Shape as Line Segments: Accurate and Flexible Implicit Surface Representation
Authors: Siyu Ren
Junhui Hou
Venue: ICLR 2025 Poster (2025)
Keywords: Implicit representation
3D geometric
surface reconstruction
Primary area: learning on graphs and other geometries & topologies
TL;DR: An accurate and efficient implicit geometry representation based on attributed line segments, which can handle both manifold and non-manifold structures.

Abstract: Distance field-based implicit representations like signed/unsigned distance fields have recently gained prominence in geometry modeling and analysis. However, these distance fields are reliant on the closest distance of points to the surface, introducing inaccuracies when interpolating along cube edges during surface extraction. Additionally, their gradients are ill-defined at certain locations, causing distortions in the extracted surfaces. To address this limitation, we propose Shape as Line Segments (SALS), an accurate and efficient implicit geometry representation based on attributed line segments, which can handle arbitrary structures. Unlike previous approaches, SALS leverages a differentiable Line Segment Field to implicitly capture the spatial relationship between line segments and the surface. Each line segment is associated with two key attributes, intersection flag and ratio, from which we propose edge-based dual contouring to extract a surface. We further implement SALS with a neural network, producing a new neural implicit presentation. Additionally, based on SALS, we design a novel learning-based pipeline for reconstructing surfaces from 3D point clouds. We conduct extensive experiments, showcasing the significant advantages of our methods over state-of-the-art methods.
The source code is available at https://github.com/rsy6318/SALS.

Published as a conference paper at ICLR 2025
SHAPE AS LINE SEGMENTS: ACCURATE AND FLEXI- BLE IMPLICIT SURFACE REPRESENTATION
Siyu Ren
Junhui Hou"
Department of Computer Science
Department of Computer Science City University of Hong Kong Hong Kong SAR, China jh www.hiiuutt....om edu hk
City University of Hong Kong
Hong Kong SAR, China
siyuren2 cêmy. cityu. edu.hk
ABSTRACT
Distance field-based implicit representations like signed/unsigned distance fields have recently gained prominence in geometry modeling and analysis. However, these distance fields are reliant on the closest distance of points to the surface, introducing inaccuracies when interpolating along cube edges during surface ex- traction. Additionally, their gradients are ill-defined at certain locations, caus- ing distortions in the extracted surfaces. To address this limitation, Shape as Line Segments (SALS), an accurate and efficient implicit geometry rep- resentation based on attributed line segments, which can handle arbitrary struc- tures. Unlike previous approaches, SALS leverages a differentiable Line Seg- ment Field to implicitly capture the spatial relationship between line segments and the surface. Each line segment is associated with two key attributes, inter- section flag and ratio, from which we propose edge-based dual contouring to extract surface. We further implement SALS with neural network, produc- ing a new neural implicit presentation. Additionally, based on SALS, we de- sign novel barning-based pipeline for reconstructing surfaces from 3D point clouds. We conduct extensive experiments, showcasing the significant advantages of our methods over otate-of-the-art methods. The source code is available at https://gihub..co//s/6/1111.1
propose
INTRODUCTION
Various implicit geometry representations have emerged as powerful approaches in 3D vision, graphics, and robotics (Park et al., 2019; Takikawa al., 2021; Wang et al., 2021; Mescheder al., 2019; Chibane et al., 2020b). Unlike explicit representations that define surfaces through point clouds polygon meshes, implicit representations describe surfaces as the isosurface of a scalar field. The signed/unsigned distance field (SDF/UDF) is among the most widely used implicit repre- sentations. Specifically, SDF captures the signed distance from given point to the closest surface, where the sign is used to distinguish between the inside and outside. The surface can be extracted from SDFs easily via Marching Cubes (MC) (Lorensen & Cline, 1987), Dual Contouring (Ju al., 2002) or their variants (Ju et al., 2002; Doi & Koide, 1991; Shen et al., 2021). However, SDF can only represent watertight shapes and cannot represent more general shapes, such as those with open boundaries and multi-layers, limiting its applications. By contrast, UDF, representing the absolute distance to the closest surface, allows for the representation of more general shapes. However, ex- tracting surfaces from UDFs is much more difficult than SDFs, although several methods (Guillard al., 2022; Ye et al., 2022; Chen et al., 2022: Zhang et al., 2023; Ren et al., 2023; Zhou al., 2022) modified from MC and DC have been recently presented extract surfaces from UDFs.
When extracting surfaces from SDFs and UDFs, MC-based methods compute the intersection point on each cube edge by interpolating the ratio of the distance values at the two endpoints of the edge. However, since this distance reflects the closest surface, inaccuracies may arise when there is another surface closer to or both endpoints, leading to erroneous intersection points, as shown
"Corresponding author. This project was supported in part by the NSFC Excellent Young Scientists Fund 62422118 18, and in part by the Hong Kong Research Grants Council under Grant 11219422 and Grant 11219324.

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Published as a conference paper at ICLR 2025
in Fig. la. Similarly, for DC-based methods (see Fig. 1b), an inaccurately calculated intersection point on the edge can lead to an incorrect solution of the quadratic error function within the cube. The differentiable properties of SDFs and UDFs, specifically their spatial gradients, are critical for surface extraction and offer notable advantages. However, the use of distance fields results in gradients that meaningless in certain regions, such at points equidistant from two surfaces (Fig. Ic), or even directly on the surface for UDFs. These limitations of SDFs and UDFs can introduce distortions during surface extraction.
Distance fields like SDF and UDF focus on the clos- est distances of individual spatial points to the sur- face. However, surface extraction from these fields often introduces distortions. As analyzed previously, both MC-based and DC-based methods rely on inter- polation along each edge (line segment) of the cubes during surface extraction. Therefore, natural ap- proach is to shift the focus from individual points to line segments in 3D space. So can we build field re- flecting the spatial relationship between the line seg- ment and the surface? Building this intuition, we propose Shape as Line Segments (SALS), a precise and versatile implicit surface representation that cen- ters on spatial line segments. SALS consists of two key components: 1) a differentiable line segment field (LSF) that captures the spatial relationship be- tween a query line segment and the surface, and 2) an edge-based dual contouring approach to extract the surface from the LSF. As illustrated in Fig. ld, method focuses on line segments, avoiding the distortions typically introduced by SDFs and UDFs when extracting surfaces. Additionally, we establish a relationship between the unoriented surface normal and the proposed LSF, which facilitates surface traction from LSFs. Based on SALS, we develop a neural surface representation technique and a surface reconstruction method from point clouds. Extensive experiments on diverse geometric data demonstrate the significant superiority of SALS over state-of-the-
P
d
d2
d
>
d
(a)
(b)
Vd(u) =?
d,=d2
8 x
(c)
(d)
Figure 1: Visual comparisons of various implicit geometry representations and surface extraction methods. The gray points and gray line segments denote the vertices and edges of the cubes, respec- tively. The blue lines indicate the ground truth surface, while the orange lines represent the re- constructed surfaces. Green points points mark the correctly interpolated points on the edges, whereas red points indicate interpolation errors. The pink arrows and pink lines represent ori- ented and unoriented normal vectors, respectively. (a) MC. (b) DC. (c) Gradient of distance fields. (d) Ours.
art methods.
In summary, we have made the following key contributions:
a novel, accurate, and efficient implicit geometry representation, as well as its neural implemen- tation, termed the line segment field (LSF), capable of handling shapes with arbitrary structures;
new surface extraction algorithm specifically designed for LSFs;
highly accurate efficient, and generalizable learning pipeline for surface reconstruction from 3D point clouds.
2 RELATED WORK
Implicit Geometry Representation. Implicit geometry representations use the isosurface of fune- tion or field to define a surface. Methods such as ONet (Mescheder et al., 2019), IF-Net (Chibane et al., 2020a), CONet (Peng et al., 2020), SPSR (Kazhdan & Hoppe, 2013). SAP (Peng et al., 2021), and POCO (Boulch & Marlet, 2022) employ the binary occupancy field (BOF) to represent surfaces by dividing the space into two regions-insse inside and outside the surface. This approach formulates the problem as binary classification, where the isosurface at value of 0.5 defines the surface. Com- pared to BOFs, signed distance fields (SDFs) offer a more accurate representation by incorporating distance and using signs to indicate inside and outside regions. DeepSDF (Park et al., 2019) and DeepLS (Chabra et al., 2020) optimize latent vectors using an auto-decoder to refine SDFs, while NeuralPull (Baorui et al., 2021) and OSP (Ma et al., 2022) focus on the gradients of SDFs, pulling
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points onto the surface to guide SDFs. Some traditional methods can also be integrated with neural networks; for instance, DeepIMLS (Liu et al., 2021) and DOG (Wang et al., 2022) adapt implicit moving lquares (IMLS) (Kolluri, 2008) to estimate SDFs. Recently, methods like (Wang et al., 2021; 2023; Meng et al., 2023) utilize SDFs in the volume rendering pipeline, and utilize image in- formation supervision optimize the SDFs.
BOFs and SDFs are limited to representing watertight shapes and cannot handle more general ge- ometries, such open surfaces with boundaries or non-manifold surfaces. The unsigned distance field (UDF), the absolute value of SDF, is capable of representing more general shapes. NDF (Chibane et al., 2020b) employs 3D CNNs regress UDFs from voxelized point clouds and gener- ates dense point clouds by adjusting points according to UDF gradients. Building on previous work (Baorui et al., 2021; Ma et al., 2022), CAP-UDF (Zhou al., 2022) applies a field consistency con- straint to achieve consistency-aware UDFs. To improve reconstruction accuracy and generalizabil- ity, GeoUDF (Ren et al., 2023) leverages the geometric properties of point clouds to predict UDFs, decoupling UDF from its gradients. Methods like NueralUDF(Long et al., 2023) incorporate UDFs to represent the objects the volume rendering pipeline. Beyond distance fields, other types of implicit fields have also been developed to represent surfaces (Ye et al., 2022: Lu et al., 2024).
al., 2023) and NeUDF(Liu
Surface Extraction from Implicit Geometry Representations. Marching Cubes (MC) (Lorensen & Cline, 1987) is the most widely used method for extracting surfaces from BOFs and SDFs. March- ing Tetrahedra (Doi & Koide, 1991) and its variant (Shen et al., 2021) further divide the space into tetrahedra, a simpler cell than cubes. However, these methods are not applicable to UDFs due to the lack of inside/outside information. NDF (Chibane al., 2020b) addresses this limitation by generating a much denser point cloud and then using the ball-pivoting algorithm (Bernardini et al., 1999) to create triangle meshes.
GIFS (Ye et al., 2022) employs a classifier to determine whether each edge of the cubes intersects the surface and then selects the closest match from the MC lookup table to extract triangles. con- trast, GeoUDF (Ren et al., 2023) uses UDF and its gradient to determine edge-surface intersections. MeshUDF (Guillard et al., 2022) and CAP-UDF (Zhou al., 2022) adopt a different approach by using one vertex of the cube as reference and computing the inner product of the UDF gradient with the remaining vertices. This converts UDF into an SDF within the cube based on the sign of the inner product, allowing triangle extraction through MC. However, this method lacks robustness, often mispredicting faces at edges and corner. DCUDF (Hou et al., 2023) does not directly extract the surface, it first utilizes MC extract an inflated surface, then the UDF gradients are used to refine the vertices.
Compared to these MC-based surface extraction techniques, dual contouring (DC) (Ju et al., 2002) preserves sharp surface features by solving a quadratic error function (QEF) in each cube. DC can be applied BOFs, SDFs, and UDFs alike. Recently, NDC (Chen al., 2022) improves DC by leveraging neural networks to solve QEFs, and DualMesh (Zhang et al., 2023) by introducing subsampling within each eube.
3 PROPOSED METHOD
s
$
u'v
As analyzed previously, existing implicit ge- ometry representations, such as SDF and UDF, primarily focus on the closest distance from individual points to the surface, which can introduce distortion during surface extraction. Moreover, their differentiable properties, such as the gradient of UDF, become ill-defined at certain critical positions, further reducing the accuracy of the extracted surfaces.
0
u
=0
P
(1 - s(v)
uv
o (uv)
(a)
(b)
Figure 2: Visual illustration the two attributes of the query segments: (a) o (b) s.
Unlike previous approaches, our method is focused on spatial line segments to capture the geometric relationship between the segments and the surface. Specifically, we introduce line segment field
(LSF) to represent this spatial relationship (Sec. 3.1). We then propose edge-based based dual contouring (E-DC) extract the triangle mesh from the LSF (Sec. 3.2). Finally, based on LSF and E-DC,
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propose neural shape representation (Sec. 3.3) and new pipeline for reconstructing surfaces from 3D point clouds (Sec. 3.4).
3.1 DEFINITION OF LINE SEGMENT FIELD
Unlike existing implicit representations such BOF and SDF, which concentrate individual points in space, our line segment field (LSF) emphasizes spatial line segments. Let uv E Ω be a query line segment with space and S 3D surface. The spatial relationship between uv and S can be classified into two types: intersecting and non-intersecting. If intersecting, the intersection point lies uv, whose position can be represented as linear combination of u and v.
R being the two end- and-points and f the set of all line segments in
Under such an intuition, we assign attributes to the line segment uv: a binary intersection flag o(uv) E {0, } and an intersection ratio s s(vv) E [0, 1]'. More precisely, as illustrated in Fig. 2a, when o(uv) 1, it indicates the line segment uv intersects the surface S, whereas o(uv) signifies non-intersection. The intersection ratio s s(uv) specifies the location of the intersection point, denoted as p E R2, under the case o(uv) = I; moreover, we have p = s(uv)u + (1 s(uv))v, as shown in Fig. 2b. Thus, by using the query line segments in Ω the surface S could be represented implicitly as
S= {s(uv)u + (1 s-vv)(())) v/ o(vv) 1, uv E 22.
(1)
Given that and characterize the local geometric properties of the sur- face S around the query line segment uv, we utilize the tangent plane of the surface at the intersection point p to represent the local geometry. Specifically, let E R be the unoriented normal vector at p. and Po any point on the tangent plane. worth noting that n and Po are fixed, independent of uv. Theorem demonstrates that the normal vector n is parallel to the spatial gradient of with respect to u and v, and provides a formal proof of this relationship.
S
Po
The unoriented normal vector n at the intersection point p between the ling segment uv and surface S is parallel the spatial gradient of Vus and Vss. Proof. According to Fig. 3, the intersection ratio is calculated
Theorem 1.
Figure 3: Local geometry the surface near the inter- section point.
n (v Po)
Then, its gradient for and v
n T Po) nn (v u) u))²
n, V.s n T Po) nn (v )
Obviously, they are parallel to n.
3.2 SURFACE EXTRACTION FROM LSFs VIA EDGE-BASED DUAL CONTOURING
C2
Building upon the principles of dual contouring (Ju al., 2002), propose edge-based dual contour- ing (E-DC) to extract the triangle meshes indicated by given LSF. Specifically, as shown in Fig. 4a, for a given cube C with its set of edges denoted as the normal vector n; at u;v;'s 's inter- section point can be calculated through Theorem 1. Specifically, we use the gradient of the endpoint nearest to the intersection point to determine the nor- mal vector's 's direction that point,
u
82
n
v
P
= (u,v. 1- 1-
V
C
(a)
(b)
Figure 4: Visual illustration of surface extraction from an LSF (a) QEF in each cube. (b) Triangle extraction in adjacency cubes.
n S V
if s(u,vi S 0.5
(2)
n
V 3/ V
otherwise
In practice, the inputs of and the endpoints of the line segments.
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The intersection point within the cube can be determined through optimizing the following quadratic error function (QEF): x argmin Σ
We refer readers to Appendix B for the solution to Eq. (3).
For any edge that intersects the surface, i.e., o(vv) 1, it belongs four cubes, denoted as {C.)= as shown in Fig. 4b. By solving the QEFs in these cubes, we obtain intersection points {&:]==1 within these four cubes, which can be connected to form a quadrilateral. However, since quadrilateral meshes are not widely used in practice, it is necessary to subdivide the quadrilateral into triangles to generate triangular mesh. Importantly, the intersection point p on the edge uv lies on the surface, enabling the quadrilateral to be divided into four triangles. By applying this process to edges intersecting the surface, we can extract triangle mesh from a given LSF.
3.3 LSF-BASED NEURAL IMPLICIT REPRESENTATION
In this section, we employ a neural network model the LSF of given shape, where a lightweight MLP is trained with set of ground tround-truth attributed line segments sampled in the space. After training, the MIPP can predict the attributes o and for any query line segment, producing the LSF,, where we can utilize E-DC to extract triangle mesh.
Technically, for typical query line segment uv, we concatenate its two endpoints and v, which are then fed into an MLP parameterized with 0, outputting the corresponding values for and 8. It is worth noting that the concatenation is in an orderly manner, and the output intersection ratio 8 always indicates the ratio of the distance from the intersection point to the endpoint at the second concatenated position to the total length of the line segment, while the intersection flag o is independent of the concatenation order. Such process is described
fo, s} MLP MLPo(ullv),
where || denotes the concatenation of two vectors. The surface normal at the intersection points the line segments can be determined by computing the gradient of & via backpropagation.
We optimize by minimizing the following loss function:
==
E. uVER*
()IVCE(O(TV)) o" (TV) A20"
UV
[vV I),
where BCW(:; :) represents Binary Cross Entropy, Ω is the of sampled line segments, and o" and s* are the ground-truth intersection flag and ratio, respectively. See Sec. 4.1 for the experimental comparisons and analyses.
3.4 LEARNING LSFS FROM 3D POINT CLOUDS FOR SURFACE RECONSTRUCTION
Built upon the LSF introduced in Sec. 3.1, we propose a novel learning-based pipeline for reconstructing surfaces from 3D point clouds. After training, this pipeline is capable of pre- dicting the LSF for any given 3D point cloud, enabling the subsequent application of our E- DC Sec. 3.2 to extract precise triangle mesh.
Pi
N.vv
MLP, &
uV
P1
Index
F
F
MLP,
(uv
Px
Denote by P a 3D point cloud. Let (uv) (p:). be patch of K points associated with a typical query line segment uv, which is de- rived by finding the K Knnearest neighbors of the midpoint of uv in P. Also, let Nu (uv)
0, S
K
MLP
N, (uv)
&
MaxPool
{p, K and NVvuv) Calculating the attributes of uv against the un-
K {p: v) -=
Figure 5: Flowehart our learning based surface re- construction method from 3D point clouds.
derlying surface of P is local geometry problem, which pertains only Nuv).. Based on this
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observation, we construct our surface reconstruction pipeline using the following network architec- ture, as illustrated in Fig. 5: two MLPs followed by a MaxPool operation are first utilized to extract features from these two localized patches:
(6) where 0, and 02 are the sets of network parameters. The resulting features F. and Fv are then concatenated and fed into another MLP parameterized with 0g to predict the corresponding o and st
F MaxPool (MLPea e, (Na(uv))), Fv = MaxPool (MLPee
(ww.eooooMMMMPMMMo (NV (uv)))
for 8) = MI.Po, HIPPo,(Fur (F. (Fv).
We minimize the loss function defined in Eq. (5) to train the network. See Sec. 4.1 for the experi- mental comparisons and analyses.
EXPERIMENTS
4.1 EVALUATION ON NEURAL IMPLICIT REPRESENTATION
Datasets. We randomly selected 50 shapes from the ABC dataset (Koch et al., 2019) to conduct ex- periments. Additionally. we constructed more complex dataset named non-manifold ABC dataset, in which each of the 50 shapes was produced by intersecting two randomly selected shapes for creating non-manifold structures. Besides, we utilized shapes from other commonly used datasets, including open-boundary clothes from the DeepFashion3D dataset (Zhu et al., 2020), and complex shapes from the Famous dataset (Erler et al., 2020). All shapes were rescaled to bounding cube with an edge length of 2 and centered at the origin.
Implementation Details, We employed an 8-layer MLP, with each layer comprising 512 neurons. All layers, except the final one, use the Softplus activation function (3 = 100, as recommended in (Atzmon & Lipman, 2020)). The final layer consists of 2 neurons, followed by Sigmoid acti- vation function. We sampled 10 million line segments within the space to optimize the MLP. The model was trained for 100,000 epochs with batch size of 10,000, using the ADAMW optimizer (Loshchilov, 2017) with an initial learning rate of 0.001. The learning rate was progressively ad- justed using cosine annealing (Loshchilov & Hutter, 2016), with minimum learning rate of 10 When extracting the surface, the resolution of grids used in E-DC was set to 128, keeping the same as the baseline methods. To measure the reconstructed quality quantitatively, we sample 10 points from each surface to calculate L1-CD, Normal Consistency (NC), and F-Score with the threshold of 0.005 and 0.01 with respect to the ground truth surfaces.
Methods under Comparison. We compared our method against three commonly used implicit geometry representations, i.e., BOF, SDF, and UDF, using MLPs with the same architectures as ours for modeling. For surface extraction methods from implicit representations, we benchmarked SALS against MC (Lorensen & Cline, 1987), NDC (Chen et al., 2022), MeshUDF (Guillard et al., 2022), DCUDF (Hou et al., 2023), and DualMesh (Zhang et al., 2023). Additionally, included two recently proposed methods, i.c., NGLOD (Takikawa et al., 2021) and UODF (Lu et al., 2024), maintaining the settings consistent with their original papers.
We quantitatively compared those methods on the ABC and non-manifold ABC datasets in Table 1, where it be seen that our method outperforms the other methods in terms of all metrics, especially on the non-manifold ABC dataset. The visual results are shown in Fig. 6, and obviously, the surfaces represented by our method can preserve detailed structures, including the non-manifold ones. Our method only built upon a simple MLP can represent shapes with higher accuracy, compared with the relatively complex methods, such as NGLOD and UODF, which credited to our novel LSF. BOF represents the probability that a query point is either inside or outside the surface, and it is less accurate than other methods. Surfaces extracted from the distance fields present distortion at the detailed regions since they rely on the closest distance of individual points to the surface. Therefore, under the same MLP settings, these methods exhibit low representation accuracy. UODF utilizes SPSR (Kazhdan & Hoppe, 2013) to extract the surface, decreasing its accuracy. When handling shapes with non-manifold structures, the representation accuracy of both BOF and SDF drops significantly, they are inherently unable to represent non-manifold geometry. While UDF can represent non-man mon-follfl structures, its reliance on gradients for surface extraction leads inaccuracies, particularly non-manifold regions, which in turn reduces representation
Results.

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Published as a conference paper at ICLR 2025
Table I: Quantitative comparisons of different neural implicit representations with various surface extraction methods on the ABC and non con-fooifold ABC datasets, The best results are highlighted in bold.
Methods
# Param
CD (10
NC
F1-0.005
F1-0.01
Neural BOF MC (Lorensen & Cline, 1987)
184M
6,385
0.911
0.528
0.941
Neural SDF MC. (Lorensen Cline, 1987)
L84M
5.810
0.966
0.618
0.946
Neural SDF NDC (Chen et al., 2022)
2.0MM
4.607
0.967
0.963
NGLOD (Takikawa
2021)
10.14M
6.070
0,919
ABC
Neural UDF MeshUDF (Ghillard
2022)
L84M
5.075
0.967
0.578
0,937
Neural UDF NDC (Chen
2022)
2.07M
4.944
0.955
0.584
0,942
Neural UDF DCUDF (Ilou al., 2023)
1.84M
4,593
0.971
0.625
0,962
Neural UDF DualMesh (Zhang al., 2023)
1.44
4.725
0.978
0.609
0,952
UODF (Lu al., 2024)
5.55M
4.794
0.977
0.603
0,953
SALS (Ours)
1.84M
4.493
0.980
0.638
0.965
Neural BOF MC (Lorensen & Cline, 1987)
L.4M
14.371
0.886
0.356
0.810
Non Non- con-mannfold ABC
Neural SDF MC (Lorensen Cline, 1987)
184M
13.567
0,929
0.422
Neural SDF NDC (Chen et al., 2022)
2.07M
8.642
0.835
NGLOD (Takikawa
2021)
10.14M
10.250
0.913
0.301
0.750
Neural UDF MeshUDF (Guillard et al., 2022)
184M
6,465
0.948
0.392
0.855
Neural UDF NDC (Chen al., 2022)
2.07M
14.893
0.932
0.382
0.849
Neural UDF DCUDF (Hou al., 2023)
1.4M
6.111
0.954
0.427
0,886
Neural UDF DualMesh (Zhang et al. 2023)
1.84M
6.706
0.961
0.367
0.823
UODF (Lu et al., 2024)
5.55M
6.746
0.953
0.367
0.835
SALS (Ours)
1.84M
5.949
0.963
0.437
0.889
accuracy. Additionally, Fig. 7 visualizes the results our method on other general shapes from the DeepFashion3D and Famous datasets. Obviously, the extracted surface can preserve detailed structures. We refer the readers to Appendix C for more results.
4.2 EVALUATION ON SURFACE RECONSTRUCTION FROM 3D POINT CLOODS
Datasets. We employed a significantly smaller training set compared to previous methods, compris- ing 100 shapes selected from the Thingi10K dataset (Zhou & Jacobson, 2016) to train network. For testing, we utilized the ABC and non-manifold ABC datasets, as well as more diverse shapes from the DeepFashion3D (Zhu al., 2020), Synthetic Rooms (Peng et al., 2020) and Waymo (Sun et al., 2020) datasets. Each shape contains 40,000 randomly sampled points.
Implementation Details. Each of the three MLPs in Fig. consists of layers, with 256 neurons per layer. All layers. except the final one, used the LeakyReLU activation function. The patch size for K-NN
K- 20. The network was trained for 400 epochs with batch size of where 5120 line segments were sampled for each shape at each iteration. The optimizer used was ADAMW (Loshchilov, 2017), with an initial learning rate 0.001, which was adjusted via cosine annealing (Loshchilov & Hutter, 2016) to minimum learning rate of 10-5 The grid resolution was set 128 when performing surface extraction, keeping the same as the baseline methods.
Methods under Comparison. We compared our method against SOTA surface reconstruction methods, including POCO (Boulch Marlet, 2022), GIFS (Ye al., 2022), CAP-UDF (Zhou al., 2022), and GeoUDF (Ren al., 2023), whose settings were maintained the same their original papers for fair comparisons.
Results. Table 2 lists the quantitative results of different methods. It is obvious that method outperform baseline methods, especially on the non mon-folifold ABC datasets. Achieving such high-
Table 2: Quantitative comparisons of 3D point cloud-based surface reconstruction methods on the ABC and non-manifold ABC datasets. The best results are highlighted in bold.
Method
Param
CD (10-3)
NC1
F1-0.005
F1-0.01
POCO (Boulch Marlet, 2022)
12.79M
8.855
0.952
0.558
0.886
GIFS (Ye et al., 2022)
3.68M
5,726
0,940
0.471
0.923
ABC
CAP-UDF (Zhou et al. 2022)
0.46M
5,308
0.958
0.554
0.924
GeoUDF (Ren et al., 2023)
0.77M
4.678
0.970
0.615
0.957
SALS (Ours)
1.07M
4.560
0.973
0.630
0.962
Non- manifold
POCO (Boulch Marlet, 2022)
12.79M
16.830
0.905
0.269
0.658
GIFS (Ye et al., 2022)
3.68M
7.078
0.927
0.305
0.826
ABC
CAP-UDF (Zhou et al. 2022)
0.46M
7.280
0.929
0.357
0.833
GeoUDF (Ren et al., 2023)
0.77M
6.230
0.950
0.410
0.869
SAL.S (Ours)
1.07M
6.033
0.951
0.425
0.883
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Neural SDF
MC
Neural SDF
NDC
NGLOD
Neural UDF
Mc
Neural UDF
NDC
Neural UDF +
DCUDF
Neural UDF
DualMesh
UODF
Ours
GT
Figure 6: Visual comparisons of different neural implicit representations. Q Zoom to see details. precision reconstruction with so little training data and such a simple network structure primarily attributed to our proposed implicit geometry representation and the effective local geometry mod- eling. The visual comparison shown Fig. 8. where our method can preserve detailed struc- tures. Furthermore, Figs. 9 and 10 present the results of our method applied to additional shapes and scenes from the DeepFashion3D, Synthetic Rooms, and Waymo datasets. Notably, the recon- structed surfaces successfully capture fine structural details, highlighting the strong generalizability and robustness of our approach across diverse datasets. We refer the reader to Appendix D for more results.
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Ours
GT
Figure 7: Visual results on the DeepFashion and Famous datasets. Q Zoom in to see details.
TTTTTTT 1o0ddd
(a) Input (b) POCO
(c) GIFS (d) CAP- UDF (e) GeoUDF
(f) Ours
(g) GT
Figure 8: Visual comparisons different point cloud surface reconstruction methods.
Q Zoom
details.
4.3 ABLATION STUDY
We conducted comprehensive ablation studies on the ABC dataset to show the rationality of design.
(a) w/o normal
(b) w/ normal
(c) GT
Figure 11: Extracted surfaces using E-DC without and with normal.
Spatial gradient of s. Theorem demonstrates that the normal vector the intersection point is parallel to the spatial gradient of s, which is crucial for accurate surface extraction. To validate this, we conducted experiments where, instead of directly utilizing the normal vectors at the intersection points, employed the center of the intersection points within each cube as the solution for the QEF during surface extraction from the LSF. As shown in Fig 11, removing normals from E-DC significantly reduces the ability capture fine surface details, as normals essential for describing detailed structures. On the other way, this also confirms the validity of Theorem
E-DC Resolution. We eval- uated the accuracy of the ex- tracted surfaces under varying resolutions of E-CD, with the numerical results presented in Table 3. At lower resolutions, such 64, the representation accuracy decreases significantly. How-
Table 3: Representation accuracy under various resolutions of E-DC.
Inf. Time
GPU Mem.
CD (10-5)
F1-0.005
F1-0.01
64
1.099x
T,55GB
4,731
0.963
0.606
0.952
128
6.682s
4.78GB
4.493
0.980
0.638
0.965
19.663s
11.99B
0.981
0.638
ever, increasing the resolution beyond 128, for instance to 192, does not yield substantial improve- ments in accuracy despite requiring more computational resources. We also refer reviewers to Fig. 15 of Appendix C for the visual results.
Table 4: Point cloud surface reconstruction accu- racy under different K-NN sizes.
K-NN Size of Query Line Segments. When com-
puting the LSF for query line segments from point cloud, the size of the K-NN patch is critical hy- perparameter. To evaluate its impact, we tested the accuracy of our method under different values of K.
K
CD (10
NC
F1-0.005 +
F1-0.01 1
10
6.367
0.903
0.489
0.848
20
4.560
0.973
0.630
0.962
50
14,370
0.657
0.164
0.415

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Input
GIFS
CAP-UDF
GeoUDF
Ours
Visual results of point cloud surface reconstruction methods DeepFashion3D and Synthetic Rooms datascts. Zoom in to see details.
Figure 9:
Input
(b) GeoUDF
(c) Ours
Figure 10: Visual results of point cloud surface reconstruction methods the Waymo dataset. Zoom in
see details.
As listed in Table 4, setting K either too large small leads noticeable decrease in recon- struction accuracy. We also refer reviewers to Fig. 20 of Appendix for visual results.
Point Cloud Density. We evaluated the reconstruc- tion accuracy of our method across different point cloud densities to verify its robustness. The numer- ical results, presented in Table 5, indicate slight
Table 5: Point cloud surface reconstruction accu-
racy under different numbers of input points.
CD(10-1)
F1-0.005
Tx 10
4.842
0.53
0.608
0.948
decrease accuracy when the point clouds become sparser. However, increasing the number of points does result in significant improvements in accuracy, suggesting its robustness.
4 10*
4.560
0.973
0.630
0.962
10
4.514
0.977
0.636
0.964
We also refer
reviewers to Fig. 19 in Appendix for visual results.
CONCLUSION
We have introduced SALS, an accurate and flexible implicit geometry representation. We began by analyzing the primary limitations of existing implicit representations, which are predominantly based on distance fields. contrast, SALS focuses on the spatial relationship between line segments and the surface, resulting in greater accuracy than prior methods. Leveraging SALS, proposed both a neural shape representation approach and a learning pipeline surface reconstruction from point clouds, achieving superior performance over oftte-of-the-rtt methods. The effectiveness and advantages of SALS underscore its potential to significantly advance the field of 3D geometric modeling and processing.
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REFERENCES
Matan Atzmon and Yaron Lipman. Sald: Sign agnostic learning with derivatives. arXiv preprint 1xXiv:2006.05400, 2020.
Ma Baorui, Han Zhizhong. Liu Yu-Shen, and Zwicker Matthias. Neural-pull: Learning signed distance functions from point clouds by learning to pull space onto surfaces. In International Conference on Machine Learning, volume 139, pp. 7246-7257, 2021.
Fausto Bernardini, Joshua Mittleman, Holly Rushmeier, Cláudio Silva, and Gabriel Taubin, The ball-pivoting algorithm for surface reconstruction. IEEE transactions visualization and com- puter graphics, 5(4):349-359 1999.
Alexandre Boulch and Renaud Marlet. Poco: Point convolution for surface reconstruction. Pro- ceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp. 6302 6314, June 2022.
Rohan Chabra, Jan E Lenssen, Eddy Ilg, Tanner Schmidt, Julian Straub, Steven Lovegrove, and Richard Newcombe. Deep local shapes: Learning local sdf priors for detailed 3d reconstruction. In European Conference on Computer Vision, pp. 608-625. Springer, 2020.
Zhiqin Chen, Andrea Tagliasacchi, Thomas Funkhouser, and Hao Zhang. Neural dual contouring.
ACM Transactions Graphics (TOG), 41(4):1-13, 2022.
Julian Chibane, Thiemo Alldieck, and Gerard Pons-Moll. Implicit functions in feature space for 3d shape reconstruction and completion. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6970-6981, June 2020a.
Julian Chibane, Gerard Pons-Moll, al. Neural unsigned distance fields for implicit function learn- ing. Advances in Neural Information Processing Systems, 33:21638-21652, 2020b.
Akio Doi and Akio Koide. An efficient method of triangulating equi-valued surfaces by using tetrahedral cells. IEICE TRANSACTIONS on Information and Systems, 74(1):214-224, 1991.
Philipp Erler, Paul Guerrero, Stefan Ohrhallinger, Niloy Mitra, and Michael Wimmer. Points2surf learning implicit surfaces from point clouds. In European Conference on Computer Vision, pp. 108-124. Springer, 2020.
Benoit Guillard, Federico Stella, and Pascal Fua. Meshudf: Fast and differentiable meshing of unsigned distance field networks. In European Conference on Computer Vision, pp. 576-592, 2022.
Fei Hou, Xuhui Chen, Wencheng Wang, Hong Qin, and Ying He. Robust zero level-set extraction from unsigned distance fields based on double covering. ACM Transactions on Graphics (TOG), 42(6):1-15, 2023.
Tao Ju, Frank Losasso, Scott Schaefer, and Joe Warren. Dual contouring of hermite data. In Pro- ceedings of the 29th annual conference Computer graphics and interactive techniques, pp- 339-346, 2002.
Michael Kazhdan and Hugues Hoppe. Screened poisson surface reconstruction. ACM Transactions Graphics, 32(3):1-13, 2013.
Sebastian Koch, Albert Matveev, Zhongshi Jiang, Francis Williams, Alexey Artemov, Evgeny Bur- naev, Marc Alexa, Denis Zorin, and Daniele Panozzo. Abc: big cad model dataset for geomet- ric deep learning. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pp. 9601-9611, 2019.
Ravikrishna Kolluri. Provably good moving least squares. ACM Transactions on Algorithms, 4(2): 1-25, 2008
Shi-Lin Liu, Hao- Hao-Xiang Guo, Hao Pan, Peng- Shuai Wang. Xin Tong, and Yang Liu. Deep implicit moving least-squares functions for 3d reconstruction. In Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition, pp. 1788-1797, June 2021.

---

Published as a conference paper at ICLR 2025
Yu-Tao Liu, Li Wang. Jie Yang, Weikai Chen, Xiaoxu Meng, Bo Yang, and Lin Gao. Neudf: Leaning neural unsigned distance fields with volume rendering. In Proceedings the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 237-247, 2023.
Xiaoxiao Long, Cheng Lin, Lingjie Liu, Yuan Liu, Peng Wang. Christian Theobalt, Taku Komura, and Wenping Wang. Neuraludf: Learning unsigned distance fields for multi-view reconstruction of surfaces with arbitrary topologies. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pp. 20834-20843, 2023.
William Lorensen and Harvey E Cline. Marching cubes: A high resolution 3d surface construction algorithm. ACM siggraph computer graphics, 21(4):163-169 1987.
I Loshchilov. Decoupled weight decay regularization, arXiv preprint
2017.
Iyy Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint 1608 03988, 2016.
Yujie Lu, Long Wan, Nayu Ding, Yulong Wang, Shuhan Shen, Shen Cai, and Lin Gao. Unsigned or- thogonal distance fields: An accurate neural implicit representation for diverse 3d shapes. In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 20551- 20560, 2024.
Baorui Ma, Yu-Shen Liu, and Zhizhong Han. Reconstructing surfaces for sparse point clouds with on-surface priors. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6315-6325, June 2022.
Xiaoxu Meng. Weikai Chen, and Bo Yang. Neat: Learning neural implicit surfaces with arbitrary topologies from multi-view images. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 248-258. 2023.
Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Oc- cupancy networks: Learning 3d reconstruction in function space. In Proceedings the the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 4460-4470, June 2019.
Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 165-174, June 2019.
Songyou Peng, Michael Niemeyer, Lars Mescheder, Mare Pollefeys, and Andreas Geiger. Convolu- tional occupancy networks. In European Conference on Computer Vision, pp. 523-540. Springer, 2020.
Songyou Peng, Chiyu Jiang, Yiyi Liao, Michael Niemeyer, Mare Pollefeys, and Andreas Geiger. Shape as points: A differentiable poisson solver. Advances in Neural Information Processing Systems, 34:13032-13044 2021.
Siyu Ren, Junhui Hou, Xiaodong Chen, Ying He, and Wenping Wang. Geoudf: Surface reconstruc- tion from 3d point clouds via geometry-guide distance representation. In Proceedings of the IEEE/CVF International Conference Computer Vision, pp. 14214-14224, 2023.
Tianchang Shen, Jun Gao, Kangxue Yin, Ming- Yu Liu, and Sanja Fidler. Deep marching tetrahedra: hybrid representation for high-resolution shape synthesis. Advances in Neural Information Processing Systems, 34:6087-6101, 2021.
Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, et al. Scalability in perception for au- tonomous driving: Wayme open dataset. In Proceedings of the IEEE/CVF conference com- puter vision and pattern recognition, pp. 2446-2454, 2020.
Towaki Takikawa, Joey Litalien, Kangxue Yin, Karsten Kreis, Charles Loop, Derek Nowrouzezahrai, Alec Jacobson, Morgan McGuire, and Sanja Fidler. Neural geometrie level of detail: Real-tima rendering with implicit 3d shapes. In Proceedings of the IEEÉ/CVF Conference Computer Vision and Pattern Recognition, pp. 1 11358-1 1138-11367, 2021.
12

---

Published as a conference paper at ICLR 2025
Peng Wang, Lingjie Liu, Yuan Liu, Christian Theobalt, Taku Komura, and Wenping Wang. Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. Advances in Neural Information Processing Systems, 34:27171-2771 2021.
Peng-Shuai Wang, Yang Liu, and Xin Tong. Dual octree graph networks for learning adaptive volumetric shape representations. ACM Transactions on Graphics, 41(4):1-15, 2022.
Yiming Wang, Qin Han, Mare Habermann, Kostas Daniilidis, Christian Theobalt, and Lingjie Liu. Neus2: Fast learning of neural implicit surfaces for multi-view reconstruction. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 3295-3306, 2023.
Jianglong Ye, Yuntao Chen, Naiyan Wang, and Xiaolong Wang. Gifs: Neural implicit function for general shape representation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12829-1222 12839, June 2022.
Congyi Zhang, Guying Lin, Lei Yang. Xin Li, Taku Komura, Scott Schaefer, John Keyser, and Wenping Wang. Surface extraction from neural unsigned distance fields. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 22531-225444 2023.
Junsheng Zhou, Baorui Ma, Yu-Shen Liu, Yi Fang, and Zhizhong Han. Learning consistency- aware unsigned distance functions progressively from raw point clouds. arXiv preprint arXiv:2210.02757, pp. I-xxx, 2022.
Qingnan Zhou and Alec Jacobson. Thingi10k: dataset of 10,000 3d-printing models. arXiv preprint arXiv:
2016.
Heming Zhu, Yu Cao, Hang Jin, Weikai Chen, Dong Du, Zhangye Wang, Shuguang Cui, and Xi- aoguang Han. Deep fashion3d: A dataset and benchmark for 3d garment reconstruction from single images. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, Au- gus! 23-28, 2020, Proceedings, Part I 16, pp. 512-530. Springer, 2020.
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Appendix
A DIFFERENCE FROM PREVIOUS METHODS
Here, we demonstrate that our approach is fundamentally distinct from prior implicit representation methods, such GIFS (Ye al., 2022) and UODF (Lu al., 2024), well previous surface extraction techniques, including NDC (Chen et al., 2022) and DualMesh (Zhang et al., 2023).
GIFS (Ye et al., 2022) employs classifier to predict whether line segment intersects the surface. an intersection occurs, the intersection point on the line segment interpolated using the UDF values of its endpoints. Then, it modifies the Marching Cubes (MC) algorithm extract surfaces based on the intersection status of each cube's 's edges. In contrast, our SALS predicts an intersection ratio for a given line segment, avoiding the interpolation errors introduced by UDFs, as discussed in Sec. 1. Furthermore, we propose E-DC to extract surfaces directly from our LSF. Experimental results demonstrate the significant advantage of our method in terms of both reconstruction accuracy and efficacy.
For query point, UODF (Lu al., 2024) only predicts its distance to the surface along three fixed directions, making it non-differentiable. Additionally, UODF does not offer surface extraction method but instead generates dense point cloud, from which masked Poisson method is used to reconstruct the mesh. In contrast, approach samples line segments across the entire space, de- velops their differentiable properties, and utilizes the custom- custom-designed E-DC for surface extraction.
Our proposed E-DC is significantly different from previous DC-based surface extraction methods, NDC (Chen et al., 2022) and DualMesh (Zhang et al., 2023). Both NDC and DualMesh focus only surface extraction from implicit representations, like SDFs USDFs. NDC utilizes trained network to predict the intersection point in each cube, and DualMesh adapts the traditional DC for neural UDFs. Differently, our proposed E-DC is an optimization-basee surface extraction method, specifically designed to extract surfaces from LSF.
B SOLUTION OF QEF
QEF can be solved using least squares method,
(ATA)-AAT
(8)
where matrix A and B are defined as
A
- -0
(9)
B
O.p n
Considering that A A A may be singular, directly calculating its inverse may get wrong results. We first perform eigenvalue decomposition on the matrix,
A
(10)
where is an orthogonal matrix and Σ is a diagonal matrix, Σ = Diag(o1, 02, os), with 01 2 02 2 03 representing the eigenvalues. Here define the inverse matrix of Σ
(n)
Σ-
0
Inv(az)
(11)
Inv(as))
0
where Inv is defined as
[1/o, if =E
Inv(o)
(12)
0,
else
In experiment, we 0.1.
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C NEURAL IMPLICIT REPRESENTATION
Network Details. The last layer of the network uses a sigmoid function to constrain the output to the range [0,1]. For the intersection flag. we apply a threshold binarize the output. If the threshold is set too low, some intersecting line segments may be incorrectly classified as non-intersecting, resulting in holes when extracting the surface using E-DC. The number of holes can reduced by lowering the threshold for the intersection flag. In our experiments, the threshold set to 0.1. We have included visual results of the extracted surfaces under different thresholds in Fig. 12.
(a) 0.8
(b) 0.5
(c) 0.2
(d) 0.1
(c) 0.05
(f) GT
Figure 12: Extracted surfaces under different thresholds for intersection flag.
Error Map. Fig. 13 presents the error maps of the reconstructed models shown in Fig. 6. The results demonstrate that the reconstruction errors of models produced by our method are significantly smaller than those of the baseline methods, particularly in regions with sharp features.
Correctness of Theorem 1. To verify the correctness of Theorem 1, utilize two different kinds of normal vectors in E-DC, random and zeros, and the visual comparison and error maps are shown in Fig. 14. It can be seen that using the gradient of the direction of normal vectors could achieve significant better reconstruction accuracy, demonstrating the correctness of Theorem 1.
Evaluation Sharp Edges. Following previous methods (Chen al., 2022), utilize Edge Chamfer Distance (ECD) and Edge F-score (EFI) to evaluate the preservation of sharp edges. The results on the ABC dataset are shown in Table 6. Clearly, under the evaluation metrics ECD and EF1, our SALS significantly outperforms the baseline methods.
Table 6: Numerical comparison of different methods under ECD and EFI metrics.
Method
ECD (10-?) 4
EF1 1
Neural BOF + MC
5,231
0.182
Neural SDF + MC
7.238
0.167
Neural SDF NDC
2.501
0.526
NGLOD
28.249
0.044
Neural UDF + MeshUDF
6,651
0.169
Neural UDF + NDC
2.632
0,476
Neural UDF + DCUDF
2,828
0,058
Neural UDF+ + DualMesh
2.870
0.609
UODF
55,823
0.034
SALS (Ours)
1.970
0.601
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0.008
Neural SDF
MC
Neural SDF
NDC
NGLOD
Neural UDF
MeshUDF
Neural UDF
+
NDC
Neural UDF DCUDE
Neural UDF
DualMesh
UODF
Ours
Figure 13: Visual comparisons of different neural implicit representations. Q Zoom in to see details.
Different E-DC Resolutions. Fig. 15 demonstrates the visual results of different E-DC resolutions on the selected shapes from the Famous dataset. When the resolution is set to 64, the extracted surfaces do not show significant distortion.
Positional Encoding. Positional Encoding (PE) is widely adopted technique in implicit repre- sentations. However, incorporating PE into our SALS framework leads to reduction in surface reconstruction quality. As illustrated in Fig. 16, the of PE introduces floating artifacts around the extracted surface, diminishing the overall representation accuracy.
Efficiency Analysis. In the task of neural implicit representation (Sec. 3.3), the time cost consists of two parts, training/overfitiigg time and surface extraction time. We have recorded the runtime for these parts, shown in Table 7. It is worth noting that the resolution of Marching Cubes, Dual
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(a) Random
(b) Zeros
0.004
Gradient of
(d) GT
Figure 14: Extracted surfaces using E-DC with different estimated normal vectors. Q Zoom in to see details.
64
128
192
GT
Figure 15: Visual results of different E-DC resolutions the selected shapes from the Famous dataset. Q Zoom see details.
Contouring, and our E-DC is 128. During training, method takes slightly longer compared to neural BOF/SDF/UDE because it processes a 6D input vector and outputs a 2D vector, unlike the latter approaches which take 3D input vector and produce scalar output. NGLOD and UODF
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GT
w/o PD
DE=6
PC=10
Figure 16: Visual comparison of shape representation with and without PE in the MLP. Q Zoom details.
employ complex architectures, resulting in longer training times. In terms of surface extraction time, method is slower than MC, MeshUDF, and DualMesh but significantly faster than DCUDF and NDC.
Table 7: Running time of different representation methods.
Method
Training Time (min)
Inference Time (s)
Neural BOF + MC
10.02
0.57
Neural SDF MC
0.60
10.34
Neural SDF NDC
7,41
NGLOD
30.14
1.02
Neural UDF + MeshUDF
1.57
Neural UDF + NDC
6.983
10.56
Neural UDF + DCUDF
68.81
Neural UDF + Dual Mesh
1.69
UODF
185.68
10.64
SALS (Ours)
11.56
2.96
Runtime of E-DC. We measured the runtime for calculating intersection points in each cube and performing triangle extraction, shown in Table 8. The results demonstrate that our E-DC is faster than NDC terms of intersection point calculating in each cube and triangle extraction. This efficiency is due to method not relying neural network to estimate the intersection points within each cube.
Table 8: Comparison of runtime between NDC and our E-DC.,
Method NDC
Calculate Intersection Points (s)
Triangle Extraction (s)
Total Time (s)
0.151
0.045
0.196
E-DC (Ours)
0.1
0.005
0.134
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D SURFACE RECONSTRUCTION FROM POINT CLOUDS
Error Maps. Fig. 17 shows the error maps of the reconstructed models shown in Fig. 8, demon- strating that the reconstruction errors the models produced by our method are significantly smaller than those of the baseline methods, especially in regions with sharp features.
0.008
(a) Input
(b) POCO
(c) GIFS
(d) CAP-UDF
(e) GeoUDF
(f) Ours
Figure 17: Visual comparisons of different point cloud surface reconstruction methods. Zoom
details.
Additional Visual Results. Fig. 18 shows additional visual results on the ABC dataset.
Input
Ours
GT
Figure 18: Additional visual results on the ABC dataset. Q Zoom in to see details.
Evaluation on Sharp Edges. We utilize ECD and EFI to evaluate the preservation of sharp edges in the reconstructed surfaces from point clouds. The results the ABC dataset are shown in Table
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Table Numerical comparison different point cloud surface reconstruction methods under ECD EFI metrics.
Method
ECD (le-2)
EFI
POCO
14.661
0.069
GIFS
7.669
0.105
CAP-UDF
11.022
0.140
GeoUDF
14.818
0.067
Ours
3.480
0.432
9. Clearly, under the evaluation metrics ECD and EFI, our SALS significantly outperforms the baseline methods.
Different Point Cloud Densities. Fig. 19 illustrates the reconstructed surfaces under varying input point cloud densities. Notably, reducing the point cloud density does not lead significant drop in reconstruction accuracy, underscoring the robustness of our method to variations in input point cloud density.
TITTTTT
10000
40000
100000
GT
Figure 19: Visual results of different input point cloud densities. Q Zoom to see details. Different K-NN Patch Sizes. Fig. 20 shows the reconstructed surfaces under different K-NN patch sizes. We also tested different K-NN patch sizes on point clouds with different densities, numerical results are shown in Table 10. It can be seen that the variation trend of K-NN patch size is similar across point clouds with different densitics and datasets, i.c., too large or small K-NV patch size lead to decrease in reconstruction accuracy.
LILL
Input
10
20
50
GT
Figure 20: Visual comparison of reconstructed surfaces under different K-NN patch sizes. Zoom in to see details.
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Table 10: Surface reconstruction of point clouds varying densities under different KNNN sizes.
Dataset
Points
K
CD (10 (10-3) J
NC
F1-0.005 t
F1-0.01
10
9.852
0.840
0.367
0.685
10000
20
4.842
0.953
0.608
0.948
50
11,504
0.609
0.163
0.524
10
6.367
0.903
0.489
0.848
ABC
40000
20
4.560
0.973
0.630
0.962
50
14.370
0.657
0.164
0.415
10
5.547
0.930
0.541
0.902
100000
20
4.514
0.977
0.636
0.964
50
29.224
0.643
0.068
0.170
10
13.956
0.754
0.188
0.512
10000
20
6.992
0.857
0.441
0.818
50
13.466
0.576
0.088
0.399
10
8.502
0.845
0.291
0.724
non mon-manifold ABC
40000
20
6.033
0.951
0.425
0.883
50
14.565
0.650
0.144
0.456
10
8.494
0.845
0.291
0.725
100000
20
4.733
0.956
0.613
0.949
50
14.562
0.651
0.144
0.456
Efficiency Analysis. In the task of surface reconstruction from point clouds (Sec. 3.4), the inference times different methods are shown Table 11. Our method comparable to POCO and GcoUDF but faster than GIFS. CAP-UDF is an optimization-bann method, which requires more time.
Table 11: Running time of different point cloud surface reconstruction methods.
Method Inference Time (s)
POCO
GIFS
CAP-UDF 906.34
GcoUDF 15.269
SALS (Ours)
13.66
45.57
14.38
LIMITATION
Unlike previous implicit representations that primarily focus on individual points in 3D space, our proposed SALS targets line segments in 3D space, which introduces greater complexity. Conse- quently, when the number of sampled line segments is limited, the network struggles to accurately model the LSF, leading distortions in the reconstructed surfaces, shown in Table 12.
Table 12: Numerical results under different numbers samples.
Samples
CD (10-3) 4
NC
F1-0.005 t
F1-0.01
100K
19.200
0.778
0.304
0.565
500K
8.086
0.923
0.542
0.862
IM
5.242
0.962
0.609
0,937
5M
4.623
0.976
0.630
0.959
10M
4.493
0.980
0.638
0.965
20M
4.691
0.975
0.629
0.956
40M
4.534
0.979
0.635
0.963
21


Reviews:
[{"review_id": "v4yh4iXxsi", "reviewer": "Reviewer_hFqZ", "summary": "The paper presents a learnable surface extraction method based on edge-based dual contouring. Similar to Neural Dual Contouring (NDC), it uses a network to predict an intersection or crossing flag per edge. However, unlike NDC, it optimizes vertex locations within each cell using a Quadratic Error Function (QEF) rather than predicting them directly through the network. This optimization requires both the position of the intersection along each edge and its normal direction. Consequently, in addition to predicting the intersection flag, the network is also trained to output a scalar value representing the intersection position on the edge, using a concatenation of the corresponding cell vertices as input. Instead of the normal direction the paper uses the gradient of this intersection scalar  which is shown to  align with the the normal direction.\n\nExtensive experiments validate the method’s effectiveness, leveraging the ABC and non-manifold ABC datasets to evaluate implicit representation in addition to DeepFashion3D, Synthetic Rooms, and Waymo datasets to illustrate generalization of  models trained with a subset of Thingi10k.", "questions": "- How good is the trade-of between performance and inference time of the proposed method compared to other methods?\n- How the K-NN value is chosen for each dataset and point cloud density, which would help to understand its impact on performance?\n- To solve the QEF optimization, the gradient the network w.r.t e one of the edge vertices should be used. Which one do you use? Also since they have different orientations how does this choice impact the stability of the optimization and the overall performance?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-written and easy to follow.\n- The focus on edges (segments) rather than points for predicting intersection positions, combined with the novel approach of linking the gradient with the surface normal, effectively leverages QEF optimization for vertex positioning.\n- **Evaluation and performance:** The evaluations are thorough, showing clear improvements over state-of-the-art methods.", "weaknesses": "- **Motivation and comparison with Neural Dual Contouring (NDC):** The paper could better clarify its approach relative to NDC, particularly why optimizing vertex locations with QEF outperforms direct network predictions.\n- **Training and inference times:** A table comparing performance, training time, and inference time would provide clarity. The method appears slower in inference than others; for instance, NDC achieves sub-second inference at a 128 resolution. Then, the question is  how good is  the trade-of between performance and inference time of the proposed method ?\n- **Dependence on K-NN size for point cloud surface reconstruction:** The paper lacks details on how the K-NN value is chosen for each dataset and point cloud density, which would help to understand its impact on performance.", "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": "YajxeTbTJc", "reviewer": "Reviewer_1tcg", "summary": "The paper proposes Shape as Line Segments (SALS), an accurate and efficient implicit geometry representation based on attributed line segments, which can handle arbitrary structures. The authors then design a novel learning-based pipeline for reconstructing surfaces from 3D point clouds. The experiments show the method produces better results than the state-of-the-art methods.", "questions": "- I suggest add normal maps, which may highlight the differences more clearly in certain results, such as in Fig. 11.\n- Some related works need to be added. [1-3].\n\n- In the network design, I’m interested in exploring the results of replacing max pooling with mean pooling, as it may provide more information of neighbors in feature aggregation.\n\n- Discuss the limitations.\n\n[1] Fast Learning of Neural Implicit Surfaces for Multi-view Reconstruction\n[2] NeUDF: Learning Neural Unsigned Distance Fields with Volume Rendering\n[3] NeAT: Learning neural implicit surfaces with arbitrary topologies from multi-view images", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper's contributions are clearly defined, with comprehensive experimental validation on several benchmark datasets.\n\n- Paper writing is clear and it’s easy to read.", "weaknesses": "- Some results exhibit only minor differences, as seen in Figure 8, where the outcomes of Ours and GeoUDF appear quite similar. Presenting additional results, such as error maps or normal maps, may provide clearer visual distinctions.\n\n- Lack of analysis of limitations. It would be interesting to evaluate how the proposed method performs on thin structures, such as leaves, or particularly complex geometric structures, like a basket, to better understand its applicability to different types of shapes.", "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": "0NofdDSvY6", "reviewer": "Reviewer_SSdJ", "summary": "This paper introduces a novel method for sharper surface reconstruction. It proposes to model the implicit surfaces as a field of line segments, in comparison to the previous fields for points.", "questions": "- For the applications to Waymo data, are the data also rescaled to a unit cube?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- A novel implicit representation of surfaces based on line segments is proposed. \n- For the novel line segments field, surfaces are extracted with edge-based dual contouring.\n- The LSF-method is extend to surface reconstruction from point clouds. \n- Experimental demonstrations are provided for manifold and non-manifold structures.", "weaknesses": "- Quantitative comparison are only provided for a limited amount of shapes in ABC dataset. \n- The improvement in surface quality appears incremental. The line segment field uses query grids of $128^3$. What is the counterpart resolution of point queries for the compared methods? \n- Given the method’s focus on sharper surface reconstruction, it is recommended that the authors also report metrics evaluated specifically around surface edges, similar to those used in ``Neural Dual Contouring (Chen et al., SIGGRAPH 2022)''.", "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": "9EaZnDHB3F", "reviewer": "Reviewer_eA2y", "summary": "The paper presents a new approach for implicitly encoding 3D surfaces. The idea is to encode how an arbitrary line segment intersects a surface, represented by a binary variable showing intersection or not, and a scalar within [0,1] showing the ratio of the intersection point along the line segment. The motivation for using such a line segment representation is that to extract surfaces from implicit fields, one usually has to find such intersection points for regular grid edges, as done in marching cubes and dual contouring. The paper does analysis of this line segmentation representation, presents a dual contouring adaptation for this representation, develops a learning based network design for generating such a representation, and conducts experiments for shape encoding and shape reconstruction from point clouds, where the method shows improved accuracy than alternative representations like SDF and UDF. \n\nThe idea makes sense intuitively by directly catering for the surface extraction step when learning the implicit rep, and is novel as far as I know. Given the novelty of the approach, some questions can be answered to better understand the approach. See the questions below.", "questions": "See weakness section.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The idea of representing shapes as intersections with respect to arbitrary edges is pursued and turned into a concrete design, and shows improved accuracy than alternative distance field representations due to its direct compliance with surface extraction algorithms like dual contouring.\n\nThe paper is well written and easy to read overall.\n\nThe paper shows the advantage of the new representation in two different tasks, i.e. shape encoding and shape reconstruction from point clouds. The dataset used is also challenging and diverse, including nonmanifold CAD shapes, scenes and open boundary shapes.", "weaknesses": "There remain some key technical questions that can be explored for this novel representation.\n\n- How to handle cases where a line segment should intersect the surface multiple times? Such cases should be common when the line segment is larger than the local feature size.\n\n- I am not sure the proof for Theorem 1 is correct. Why are p and n not functions of u/v? What assumptions have to be made about the surface, to ensure this invariance to u/v?\n\n- Eq(2), should there be a square of the dot product? Otherwise the minimization is not bounded from below.\n\n- Sec.3.3, it's said that surface normal is determined as the gradient of s. But the gradient is taken with respect to which variable? u or v? What if they give different gradients?\n\n- Related to the above question, the network producing the line-segment field should have certain invariance and equi-variance. That is, with respect to the switch of u/v, its output o (binary intersection) should be invariant, and its output s (ratio) should be equivariant. But the network design in its current form seems not preserve such properties.\n\nThe results frequently show holes that are undesirable. Such holes would not appear for SDF based methods. What can be done to fix such issues?\n\nThe ablation about normal vectors (Sec.4.3) is quite obscure, with only one visual example and minor differences. Can quantitative differences be observed with or without normal vector for dual contouring?", "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)