stage coarse-to-fine solver: diffusion estimates coarse correspondences, then a conditional bridge refines their hierarchy-preserving high-resolution lift. Deterministic endpoint-CFM and stochastic Brownian-bridge variants share the high-resolution state, geometry conditioning, and endpoint prediction while differing in path variance and integration.
- 3.1 METHOD OVERVIEW: Stage I performs reverse diffusion in compact coarse matching-matrix space, producing a complete global estimate for later high-resolution refinement.The coarse space reduces matrix size and uses features with larger receptive fields; DDIM is used during inference.
- 3.4 UNIFIED STAGE-II CONDITIONAL-BRIDGE MODEL: Stage II transports the lifted high-resolution matrix toward the ground-truth endpoint through either endpoint-CFM ODE or Brownian-bridge SDE dynamics.Both variants share the same endpoints and geometric conditioning, but use zero versus finite path variance.
- 3.3 HIERARCHY-PRESERVING LIFTING TO HIGH-RESOLUTION SPACE: Hierarchy-preserving lifting copies every coarse correspondence score to high-resolution pairs sharing the same source and target parent points.The lift is fixed, gradient-free, and requires no additional learned mapping.
- 3.3 HIERARCHY-PRESERVING LIFTING TO HIGH-RESOLUTION SPACE: The lifted source is rank-bounded and block-constant, providing a structured low-dimensional prior while Stage II recovers within-block correspondence details.Its rank satisfies rank(X0) ≤ rank(Y_hat(cr)) ≤ min(Ncr, Mcr).
- 3.4 UNIFIED STAGE-II CONDITIONAL-BRIDGE MODEL: A shared geometry-conditioned transformer predicts the matching-matrix endpoint from the evolving state, warped source, target geometry, features, and time embedding.Sinkhorn-normalized weights and soft Procrustes provide alignment conditioning, which is recomputed at every solver step.
- 3.5 STAGE II-A: DETERMINISTIC ENDPOINT-CFM SOLVER: The stochastic branch adds a low-rank pairwise noise head, while endpoint-CFM sets path noise to zero and integrates its induced velocity with explicit Euler steps.The Brownian branch uses Euler–Maruyama on a truncated interval followed by a final endpoint prediction.
4 EXPERIMENTS
Across correspondence, downstream registration, and zero-shot transfer, both BridgeMatch bridge variants outperform Diff-Reg, with larger gains under low overlap and deformation. The evaluation uses matched deformation solvers where specified, while protocol and pretraining differences constrain some comparisons.
- 4.2 NON-RIGID CORRESPONDENCE: Both Stage-II variants improve correspondence quality over Diff-Reg, with generally larger gains on the low-overlap 4DLoMatch benchmark.The stochastic bridge achieves the best NFMR and IR on both datasets, including gains of 5.26 and 11.79 points on 4DLoMatch.
- 4.2 NON-RIGID CORRESPONDENCE: The correspondence comparison does not isolate numerical integration because the deterministic and stochastic variants differ in paths, auxiliary predictions, objectives, and dynamics.This limits attributing their performance difference specifically to the integrator.
- 4.2 NON-RIGID CORRESPONDENCE: With GraphSCNet fixed, the stochastic bridge improves all eight registration metrics over Diff-Reg, including 4DLoMatch-F EPE from 0.095 to 0.089 and AccR by 4.3 points.Endpoint-CFM improves seven of eight metrics; RGGT with TRELLIS features remains stronger on several metrics but uses an additional prior.
- 4.2 NON-RIGID CORRESPONDENCE: The qualitative comparisons visualize correspondences on the left and GraphSCNet-estimated deformations on the right, using green and red lines for correct and incorrect matches.The stochastic bridge shows the most consistent qualitative results, with fewer long-range incorrect matches and better alignment around limbs and animal extremities.
- 4.3 ZERO-SHOT GENERALIZATION ON CAPE: Both variants improve all four CAPE metrics over Diff-Reg without target-domain adaptation, with the stochastic bridge reducing EPE from 0.018 to 0.016 and increasing AccS by 2.0 points.On the large-deformation subset, the stochastic bridge reduces EPE by 13.8% relative to Diff-Reg.
- 4.4 ZERO-SHOT GENERALIZATION ON DEEPDEFORM: On shared DeepDeform pairs, both variants improve all four metrics over Diff-Reg; the stochastic bridge reduces EPE by 17.1% and increases AccS by 9.6 points.SyNoRiM remains much stronger because it is trained on the target domain, and results from different protocols are not directly ranked against these zero-shot results.
5 CONCLUSION
BridgeMatch combines coarse diffusion with high-resolution matching-matrix refinement through deterministic or stochastic transport dynamics. Both variants improve correspondence estimation and downstream registration, with the stochastic bridge strongest under low overlap.
- Both BridgeMatch variants improve correspondence estimation over Diff-Reg and strengthen downstream registration with a fixed GraphSCNet solver.
- The stochastic bridge gives the largest gains under low overlap.
- BridgeMatch lifts a complete coarse soft matrix into a rank-bounded, block-constant high-resolution starting point.
- Stage II refines the lifted matrix using either a deterministic endpoint-CFM ODE or a stochastic Brownian-bridge SDE.
- Models trained on 4DMatch also improve matched Diff-Reg pipelines on CAPE and real-world DeepDeform without target-domain adaptation.
- Ablations identify high-resolution refinement as useful and expose a trade-off between accuracy and runtime.