Guaranteed Simply Connected Mesh Reconstruction from an Unorganized Point Cloud
Authors: Liyan Chen
Jingyi Li
Qixing Huang
Venue: ICLR 2026 Poster (2026)
Keywords: 3D Reconstruction
spectral techniques
topological guarantee
Laplacian
Hodge decomposition
Primary area: optimization
TL;DR: Reconstruct a simply connected manifold mesh from a noisy 3D point cloud with the Helmholtz-Hodge Theorem

Abstract: We introduce an approach that reconstructs a closed surface mesh from a noisy point cloud, where the topology of surface is guaranteed to be simply connected, i.e., homeomorphic to a topological 2-sphere. This task enjoys a wide range of applications, e.g., 3D organ and vessel reconstruction from CT scans. Central to our approach is a robust module that takes a collection of oriented triangles in a 3D triangulation as input and outputs a simply connected volumetric mesh whose boundary approximates the input triangles. Starting from a 3D Delaunay triangulation of the input point cloud and initial triangle orientations obtained through a spectral approach, our approach alternates between applying the module to obtain a reconstruction and using that reconstruction to reorient the input triangles.  Experimental results on real and synthetic datasets demonstrate the effectiveness of our approach.

Published as a conference paper at ICLR 2026
GUARANTEED SIMPLY CONNECTED MESH RECON- STRUCTION FROM AN UNORGANIZED POINT CLOUD
Liyan Chen'
Jingyi L
Qixing Huang
'hee University of Texas at Austin {liyans, huangqx) (cs. utexas edu jingyi. 11 www.11eppptecchiiiieeeee edu
"École Polytechnique
3 Inria Saclay
ABSTRACT
We introduce an approach that reconstructs a closed surface mesh from a noisy point cloud, with a guarantee that the reconstructed surface is simply connected (homeomorphic
2-sphere). This task enjoys wide range of applications, e.g.. 3D organ and vessel reconstruction from CT scans. Central to our approach is a robust module that takes a collection of oriented triangles in 3D triangulation as input and outputs a simply connected volumetric mesh whose boundary approxi- mates the input triangles. Starting from a 3D Delaunay triangulation of the input point cloud and initial triangle orientations obtained through a spectral approach, approach alternates between applying the module obtain reconstruction and using that reconstruction to reorient the input triangles. Experimental results real and synthetic datasets demonstrate the robustness and accuracy of our ap- proach.
INTRODUCTION
Reconstructing a 3D mesh surface from noisy point clouds is fundamental problem in com- puter graphics and related domains |Berger et al.] (2017). Existing methods caaiodngg-rniiggggg from com- putational geometry to implicit function and neural Spproaches-achevve achieve strong geometric fidelity but generally lack topological control. In particular, no prior method guarantees simply con- nected reconstruction, a property essential for modeling structures such as human organs and ves- sels (cf. CrossSDF Walker et al. (2025)). This gap stems from the absence of computational tools for effective topological control. We review these approaches in detail in Section 2
In this paper, we introduce a mesh surface reconstruction approach that approximates the under- lying surface while guaranteeing that the reconstruction is simply connected and closed, as shown in Figure 2 We present a nontrivial generalization of the winding number field paradigm Feng al. (2023). extending topological control from ID curves on 2D surfaces to 2D surfaces embedded in 3D. Our formulation takes input collection of oriented triangles from 3D triangulation and outputs a triangle mesh that is closed, simply connected, and manifold. Motivated by IPSR Hou let al. (2022), we introduce approach that combines initialization phase and alternating phase. The initialization phase computes a 3D Delaunay triangulation of the input point cloud and establishes consistent orientations of the triangles. A key contribution of this paper is spectral method for computing these orientations. The algorithm then alternates between reconstructing a surface from the current triangle orientations and using the resulting surface to select and reorient the input triangles. This procedure robust and effectively removes outlier triangles from the input.
We evaluated our approach on the 3D medical reconstruction benchmark introduced in DoosSDFWalkers et al. (2025), as well as on a synthetic benchmark with controlled noise levels and sampling densities. The experiments demonstrate that our method is the only one that guarantees simply connected reconstructions while matching state-of-the-art methods in geometric accuracy. An ablation study further justifies the design choices underlying our approach.
2 RELATED WORK
Existing surface reconstruction approaches can be broadly divided into two categories. The first category consists of computational geometry approaches Amenta al. (1998); Cheng et al. (0033)

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(a) Input CT Scan
(b) CrossSDF
(c) Our Reconstruction
(d) Ground Truth
Figure 1: Visual comparison of pulmonary vascular reconstruction. The top row displays the full rendered results, while the bottom row presents corresponding zoomed-in views. From left to right: (a) the input CT scan; (b) the reconstruction of Walker et al., (2025) with 6 connected components and numerous spurious handles; (c) our simply connected reconstruction; and (d) the simply con- nected ground truth.
The advantage of these approaches that they offer theoretical guarantees when points are clean and their distributions are adaptive to the curvature of the underlying shape. However, their performance degrades significantly in the presence of outliers in the input data Dey & Goswami (2004). Our method belongs to this category but remains robust to noisy input. The work most closely related is the spectral approach of Kolluri et al. (2004), which, unlike ours, does not guarantee the reconstruction of a simply connected surface mesh.
Another category of surface reconstruction methods first fits implicit function to the input points and then extracts a level-set using an algorithm like Marching Cubes Lorensen & Cline (1987).
Among classic techniques, Poisson Surface Reconstruction (PSR) Kazhdan et al. (2006) remains highly influential but, like many early methods Hoppe et al. (1992); Carr et al.] (2001). requires oriented normals as input. This requirement extends to many modern deep learning approaches such as DeepSDF Park et al. (2019) and SIREN Sitzmann et al. (2020). While methods such as
IPSR Hou et al. (022)) SAL Atzmon & Lipman (2020), and SALD Atzmon & Lipman (202 have been developed handle unoriented point clouds, none provides explicit control over the topology of the reconstructed surface, which is the primary focus of this paper.
Our approach inspired by the winding number Jacobson et al. 2013 concept recently used to guide normal orientation in iterative frameworks Xu et al. (2023): ) Liu et al. 2025). Our work builds most directly on Feng et al. (202)), which introduces Helmholtz- Hodge Decomposition (HHD) of discrete winding number heeld to capture the non-trivial topology of curves on a surface. In this paper, generalize this framework from 2D curves 3D surfaces and address several critical challenges. First, since the HHD framework requires consistently oriented triangulation as input, we propose method to initialize triangle orientations spectrally and then refine them in an alternating scheme alongside the surface reconstruction, akin to IPSR Hou et al. (2022). Second, employ robust optimization to suppress spurious topological handles, which naturally promotes outlier removal and enforces a desired simple topology.
3
OVERVIEW
We begin with the problem statement in Section 3.1 followed by an overview our approach in Section 3.3
3.1
PROBLEM STATEMENT
The input is a collection of n points P = (p: E RR3) sampled from some underlying 3D object. Typical examples include data acquired from 3D sensors such LiDAR, depth cameras, and CT
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(a)
(b)
(c)
Figure 2: Overview our approach. (a) Input unoriented point cloud. (b) The initialization stage computes a base triangulation (left) and an initial orientation of the boundary faces Fo- (c) The alternating stage iterates between computing the correction surface A and updating the current re- construction, represented as a collection of oriented faces.
scans. Such point sets are often irregularly distributed and may contain outliers. The output is closed triangle mesh M2 (V,E,F) where V partially overlaps with A and M2 is simply connected - that is, path-comnected and free of nontrivial loops. In other words, V/ excludes outliers present in P and introduces additional points to satisfy the simply connected topological constraint and fill missing regions.
3.2
ALGEBRAIC FORMULATION FOR TOPOLOGICAL GUARANTEES
core contribution of our method the provision of global algebraic guarantee for simply con- nected reconstruction, effectively shifting the topological constraint from a complex combinatorial domain into an efficient linear domain. We formally derive this guarantee using De Rham's 's Theorem and the Hodge Isomorphism Theorem Warner 1983) (see Appendix B.3 for the further statements).
The theoretical mechanism relies the Helmholtz-Hodge Decomposition (HHD), which separates an input 1-form into exact, co-exact, and harmonic components. Crucially, the Hodge Isomorphism Theorem states that for smooth manifold, the space of harmonic 1-forms is strictly isomorphic the first cohomology group H' which is the algebraic generator of topological handles. By algo- rithmically removing these harmonic 1-forms via a linear solve, our formulation strictly enforces trivial H The final surface, constructed as the level set of this optimized field, is therefore guaran- teed to be simply connected by construction rather than by probabilistic or heuristic approximation.
3.3 APPROACI OVERVIEW
Figure illustrates our reconstruction pipeline, which consists of an initialization stage and an al- ternating reconstruction stage. The initialization stage first computes a 3D Delaunay triangulation Ms (V. (,,, together with subset of oriented faces Fo F from the input points. Each face in Fo assigned an orientation value in 13. To avoid elongated tetrahedra, additional points inserted into the triangulation under the guidance of octree built from the input. The subset Fo is then defined as the set of tetrahedral faces whose vertices belong to the input points. We then assign orientation each face in Fu using spectral formulation. This initialization stage is described in Section 5
In the alternating reconstruction stage, we first apply the core module (see Section This module takes as input the 3D triangulation M3 together with the oriented faces Fo defined above, and outputs closed, simply connected mesh whose faces F approximate Fo. The resulting mesh M2 then induces an optimized of oriented faces F F. We update Fo Fo J and reapply the core module. This alternating process terminates once F stabilizes, typically within 1-4 iterations.
CORE MODULE
We begin with an overview of the core module in Section which generalizes the winding number field approach of Feng al. (2023), and then describe its details in Section 4.2
We assume that input inliers accurate, setting that applies many sensors. See Section 7 for further discussion.
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4.1 MODULE OVERVIEW
The foundation of our approach is the winding number field u : R - Z associated with an oriented 2D surface S embedded in R2, where u(p) denotes the number of times S encloses point pe R is clear that u is piecewise constant with discontinuities on the surface S.
A key property of u is that it encodes all topological features of S throuuh the harmonic component of its Hodge-Helmholee decomposition (HHD). To exact this information, we first take the discrete 1-form w = Du, where D) denotes the discrete gradient operator mapping the vertex-based scalar field to a I-form defined on edges. Then the Helgee- decomposition states that any discrete 1-form w can be uniquely and orthogonally decomposed into three components: an exact part, a coexact part and a harmonic part:
= doa 5z, + 7.
(1)
Here, do is the discrete gradient operator and 62 is the discrete curl operator (the adjoint of the 2-form exterior derivative dj). The component a is curl-free, 82, is divergence- free. and the harmonic component y is both curl-free and divergence-free The harmonic part 7 represents the de Rham cohomology class of w, which is isomorphic to the non-trivial 1-homology of the domain. In our context, 7 serves
the dual representation of all nonbounding 2-cycles implicitly defined by the input Fo. A more formal introduction of HHD and the associated discrete operators is provided in Appendix A and Appendix B
Another attractive property of u is that it can be recovered, up to a global constant, by evaluating its jump across the surface S. However, this recovery procedure implicitly assumes that the inside- outside labeling at each point of S is known, so that the absolute 'jump' [eng et al. (2023) in u cross S can be properly defined. Such orientation information is usually unavailable from unorganized point cloud, leading to the well-known consistent normal orientation problem. In Section 5.2 we will introduce how to compute such orientation information.
4.2 MODULE PIPELINE
The first step in discretizing the winding number field u on the tetrahedral mesh M3 is to determine an appropriate representation. A natural approach is model u as piecewise-constant function that assigns constant value each tetrahedron T and enforces jump constraints of the form u²" - === across shared faces of adjacent tetrahedra. However, this formulation leads to an underconstrained system and does not offer effective way to extract topological signatures. Motivated by Feng cet al. (2023), we instead represent u in a reduced coordinate system by defining ui!" := u² u²'(v) for each vertex v, and expressing
T (uo) + cf:
(2)
where un denotes an underlying vertex-based field, and e encodes the jump information across Fo- Specifically, of is computed by accumulating local jump updates along path of face-adjacent tetrahedra from a local reference tetrahedron Ter (where e = 0) to T.. Consequentl, eg is constant within each tetrahedron; that is, cf cT. for any vertices v, of of T.
Consider two adjacent tetrahedra Tj and Tl sharing face f (i.j. k). For any vertex e k). the difference of the potential u can be written as
To ((u)), + c..)) ((uo), + 2.
(3)
Since the base field (uo), cancels, this difference is independent of the vertex v and is therefore constant across the face f. By construction, the path-integral definition of c enforces
r (-1,1),
where the sign depends on the orientation of f. Hence, this parameterization satisfies the prescribed jump conditions.
We define the I-form derived from the Darboux derivative of uf as follows. For each edge (i.j),
Wij = ( (u) - ) = (w: -u;) + Wijs
(4)
where
Wij AVST,etan(s))(5 -
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and fan(ij) denotes the set of all tetrahedra incident to the edge (i.j).
Our approach is based on the HHD of the I-form w. Since and differ only by the exact form du (the Darboux derivative of u), their harmonic components coincide. Therefore, the harmonic component can be computed from w, which can be directly obtained from the input.
4.2.1 COMPUTATION OF HELMIJOLTZ-HODGE DECOMPOSITION
In the HHD formulation shown in Equation ( the operators d; 6; presented sparse matrices, while w, a, G3 and y are expressed in vector form. We next describe how to compute a and B, both of which require solving linear systems.
Specifically, the exact component or is given by the linear system
Ta S1w,
(5)
where Lo is the Dihedral Angle Laplacian (DAL), and 81 is the discrete divergence operator.
The weight of an edge (i in the Dihedral Angle Laplacian Lo is defined by the of contribu- tions from all tetrahedra incident on that edge:
T
Wij
Wiy
TEfan(i))
where the contribution w from single tetrahedron is determined by its dihedral angle geometry (see Appendix C}. With these weights, the DAL takes the form
Lo F diag(w:) E,
where E E 1, [|XV is the matrix representation of the discrete gradient operator dg. Simi- larly, the coexact component B3 is obtained by solving the linear system
L2B = diw,
where di is the discrete curl operator and
Lg diag(wa))
the matrix representation of the 2-form Laplacian. Here, E (-1,0, Fi1 encodes the operator di.
The harmonic 1-form is given by
7 a 5z,3'
In the special case du, the decomposition yields a u, u = 0, = -. Because the HHD is linear, and w share the same harmonic component.
4.2.2 CORRECTION 2-CHAIN CONSTRUCTION
After extracting 7, which encodes the dual representation of all nonbounding topological features, we construct correction 2-chain that captures these features. The chain A identifies the faces to be added to removed from Fo in order to obtain a simply connected surface.
One challenge that the harmonic form y determines only a homology class for the correction surface A. Indeed, subtracting the boundary of any 3-chain W yields another valid surface A- 23 W, since A and A 2 W are homologous. To resolve this ambiguity, we seek correction surface A that is consistent with while minimizing suitable objective function.
To this end, we parameterize A using per-tetrahedron potentials oT for T' T. For face shared by two adjacent tetrahedra Ti and T2, we define
ATy. := σT, OT,
For any edge (i,j) e E, the consistency constraint is given by
(Do) = "/a+
(7)
where D is the Darboux gradient operator.
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To determine the correction surface, promote sparsity of A while accounting for the geometric importance of each face through its area. Unlike Feng et al. (2023), which formulates this problem as línear program, we use the L norm for minimization:
min Area(f)lAy])
(8)
JEF
This formulation is motivated by two considerations. First, the L penalty promotes sparse correc- tions and yields improved robustmess in the presence of outliers in Fo. Second, it admits an efficient solution iteratively reweighted least squares, which scales more favorably to large problems than solving a linear program. Following this strategy, we approximate Equation (8) by the weighted quadratic objective
min Σ Area( JEF
and solve the linearly constrained quadratic program defined by Equation ( to obtain updated solution of. Note that for an interior face f sshared by tetrahedra Ti and T2, we use the equivalent notations wy and WT- The weights then updated according
WT To
+ (o)
0 )
where e is a small positive constant introduced to avoid division by zero. This procedure is repeated until o.j. converges.
4.2.3 FINAL SURFACE RECONSTRUCTION
Let F. denote the 2-chain defined by Fo- After computing the correction 2-chain A, construct a "clean" 2-chain I' T5, A, which removes the nonbounding components of Ty.- Because T' may introduce faces that bridge previously disconnected components, we improve the smooth- ness of the final reconstruction by solving the Laplacian system once more:
Lwg b',
(10)
where b' is the source vector derived from 1" The resulting function
Wo
(11)
a jump harmonic field whose discontinuities correspond only to homologically trivial surfaces.
The final reconstruction is obtained by rounding the values of within each tetrahedron to produce an integer-valued 3-chain W. The reconstructed surface S is then defined as the boundary of W:
S = 6.W.
(12)
By construction, the boundary of a connected volume and therefore forms a closed 2-chain that is homologous to zero, guaranteeing that S has simply connected topology.
INITIALIZATION STAGE
This section presents the initialization stage of our approach. In Section 5.1 describe how to
obtain tetrahedral mesh M₁ EV,E, F. T) where P V. In Section 5.2 contribution of this paper, a spectral approach that computes consistent oriented faces from P.
present key
5.1 DELAUNAY TRIANGULATION
simple initialization approach to compute the 3D Delaunay triangulation of to obtain M3- However, this approach may introduce tetrahedra with very long edges and fails to provide tetrahedra on both sides of the underlying boundary triangles, a property that is critical for our framework. Motivated by Kolluri et al. (0004). we therefore augment the input point set P with the cell centers of a sparse grid enclosing P, and perform 3D Delaunay triangulation on the augmented set construct the tetrahedral mesh M3 used in algorithm.

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(a) Ground truth
(b) OReX
(c) CrossSDF
(d) Screened PSR
(e) Ours
Figure 3: Qualitative comparisons between our approach and baseline approaches.
5.2 TRIANGLE ORIENTATION
Let Fo C F denote the set of faces in M3 whose three vertices lie in P. Our goal is determine a consistent orientation of each face f E Fo for subsequent optimization. To this end, we associate with each face f Fo an indicator variable 2 {-1. 1), and collect these variables into vector We then optimize æ using the formulation derived from the framework in Section At an abstract level, the harmonic component can be written
Ax,
(13)
where A is a linear operator determined by the coefficients c² the Darboux derivative operator, and Modge-Helmhhole decomposition. We seek an orientation æ such that y vanishes.
A technical challenge is that a is an integer vector, making direct minimization of ||Ax] infeasible. To address this, we relax æ to be a real-valued vector and minimize the ratio |
whose optimal
|
solution un (AT A) the eigenvector associated with the smallest eigenvalue of A.A We
then recover discrete orientation by rounding { {-1, 13 according the sign of each entry. In the following, we describe the construction of A and the computation of the smallest-eigenvalue eigenvector A
First, note that each cT depends on æ. Therefore, we can encode
Ba, where [ is sparse matrix. Based on Equation 1.. we have
(14)
= daa - 523
-o 82 L/d1),
(15)
where L' and L denote the Moore-Penrone pseudoinverses of the O-form and 2-form Laplacians.
Combining Equation (13). Equation (14) and Equation (15). we have
A = (I do L - LBd.)
(16)
We use the Lanezos algorithm to compute the eigenvector associated with the smallest eigenvalue of A Each iteration requires solving six sparse linear systems (four involving and two involving L2).
EXPERIMENTAL RESULTS
This section presents an ablation study of our approach. We begin with the experimental setup in Section 6.1 We then present results real and synthetic datasets in Section 6.2 and Section 6.3 respectively. Finally, Section 6.4 presents an ablation of two components our approach.
EXPERIMENTAL SETUP
Datasets. Our evaluation utilizes six challenging models from CrossSDF Walker al. (2025)
Cross-sections of CT scans are treated as input point clouds. We follow the CrossSDF protocol in using both aligned and non-aligned cross-section orientations. All models consist of single

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Alvealis
Corebmi Align NonAlign
Coromaries Align
Coro
Heart NonAlign
Pulmonary NonAlign
Method
Align
NonAlign
NonAlign
Align
NonAlign
Align
Align
CD
CrossDF
0.35
0.56
0.24
0.26
0.28
0.46
0.21
0.29
0.38
0.82
0,24
0.34
ORcx
3.6
488
12
2
10
1.6
0.48
0.33
4.6
8.4
12
5.6
ScreenedPoisson
0.44
0.57
0.39
0.44
0.46
0.67
1.3
0.95
1.0
0.85
0.52
0.59
POCO
4.1
4.5
0.36
0.79
1.1
0.72
1.1
3.4
2.7
4.2
2,4
2.0
Neural-IMLS
13
15
Ours
D.41
0.53
0.21
0,22
0.26
0.33
0.16
0.18
0.32
0.66
0.21
0.28
HD
14
6.1
3.5
4
4.7
4.5
11
7.3
ORex
21
35
8.1
19
27
26
36
21
ScreenedPoisson
17
8.1
16
25
5:4
10
26
9.2
7.7
POCO
32
155
41
9.2
8.8
24
25
16.4
Neural IMI.
68
33
48
67
37
59
61
41
Ours
118
2.8
2.4
5.2
5.3
6.2
7.8
8.2
9.3
CC
GT
3
1
CrosSDD
OReX
939
2119
40
421
219
94
13
223
511
914
261
ScreenedPoisson
53
160
923
POCO
67
91
78
14
46
279
70
134
134
Neural-IMI.S
Ours
Table 1: Quantitative results thin structures across different methods and metrics. The table com- pares the Chamfer Distance(CD)xx00) Hausdorff Distance(HD)x110) and number of Connected Components(CC), under both aligned and non-aligned versions of each structure. Note the Coro- naries dataset: because our core algorithm designed to reconstruct single simply connected component, leveraged prior knowledge pre- pre-segment the Coronaries point cloud into three anatomical components. Each component is reconstructed independently, and the results were ag- gregated.
connected component, except for the Coronaries model, which has three distinct segments. These segments are separated using single-linkage clustering and processed individually. For synthetic data, we add varying levels of noise and outliers to the Stanford Bunny model compare our method against baselines.
Following CrossSDF, assess reconstruction quality using three metrics. Chamfer Distance (CD) and Hausdorff Distance (HD) are reported to measure geometric fidelity against the ground truth mesh. For thin structures, also report the number of Connected Compo- nents (CC) to verify topological consistency and structural integrity.
Evaluation Protocol.
Implementation. We implemented high-perffrmance solver for the sparse linear system in Equa- tion (16) on tetrahedral meshes. Our approach is a direct application of algorithm-sysaat systtmmmsttm co-design: fused the conjugate gradient (CG) method and its geometric multigrid preconditioner into a sin- gle persistent GPU kernel. This design contains all CG iterations and multigrid propagations Baaaato Bolitho et al. (2007), eliminating costly memory round-trips. By leveraging CUTLASS Thakkar et al. (2025). ,. we further optimized sparse matrix arithmetic and memory access patterns for the NVIDIA GH200 GPU. As result, a single iteration of our solver for Equation (16) on 100k point cloud completes 0. milliseconds.
6.2 ANALYSIS OF 3D MEDICAL RECONSTRUCTION
We compare our method against five baselines: CRossSDFWallee et al. (2025) and OReX (Swdaaee et al. (2023), which are tailored for reconstruction from cross-sections, and three state-of-the-an general-purpose methods, ScreenedPoisson Kazhdan Hoppe (2013), POCO Boulch Marlet
2022 and Neural-IMI.S Wang et al. (2024).
Quantitative results are presented in Table Our approach consistently outperforms all baselines across all three metrics (CD, HD, CC). Notably, it is the only method to correctly reconstruct the ground truth number connected components. While Neural-IMLSS is the second-best in topolog- ical accuracy, our method achieves significantly better geometric fidelity. Compared to CrossSDF, the closest competitor in geometric terms, our approach reduces CD and HD by 15.8% and 9.62%, respectively. In contrast, CrossSDF produces a large number of disconnected components for com- plex structures like the Alveoli and Heart.
Qualitative comparisons in Figure 3 visually confirm these findings. Our reconstructions are the only ones that simultaneously achieve high geometric detail comparable to the ground truth while
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(a) 8K+200
(b) 8K+400
(c) 8K+600
(d) 8K+800
(e) 8K+1000
Figure 4: Reconstruction of a synthetic shape at various noise. Top row:
Screened PSR:
Middle
row: SALD; Bottom row: Ours. Each column corresponds to different number of outliers.
Method
8K+200
8K+100
8K4800
4K+200
4K+100
4K+300
2K+200
2K-400
2K+800
1K+200
1K+400
IK+800
CD ScreenedPoisson
0.35
0.56
0.24
0.26
0.28
0.46
0.21
0.29
0.38
0.82
0.24
0.34
SALD
3.6
4,8
12
2
10
1.6
0.48
0.33
4.6
8.4
12
5.6
HD ScreenoPPoision
14
6.1
3.5
5.8
4.7
4.5
2.8
11
10
7,3
21
18
11
Ours
12.1
9.6
28
3,2
4.3
3.8
3,1
3.5
8.9
7.6
12.6
13,2
CC ScreenedPoisson
33
5
176
SALD
939
2119
219
Ours
Table 2: The evaluation protocol is the same Table #outliers.
1 Each dataset is denoted as #inliers
preserving the correct topology. This demonstrates the superiority of our method maintaining both geometric and topological integrity.
6.3 ANALYSIS OF 3D SYNTHETIC SHAPE RECONSTRUCTION
We compare our approach with two general- purpose 3D reconstruction approaches, namely ScreenedPoisson and SALD Atzmon & Lipman (2021). For each clean model, we generate an input by varying two parameters: the number points on the original shape and the number of outliers perturbed from surface points along their normals through a random offset between [-b, b], where set as 0,25 of the length of the bounding-bue diagonal.
Table 2 shows the quantitative performance of our approach and the baseline approaches. We can see that our approach significantly outperforms baseline approaches in terms of robustness against outliers. In the presence of high outlier ratios, our approach still achieves correct topology and high geometric fidelity. This is attributed the robust optimization module of our approach for removing outliers. In contrast, both SALD and ScreenedP require clean inputs for successful reconstructions. Figure shows qualitative results that are consistent with quantitative results.
6.4 ABLATION STUDY
We conduct an ablation study in 2D to demonstrate the importance of each component of our frame- work.
No initial orientation. Without initial orientation, our reconstruction fails completely, as shown in Figure Inconsistent orientations lead poor initial surface, causing essential inlier edges to be incorrectly pruned. Recovering from this state is difficult, as topological priors alone do not enforce geometric correctness.

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(a)
(b)
(d)
(e)
Figure 5: (a) Input. (b) Full approach. (c) No initial orientation. (d) No robust optimization. (e) No alternating optimization.
No robust optimization. The L norm in the optimization of the correction surface A is critical for promoting sparsity and smoothness in the reconstruction. Replacing with a less robust regularizer, such as the I norm, can result in non-smooth artifacts.
No alternating optimization. Alternating optimization is essential for handling significant numbers of outliers. A single optimization pass is insufficient to identify all outliers. Our iterative approach allows the correction surface A to progressively refine the inlier set Fo across iterations, gradually removing all outliers and converging to the correct reconstruction.
7 CONCLUSIONS, DISCUSSIONS, AND FUTURE WORK
In this paper, we introduce an approach that guarantees the reconstruction of a closed, simply connected mesh from noisy point cloud. Our method generalizes the winding number field of curves Feng et al. 2023) to three dimensions: it takes as input a collection of oriented triangles from 3D triangulation and produces closed, simply connected mesh. We develop strategies to infer consistent triangle orientations from a tetrahedral mesh and to remove input outliers through an alternating optimization scheme. Experimental results on both real and synthetic benchmark datasets demonstrate the effectiveness our approach.
Our approach assumes that input inliers are accurate, they are vertices of the reconstructed mesh. This assumption reasonable because most sensor noise in LiDAR CT scans is white or salt-and- pepper noise. In the presence of Gaussian noise, however, better strategy is to place vertices along the edges of the underlying 3D tetrahedral mesh, similar to the MarchingCuhe approach Lorensen & Cline (1987) in Poisson surface reconstruction Kazhdan et al. (2006). In future work, we plan to develop strategies smooth the final function in Equation (II). which can be used with March- ing Tngrehhheerron DOI & KOIDE (191]) for surface reconstruction.
There are ample opportunities for future research. While this paper has focused on closed, simply connected shapes, we plan to extend our framework to handle arbitrary topologies by developing tools to control the harmonic component y in the HHD. Another future direction is to leverage this representation to learn generative models of 3D shapes with controllable topologies.
ACKNOWLEDGMENTS
This work was supported by NSF-2047677, NSF-2413161, NSF-2504906, NSF-2515626, and GIFTs from Adobe and Google. This work supported by computing support on the Vista GPU Cluster through the Center for Generative AI (CGAl) and the Texas Advanced Computing Center (TACC) at UT Austin.
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A DISCRETE EXTERIOR CALCULUS AND HODGE OPERATORS
We briefly summarize the discrete exterior calculus (DEC) operators used throughout the paper. In DEC, discrete k-form quantity assigned oriented k-dimensional mesh elements that can be integrated over them. On tetrahedral mesh, discrete differential forms are associated with mesh elements: 0-forms with vertices, 1-forms with edges, and 2-forms with faces. These operators provide discrete analogues of the gradient, curl, and divergence from vector calculus.
Exterior derivative and codifferential. The exterior derivative operators d; Q 22* map k-forms to (k + 1)-forms In particular, dg corresponds to the discrete gradient and d1 to the discrete curl. The codifferential operators 6ç Ω - 22 are adjoints of dg. Here, 8n corresponds to divergence, while 62 is the adjoint of the curl operator.
Definition A.1 (Discrete Curl and Divergence) Let w be a discrete 1-form defined on oriented edges.
Discrete curl dı maps edge-based values faces and measures the circulation of w around each face. For an oriented face f (1,j,k] k),
(d]w), Wiy + wyk + Wki-
Discrete divergence 82 maps edge-based values vertices and measures the net flow leav- ing each vertex:
(o\w);
Σ
Wij Wijs
jEadj(()
where adj(i) denotes the vertices adjacent to i and Way are geometry-dependent edge weights.
In DEC terminology, dj is the exterior derivative and 61 is its adjoint.
Definition A.2 (Discrete Hodge Laplacians) The scalar potentials in the composition are obtained by solving Poisson equations involving discrete Hodge Laplacians:
de-
The 0-form Laplacian
Ao = 8,
operates vertex-based scalar functions.
The 2-form Laplacian
D2 = d1
operates on face-based quantities.
These operators are sparse and can be assembled efficiently.
B HODGE-HELMHOLTY DECOMPOSITION
This section presents formal introduction the Hodge-Helmholtz decomposition of the discrete I-form w Du, where D denotes the discrete gradient operator mapping the vertex-based scalar field to I-form defined on edges.
The Hodge-Heimhon decomposition states that any discrete I-form w can be uniquely and orthog- onally decomposed into three components: an exact part, coexact part, and harmonic part:
= a 8,3 +
(17)
Here, a is discrete O-form (a scalar potential on vertices) and / is a discrete 2-form (a scalar potential on faces). The component da is curl-free, dg,3 divergence-free, and the harmonic com- ponent 7 is both curl-free and divergence-free. This harmonic part represents the de Rham coho- mology class of w. By de Rham's theorem, this class is isomorphic to the nontrivial I 1-homology of
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the domain. In context, Y provides the dual representation of all non-bounding cycles implicitly induced by the topological errors in the imput.
To compute the Glodge-Helmhor decomposition, we first compute the potentials and (3 by solv- ing two decoupled Poisson systems, as described in Appendix [BT and Appendix B.2 respectively. Then the harmonic component is obtained as the residual.
B.1 COMPUTING THE EXACT (CURI--FREE) COMPONENT
To find the exact (curl-free) component do we first solve for the scalar potential o. Our strategy to eliminate the other components of the Hodge-Helmholtz decomposition by applying the discrete divergence operator ò1 Equation
m 51dga 5182,3 817.
The last two terms vanish: 5102 because the divergence of a curl-like field zero and i because harmonic components are divergence- Therefore,
mu8 òjdgex.
(18)
Since 81 do is the discrete O-form Laplacian Ag. we obtain the Poisson equation
Dp d1w.
In matrix form, this becomes
Loce ba.
where Lo is the dihedral-angle Laplacian, a is the vector of unknown vertex potentials, and b,, muf computed using the discrete divergence operator.
This sparse linear system is solved subject to constraint ensuring uniqueness (c.g.. fixing one vertex value enforcing zero-mean solution). Once a is obtained, the exact component is recovered as d
B.2 COMPUTING THE Co-EXACT (DIVERGENCE-FREES COMPONENT
To extract the co-exact (divergence- component 62,3, solve for the 2-form potential B. Anal- ogous to the previous subsection, we eliminate the other components of the decomposition by ap- plying the discrete curl operator dj Equation (17)
dj w = dp a d, d1y
The first and last terms vanish: di 0 because the curl of gradient is zero, and diy because harmonic components are curl-free. Thus,
d =1
Assuming the co-closed component of /3 is zero, d, 8g reduces to the discrete 2-form Laplacian Ags yielding the Poisson equation
Ag3 diw.
In matrix form, this becomes
T.2.3 = bR: where Ly is the matrix representation of Ag d2 8z. B is the vector of unknown face-based poten- tials, and by = diw is computed using the discrete curl operator.
This sparse linear system is solved for B, after which the co-exact component is recovered as 8g,9.
B.3 HOMOLOGICAL GUARANTEES HELMHOLTTZ-HOGGE DECOMPOSITION
De Rham's theorem states that for a smooth manifold, its k-th de Rham cohomology group is iso- morphic to its singular k-th homology group with real coefficients De Rham (931). In discrete setting, this means that the space of harmonic k-forms is isomorphic to the space of non-trivial k-cycles. For our 3D problem, the harmonic 1-form 7, obtained from the HHD of the field Du,
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provides dual representation of the nonbounding 2-cycles in г. A 2-cycle is nonbounding if it does not arise as the boundary of any 3-chain, i.c., if represents a topological hole or handle. By algorithmically isolating and removing the influence of y, construct final jump surface that is guaranteed to bound volume and hence homologically trivial, i.e., a disjoint union of spheres. In RR every sphere is simply connected.
C VOLUMETRIC LAPLACIAN
This section details the definition of w.. where (i.j) is an edge in tetrahedron T' As shown in Figure 6 let the four vertices of T be U, Uj. Uk. and vt. Define aij as the angle between edges v,U, and 0, and Big as the angle between edges vy and v Let OK! denote the dihedral angle between faces 0, and vr Following Liao (2025), the edge weight defined as
A
81
0yl ccot any cot B.g cos 8Nj cot² asy asot Bis))
8
Figure 6: Edge weight w
16


Reviews:
[{"review_id": "P6NQnHyBZI", "reviewer": "Reviewer_y33m", "summary": "The paper presents a method to reconstruct closed, simply connected 3D meshes from noisy point clouds by first computing a 3D Delaunay triangulation and assigning triangle orientations via a spectral method, then iteratively applying a winding-number–based module to correct topology and refine the mesh. Experiments on medical and synthetic datasets show it produces topologically correct reconstructions with high geometric fidelity, outperforming existing methods", "questions": "- Is it guaranteed that this procedure will always converge? \n- In line 179: Is it assumed that $f \\in \\mathcal{F}_0$ ? I haven't seen this to be explicitly stated, but I was wondering if it is not true, then (in line 183) shouldn't be $\\Gamma_f \\in \\{ -1, 0, 1 \\}$ ?\n- Is there any reason for not including results on thick structures (as defined in the CrossSDF paper)?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- The proposed method guarantees that the final 3D mesh is free of holes (topologically equivalent to a 2-sphere i.e., simply connected).\n- It is a non–deep-learning approach that runs very efficiently and shows strong robustness to noisy point clouds, outperforming previous methods.\n- The paper is exceptionally well presented, with clear overview sections that make the core ideas accessible even to readers outside this research area.", "weaknesses": "- The ablation study lacks quantitative results. The authors only include a qualitative 2D example (Figure 5) to demonstrate the importance of each component. It would strengthen the paper to report numerical metrics on full 3D models, similar to Tables 1 and 2.\n- The evaluation metrics are described only briefly, authors can further elaborate why these metrics are suitable. In addition, highlighting the best metric values in bold would make the tables clearer and easier to interpret. It would help the reader if the best results in the tables were highlighted (e.g. in bold).\n- It would be beneficial to include runtime comparison of the methods in Tables 1 & 2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XpmyokGAOP", "reviewer": "Reviewer_GGPn", "summary": "This paper proposes an approach that guarantees a closed, simply connected mesh reconstruction from a noisy point cloud. It is inspired by the winding number field. The method takes a collection of oriented triangles of a 3D triangulation as input and outputs a closed, simply connected mesh. Besides, it obtains oriented triangles from a tetrahedral mesh and removes input outliers via alternating optimization. Experimental results on real and synthetic benchmark datasets demonstrate the effectiveness of the\napproach.", "questions": "The experiments lack a comparison on the computation cost.\n\nIn terms of ablation study, it is important to provide more quantitative results apart from the comparison in Figure 5.\n\nAccording to the quantitative comparisons in Table 1, why other competitors perform much poorly than the proposed method. Whether the implementation setups are fair properly?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper is well written with good presentation.\n\nThe implementation details have been elaborated clearly. It might be easy to reproduce the results reported in the paper.\n\nThe formulations are defined and explained clearly.\n\nThe improvements over prior methods are promising and remarkable.", "weaknesses": "It is important to clarify the new contributions more clearly, as the prooposed method is built on several existing methods like (Jacobson et 2023, Feng et al 2023). As claimed in the paper, the novelty is generalizing the existing framework from 2D curves to 3D surfaces. \n\nApart from textual descriptions and formal equations, it is common to provide a figure which can introduce the overal pipeline in this method. \n\nSome presentations need more refinement, such as the captions in Figure 3 and 4 should not be on top.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8XXyfv4kxS", "reviewer": "Reviewer_9oMt", "summary": "This paper suggests a method to reconstruct 3D mesh surfaces from point clouds. It is claimed that the method is guaranteed to reproduce simple connected surfaces. Empirical experiments show that the method compares favorably with other methods", "questions": "1. Line 152: you say u is piecewise constant, and then talk about the 1-from du. But a piecewise constant function is not smooth right? What is the meaning of du?\n\n2. In Line 87 you say existing methods do not provide control on surface topology, but there are some such methods [1,2,3], it would make sense to discuss them somewhere and explain how these methods relate, or what is the advantage of your method.\n\n3. The discussion of the results is a bit misleading: in line 416 you say \"our approach consistently outperforms all baselines\" and the ours method is highlighted in gray, which could be interperted as winning a task. However, the \"Ours method\" does not win in all tasks in table 1 or in table 2. It would be good to highlight the winner in each column in bold, and to add some discussion and to replace \"our approach consistently outperforms all baselines\" with our approach outperfroms baselines in X out of Y instances. \n\nMinor point which you do not need to address in rebuttal. There are some spelling mistakes you should address:\n\n* Line 432 you say \"too\" but I think you mean \"two\"\n\n* Line 478  pepper is missing a 'p'\n\n\n\n\n[1] Topology-Aware Surface Reconstruction for Point Clouds, Gabrielsson et al. Computer Graphics forum 2020\n[2] Robust Optimization for Topological Surface Reconstruction. Lazar et al. TOG 2018\n[3] Topology-controlled Re-\nconstruction of Multi-labelled Domains from Cross-sections. Huang et al. TOG 2017", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The method gives good results on the surface reconstruction task\n2. The method seems novel, though I am not sufficient familiar with computer vision/graphics literature to make an informed judgement.", "weaknesses": "1. Wrong venue: This paper does not really include any learning aspects. And it assumes knowledge of geometry processing (e.g. vector forms, Helmholz Hodge Decomposition) which is not common in ICLR. It seems to me the paper would get a more meaningful review, and a move understanding audience, in a venue like Siggraph or CVPR\n\n2. I'm a bit concerned re mathematical correctness: \n(a)  The paper claims a guaranteed simply connected mesh reconstruction, but there is no formal analysis of the algorithm (of the form of Theorem-Proof).\n\n(b)  The paper contains many advanced math statements without a source e.g., in Line 152 \n\"S is simple connected when the harmonic component of the 1 form du vanishes\" but I think this is true of most of section 4. \n\n(c) I have some specific questions on the math, detailed below", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hmFPDBnati", "reviewer": "Reviewer_jxjb", "summary": "The paper presents an approach to reconstruct a closed and connected mesh from noise point cloud. The results guarantee to be simply connected and homeomorphic to a sphere. The method is validated using a few models.", "questions": "What are the efficiency performance of the method? The number of points for evaluation. The running time and memory consumption.\n\n(Feng 2023) evaluates winding numbers on surface instead of 3D volume space. Since u(p) is the number of times that the surface  S encloses p, why isn't u(p) the space winding number (Jacobson et al. (2013))?\n\nCan you give more explanations on HDD?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The method is interesting with deep math foundation.", "weaknesses": "The topology is limited to genus 0. It cannot deal with models of higher genus.\n\nThe convergence condition of the iterations are not clear.\n\nThe time performance is not reported.\n\nThough the paper claims to reconstruct noise point cloud. The noise-level is not clear.\n\nThe data set for evaluation is few. More data should be evaluated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)