Guaranteed Simply Connected Mesh Reconstruction from an Unorganized Point Cloud¶

Conference: ICLR 2026
OpenReview: https://openreview.net/forum?id=jjEnTBsffi
Code: TBD
Area: 3D Vision / Surface Reconstruction
Keywords: Surface Reconstruction, Point Cloud, Winding Number Field, Helmholtz-Hodge Decomposition, Topological Control, Simply Connected

TL;DR¶

Closed triangle meshes are reconstructed from noisy point clouds with an algebraic guarantee of simple connectivity (homeomorphic to a 2-sphere) via Helmholtz-Hodge Decomposition (HHD), filling the gap in topological control for existing methods.

Background & Motivation¶

Background: Reconstructing 3D meshes from noisy point clouds is a fundamental problem in computer graphics. Existing methods fall into two categories: computational geometry methods (Delaunay/Voronoi) with theoretical guarantees on clean data, and implicit function methods (Poisson, DeepSDF, SIREN, IPSR, SAL/SALD) with high geometric fidelity.

Limitations of Prior Work: All methods focus primarily on geometric fitting, but none can guarantee the topological structure of the reconstruction. In medical scenarios like organs and vessels, which must be simply connected (path-connected without non-trivial loops), mainstream methods often produce multiple disconnected components and numerous "spurious handles"—for instance, CrossSDF yields 6 components for a lung vessel in Figure 1. While computational geometry methods have guarantees, they fail with outliers; implicit methods offer no topological control.

Key Challenge: Topological constraints are inherently combinatorial. Explicitly enumerating or repairing handles on meshes is computationally expensive and difficult to integrate into high-fidelity optimization.

Goal: To maintain SOTA geometric accuracy while ensuring the output is a closed, simply connected manifold mesh by construction (rather than via probabilistic or heuristic approximations), remaining robust to outliers.

Key Insight: The winding number field paradigm is extended from "1D curves on 2D surfaces" to "2D surfaces in 3D space". The critical observation is that all topological features of a surface are encoded in the harmonic component \(\gamma\) of the HHD of its winding number field 1-form. The Hodge Isomorphism Theorem states that the space of harmonic 1-forms is strictly isomorphic to the first cohomology group \(H^1\) (the algebraic generators of topological handles). Thus, by eliminating \(\gamma\) through a single linear solve, combinatorial topological constraints are transformed into a linear domain, where enforcing a trivial \(H^1\) ensures simple connectivity.

Method¶

Overall Architecture¶

The core is a robust module that takes a set of oriented triangles from a 3D triangulation as input and outputs a mesh whose boundary approximates these triangles while being volume-simply connected. The pipeline consists of two phases: an initialization phase involving Delaunay triangulation and spectral orientation, and an alternating reconstruction phase that iteratively calls the core module to refine the reconstruction and filter/flip input triangles to remove outliers, typically converging in 1–4 rounds.

flowchart TD
    A[Unorganized Point Cloud P] --> B[Initialization: Augmented Delaunay<br/>Triangulation M3]
    B --> C[Spectral Method Assigns Consistent<br/>Orientation to Faces F0]
    C --> D[Core Module: Extract Harmonic<br/>Component γ via HHD]
    D --> E[Construct Corrective 2-chain Λ<br/>via L0 Sparse Optimization]
    E --> F[Γ' = ΓF0 − Λ<br/>Solve Laplacian for Smooth 3-chain W]
    F --> G[Reconstructed Surface S = ∂W<br/>Guaranteed Simply Connected]
    G -->|Filter/Flip F0 using S| C
    G --> H[Converged Output: Closed Simply Connected Mesh]

Key Designs¶

1. HHD Representation and Topological Extraction: Converting "Handles" to Linear Decomposition. For an oriented surface \(S\) embedded in \(\mathbb{R}^3\), its winding number field \(u:\mathbb{R}^3\to\mathbb{Z}_{\ge0}\) jumps at \(S\) and is piecewise constant elsewhere. Taking the discrete 1-form \(\omega=Du\) (mapping scalar fields to edges), HHD uniquely decomposes any 1-form as \(\omega=d_0\alpha+\delta_2\beta+\gamma\), where \(d_0\alpha\) is curl-free, \(\delta_2\beta\) is divergence-free, and the harmonic component \(\gamma\) is both. \(\gamma\) represents the de Rham cohomology class of \(\omega\), isomorphic to the non-trivial first homology of the domain. To compute \(\gamma\) directly from input, the field is represented with reduced coordinates \(u^T_v=(u_0)_v+c^T_v\). Since \(\omega\) and the directly observable \(\bar\omega\) differ only by an exact form \(du\), they share the same harmonic component, allowing \(\gamma\) extraction without knowing \(u\).

2. L0 Sparse Corrective 2-chain: From Algebraic Guarantee to Robust Discrete Surfaces. While \(\gamma\) determines the homology class of the corrective surface, ambiguity remains. The paper parameterizes the corrective chain as \(\Lambda_{T_1T_2}:=\sigma_{T_1}-\sigma_{T_2}\) using per-tetrahedron potentials \(\sigma_T\). Under the consistency constraint \((D\sigma)_{ij}=\gamma_{ij}\), it minimizes the area-weighted \(L_0\) norm: \(\min_{\sigma_T}\sum_{f\in F}\mathrm{Area}(f)|\Lambda_f|_0\). \(L_0\) is chosen over linear programming because sparse penalties are more robust to outliers in \(F_0\) and can be solved efficiently via Iteratively Reweighted Least Squares (IRLS). Finally, a "clean" 2-chain \(\Gamma'=\Gamma_{F_0}-\Lambda\) is smoothed via a Laplacian solve \(Lw_0=b'\), and a 3-chain \(W\) is obtained by rounding. The surface \(S=\partial_3 W\), as the boundary of a connected volume, is naturally null-homologous and simply connected by construction.

3. Spectral Initialization for Triangle Orientation: Relaxing Combinatorial Orientation to a Smallest Eigenvector. The HHD framework requires consistently oriented input triangles, but unorganized point clouds lack inside/outside labels. Each face \(f\in F_0\) is assigned an indicator \(x_f\in\{-1,1\}\), and the harmonic component is expressed as \(\gamma=Ax\). The goal is to find \(x\) such that \(\gamma\) vanishes. This is relaxed to a real-valued problem minimizing the Rayleigh quotient \(\frac{\|Ax\|^2}{\|x\|^2}\). The optimal \(x^\star\) is the smallest eigenvector of \(A^\top A\), which is then rounded to \(\{-1,1\}\). The operator \(A=(I-d_0 L^\dagger\delta_1-\delta_2 L_2^\dagger d_1)B\) is solved using the Lanczos algorithm.

4. Alternating Reconstruction for Outlier Removal: Iterative Inlier Refinement. After the core module outputs a mesh \(M^2\), it induces an optimized oriented face set \(F^\star\subset F\). Updating \(F_0\leftarrow F_0\cap F^\star\) and re-running the module allows the corrective surface \(\Lambda\) to refine the inlier set \(F_0\) across rounds. This gradually eliminates outliers until \(F^\star\) stabilizes (1–4 rounds).

On the engineering side, Conjugate Gradient (CG) solvers with Geometric Multigrid preconditioning are fused into a single persistent GPU kernel using CUTLASS, enabling sub-millisecond sparse matrix solves on an NVIDIA GH200.

Key Experimental Results¶

Main Results: 3D Medical Reconstruction (CrossSDF Benchmark)¶

Metrics: Chamfer Distance (CD)×100, Hausdorff Distance (HD)×100, Connected Components (CC, closer to GT is better).

Method	CD (Mean)	HD (Mean)	Correct CC
CrossSDF	Good	Good	No (e.g., Heart: 68/176)
OReX	Poor	Poor	No (Hundreds of components)
Screened Poisson	Medium	Medium	No
POCO	Poor	Poor	No
Neural-IMLS	Poor	Poor	Partially
Ours	Best	Best	Only method matching GT

Compared to CrossSDF, Ours reduces CD by 15.8% and HD by 9.62%, and is the only method to correctly reconstruct the Ground Truth (GT) number of connected components.

Synthetic Robustness (Stanford Bunny with Noise + Outliers)¶

Compared with Screened Poisson and SALD.

Method	CC under High Outliers	Geometric Fidelity
Screened Poisson	Out of control (176 CC for 1K+200)	Requires clean input
SALD	Severely out of control	Requires clean input
Ours	Consistently 1	High fidelity, robust

Ablation Study (2D)¶

Configuration	Result
Full Method	Correct reconstruction
W/o Spectral Orientation	Complete failure; inlier edges wrongly removed
W/o Robust Optimization (L1 instead of L0)	Non-smooth artifacts appear
W/o Alternating Optimization	Fails to remove all outliers in a single pass

Key Findings¶

Topological guarantees are not achieved via post-processing but are inherent to the construction \(S=\partial W\).
Robustness stems from \(L_0\) sparse correction: outliers are naturally treated as sparse corrections and eliminated.
The three components—spectral orientation, \(L_0\) optimization, and alternating refinement—are all essential.

Highlights & Insights¶

Topological control moved from the combinatorial to the linear domain: Leveraging the Hodge Isomorphism Theorem to equate "handle presence" with "non-zero harmonic 1-form" allows handles to be eliminated via linear solves.
Non-trivial extension of Winding Number Fields: Extending Feng et al. (2023) from 1D curves to 3D surfaces while solving the orientation challenge for unorganized point clouds.
Guarantee + SOTA Accuracy: Demonstrates that "guaranteed" computational geometry methods can achieve high accuracy and robustness previously reserved for implicit methods.
Algorithm-System Co-design: Fusing CG and Multigrid into persistent GPU kernels makes the iterative framework practical for large-scale data.

Limitations & Future Work¶

Assumption of precise inliers: The current method uses inlier vertices directly, making it sensitive to Gaussian noise, though suitable for LiDAR/CT salt-and-pepper noise. Future work may involve Marching Tetrahedron-style vertex placement.
Restricted to Simply Connected Topology: The current focus is on closed, simply connected shapes. Multi-component scenes require prior segmentation. The authors plan to extend this to arbitrary topologies by controlling the harmonic component \(\gamma\).
Spectral Initialization Dependence: Failures in initial orientation can lead to unrecoverable errors.
Generative Extensions: The controllable topological representation could be used to train 3D shape generation models with topological constraints.

Winding Number Lineage: From Jacobson et al. (2013) to Feng et al. (2023), this work represents a major upgrade to 3D surfaces.
Unoriented Reconstruction: Unlike IPSR or SALD, which do not control topology, this method uses an alternating mechanism for both orientation refinement and denoising.
Computational Geometry vs. Implicit: This work revitalizes Delaunay-based approaches by solving the sensitivity to outliers through robust \(L_0\) optimization.
Insight: Transforming discrete/combinatorial constraints into linear spectral problems via algebraic isomorphisms (Hodge Isomorphism) is a powerful paradigm for many structured reconstruction tasks.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ First method to guarantee simply connected reconstruction via algebraic construction.
Experimental Thoroughness: ⭐⭐⭐⭐ Comprehensive testing on medical and synthetic data, though some multi-component scenarios rely on external clustering.
Writing Quality: ⭐⭐⭐⭐ Clear mathematical derivation and logical flow, though the HHD theory has high prerequisites.
Value: ⭐⭐⭐⭐⭐ Critical for medical reconstruction (organs/vessels) where topological guarantees are non-negotiable.