Aligning Gradient and Hessian for Neural Signed Distance Function
Authors: Ruian Wang
Zixiong Wang
Yunxiao Zhang
Shuangmin Chen
Shiqing Xin
Changhe Tu
Wenping Wang
Venue: NeurIPS 2023 poster (2023)
Keywords: implicit neural representation
signed distance function
shape operator

Abstract: The Signed Distance Function (SDF), as an implicit surface representation, provides a crucial method for reconstructing a watertight surface from unorganized point clouds. The SDF has a fundamental relationship with the principles of surface vector calculus. Given a smooth surface, there exists a thin-shell space in which the SDF is differentiable everywhere such that the gradient of the SDF is an eigenvector of its Hessian matrix, with a corresponding eigenvalue of zero. In this paper, we introduce a method to directly learn the SDF from point clouds in the absence of normals. Our motivation is grounded in a fundamental observation: aligning the gradient and the Hessian of the SDF provides a more efficient mechanism to govern gradient directions. This, in turn, ensures that gradient changes more accurately reflect the true underlying variations in shape. Extensive experimental results demonstrate its ability to accurately recover the underlying shape while effectively suppressing the presence of ghost geometry.

Aligning Gradient and Hessian for Neural Signed Distance Function
Ruian Wang" Shandong University wra. time@gmail. com
Zixiong Wang" Nankai University
Yunxiao Zhang Shandong University
zixi waw.goooo.. com
zhangyunxiaox0@maiil com
Shuangmin Chen Qingdao University of Science and Technology csmqq@163.c com
Shiqing Xin ShanDong University xinshiging@adu. edu. en
Changhe Tu
Shandong University chtu@sdu edu. cn
Wenping Wang
TEXAS A&M UNIVERSITY wenpingecs hku .hk
Abstract
The Signed Distance Function (SDF), as implicit surface representation, provides a crucial method for reconstructing a watertight surface from unorganized point clouds. The SDF has fundamental relationship with the principles of surface vector calculus. Given a smooth surface, there exists a thin-shell space in which the SDF is differentiable everywhere such that the gradient of the SDF is an eigenvector of its Hessian matrix, with a corresponding eigenvalue of zero. In this paper, we introduce a method to directly learn the SDF from point clouds in the absence of normals. Our motivation is grounded in a fundamental observation: aligning the gradient and the Hessian of the SDF provides more efficient mechanism to govern gradient directions. This, in turn, ensures that gradient changes more accurately reflect the true underlying variations in shape. Extensive experimental results demonstrate its ability to accurately recover the underlying shape while effectively suppressing the presence of ghost geometry.
Introduction
In recent years, the neural signed distance function (SDF) has demonstrated its capability to represent high-fidelity geometry [37 16 53 19]. Existing approaches primarily employ neural networks to map coordinates to their corresponding signed distance values. Depending on whether or not supervision is used, these approaches be classified into categories: learning-based methods methods, where the former fits data samples to their corresponding ground-
and optimization- truth implicit representations
22 21 and the latter directly infer the underlying SDF 48 or multi-image
16
g
45 29 35 Despite significant advancements in SDF-based surface reconstruction, both types of methods have their drawbacks. The supervised reconstruction methods may not generalize well 42 on shapes or point distributions that are not present the training data. The optimizing- based methods, on the other hand, struggle resolve the ambiguity of the input point cloud, particularly when normal information is absent. In this paper, our
from point clouds 19 5
"These authors contributed equally. ts. Chen the corresponding author.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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focus is primarily on optimization- based reconstruction techniques for point clouds without normal information.
The majority of optimization- approaches utilize first-order constraints to regulate the variation is commonly employed to ensure that the
of the SDF. For example, the Eikonal term 19 gradients are unit vectors. However, as pointed out in [6, the Eikonal term is weak in its ability to regulate the direction of the gradients. As a result, it is challenging to prevent the emergence of ghost geometry (the level sets of the SDF are highly disordered) and unnecessary shape variations even when the Eikonal term is enforced. To address these issues, various second-order smoothness energy formulations [34, [ are proposed to steer the direction of the gradients to change toward a desirable configuration. However, it entails considerable difficulty to regulate the extent to which the smoothness term is enforced. A commonly seen artifact is that the resulting surface may be excessively smooth.
40
53
In this research, we re-examine the problem of surface reconstruction based on surface vector calculus, and introduce novel loss to facilitate the inference of faithful SDF directly from raw point clouds without oriented normals. An interesting observation is that within the narrow thin-shell space of the underlying surface, where the real SDF differentiable everywhere in the narrow thin-shell space, the gradient of the SDF is an eigenvector of its Hessian matrix and the corresponding eigenvalue is zero. Specially, when a point is situated on surface, the normal vector transforms into an eigenvector of the Hessian of the SDF, with the corresponding eigenvalue being zero. In essence, the gradient and Hessian of the SDF must be aligned within a thin-shell region surrounding the underlying surface. This alignment allows for more effective control over the direction of the gradients. Building on the key observation, we develop a new loss function that promotes the alignment of the gradient and Hessian of the SDE, rather than relying on smoothness energy. We have conducted comprehensive evaluation of our proposed approach on a variety of benchmarks and compared it to recently proposed reconstruction methods. Extensive experimental results show that our approach outperforms the state-of-the-ard techniques, whether overfitting a single shape or learning a shape space. It can not only accurately recover the underlying shape but also suppress the occurrence of ghost geometry, which verifies the effectiveness of the gradient- Hessiin alignment.
2 Related Work
2.1 Traditional Methods
Traditional reconstruction methods can be classified into two categories: explicit and implicit, Explicit methods focus on establishing direct connections between input points, while implicit methods aim to fit an implicit field that conforms to the given points and normals. Roughly speaking, popular explicit reconstruction techniques utilize computational geometry methods, such as Delaunay triangulation or Voronoi diagrams, to infer connections between points. These methods can produce well-tessellated
triangle mesh surfaces 26 [4 47. However, they may struggle to ensure manifoldness, particularly when the input point cloud contains defects such as irregular point distribution, noise, or missing parts. In contrast, implicit reconstruction techniques [25] 23 39 are capable of generating manifold and watertight surfaces. However, the majority of these methods necessitate that the input point cloud is equipped with normal information.
In the event that the provided point cloud is devoid of normals, it is necessary to either estimate the normals and orientations prior [52) to surface reconstruction or devise a novel reconstruction framework that can operate in the absence of normal information 120 227. In essence, all of them have to estimate normals either prior to during the process of surface reconstruction. As result, these approaches remain heavily reliant on the quality of the estimated normals. In cases where the provided point cloud is of poor quality, it becomes challenging for these methods to overcome the inherent ambiguity introduced by the absence of normal information.
2.2 Supervision- Learning Methods
Learning-bassed methodologies have demonstrated a superior capacity for reconstruction, particularly in the context of neural implicit function [51]. They optimize the network to implicitly encode a signed distance field or occupancy field with the supervision of ground truth. Preceding methodologies
primarily employ object-level priors 377 34 [8 to encode shapes, which are referred to as global

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priors. However, the utilization of global priors restricts the generation ability as the network is unable to deal with shapes that are absent from the training set. Subsequently, numerous methodologies
concentrate on local data priors [2 16 g 21 to enhance generalization capacity by utilizing small receptive fields. The approaches for acquiring local data priors encompass regular grid [22], KNN [17 16 8], and octree [46) 44] and so on. Although local data priors enhance the generation ability of learning-bast based methodologies, the priors derived from limited data may result in excessively smoothed surfaces. At the same time, the reconstruction performance may further deteriorate if the distribution of the test point cloud significantly deviates from the training data 42).
2.3 Optimization-basee Learning Methods
To enhance the generalization ability, the direct fitting of 3D representations from raw point clouds without supervision has been extensively investigated recent years, enabling end-to-end prediction of the target surface. The majority of optimization-basel reconstructions necessitate the fitting of individual shapes with specific network parameters, where the parameters are acquired by imposing additional constraints. SAL/SALD [1] 2 employs unsigned distances to facilitate sign-agnostic learning. IGR [99] and Neural-Pull [5] primarily utilize Eikonal terms to constrain the field be an SDE Additionally, several methodologies 3 55 have been modified from Neural-Pull in pursuit of enhanced quality. DiGS (6] incorporates Laplacian energy into neural implicit representation learning for unoriented point clouds. In summary, the preponderance of existing methodologies constrain the norm of gradients or minimize smoothness energy to attain equilibrium between geometric details and overall simplicity, but this inevitably results in ghost geometry or over-smoothed reconstruction outcomes. In this paper, our emphasis is on the alignment of the gradients and the Hessian of the SDF to more effectively regulate the direction of the gradients.
3 Method
3.1
Neural signed distance function
Given unoriented point cloud P restricted in the range Ω: [- x [-1,1] x [-1, our task is to find a neural signed distance function (SDF) fo R3 + R, such that fo predicts a signed distance value for an arbitrary query point q E Ω where the neural function fe parameterized by 0 and can be assumed to be Co-coniinuous everywhere in the domain. We use St to denote the level-set surface the value of i.e.
S = fq E R (1
(1)
The underlying surface can be naturally obtained by extracting the zero level-set surface So, through the use of contouring algorithms, such as Marching cubes [30] The task of this paper is to fit the high-fidelity implicit representation without supervision.
3.2 Surface vector calculus related to SDF
In order to fit the SDF, there are typically three types of boundary conditions for constraining fo: (1) Dirichlet condition fp) = 0 that requires each point p
to be situated on Sp as far as possible, (2) Eikonal condition fo that enforces fo be a distance field with unit gra- dients, and (3) Neumann condition V. = / that aims to align the gradients with the normal field.N: Existing neural implicit representations incorporate the aforementioned boundary con- ditions constraints, either explicitly [99 40 or implicitly 5 [. However, it is important to note that the Neumann condition cannot be enforced due to the absence of oriented normals.
1
Figure 1: 2D visual for differential property of SDF for circle. From left to right: SDF with gradient, mini- mum (0) and maximum cigenvalues of the Hessian ma- trix with their corresponding cigenvector.
According to the surface vector calculus (32). the SDF has fundamental relationship with the differential properties of a smooth surface. Let J Jv denote the Jacobian matrix of the gradient V fe.
3

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or equivalently the Hessian matrix Hi, of fe. Since this matrix is symmetric and diagonalizable, we can find the eigenvectors and eigenvalues of H,, for any point where fo is differentiable (a narrow thin-shell space surrounding the base surface). By differentiating both sides of the identity [olle = 1, we get:
H ()) 0,
(2)
which implies that (q) is exactly an eigenvector of the Hessian f (q), with a corresponding eigenvalue To more detailed, when q lies on the surface, the gradient represents the normal vector at q. Consequently, we have
H (a.NV -.
36), with the eigenvalues respectively being the opposite of the corresponding principal curvatures. If q is off the surface but in the differentiable region, the three eigenvectors reflect the differential properties of the closest surface point of q. i.e., the projection of q onto the surface. In summary, the eigenvectors of the Hessian can disclose the fundamental properties of the underlying surface.
Additionally, the other two eigenvectors H (q) define the principal directions q
A similar property holds in the 2D setting. In Fig. I we provide a 2D circle as a toy example to illustrate this property. It's evident that in the vicinity of the base surface, the Hessian matrix of the SDF consistently exhibits two eigenvectors: one aligned with the gradient, and the other orthogonal the gradient. Moreover, we visualize the distribution of the eigenvalues using color-coded style, with the eigenvalue corresponding to the gradient being O.
3.3 Gradient-Hessial Alignment
It should be noted that even if the underlying surface is infinitely smooth, the SDF may not necessarily be smooth everywhere. From a geometrie perspective, the SDF is non-differentiable at medial-axit points with at least two nearest projections onto the surface. The neural implicit function fo, being at least -m-smooth, unlikely to be identical the real SDF. We can make reasonable assumption that So resembles the real SDF in the differentiable region that in the non-differentiable region. In light of this, is reasonable to enforce the alignment between the gradients and the Hessian for points within a thin-shell space of appropriate width.
Based on the aforementioned analysis, propose a loss function promote the alignment of the Gradient-Hesse
Tralign (q) = |HH
with
fo(())
g(4)
(5)
bx m
(DNS5
Figure 2: The level-sets, from left to right, show the distance fields learned by SIREN 40) SIREN with dirichlet energy (28), laplican energy [6] and hessian energy (54), DiGS 66) and ours with 100 points (black) as input. The black lines represent zero-isosurface. Our methods effectively suppress the ghost geometry with the concern of the gradient directions.

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Chamfer
F-Score
mean J std. 4
mean
std. :
SPSR
23
4.36
1.56
75.87
18.57
SIREN 40
18.24
17.09
38.74
31.26
SAP 3N
6.19
1.75
57.21
11.66
iPSR 20
4.54
1.78
75.07
19.18
PCP 5
6.53
1.75
47.97
14.50
CAP-UDF
55
4.54
1.82
74.75
18.84
DIGS 0
4.16
1.44
76.69
18.15
Ours (SIREN)
3.86
1.10
78.80
16.01
CAP-UDE
DIGS
Ours
Table 1: Quantitative comparison SRB
[49
The methods marked with ** require point nor- mals.
Figure 3: Visual comparisons on SRB
Difference from the Eikonal term. By enforcing the Eikonal term, we can prevent the SDF from degenerating into a trivial field, such as fe = 0 almost everywhere. But for a distance field rooted at always holds except at the non-differentiable points. In practice, fell = 1 is evaluated at discrete points, and the number of network parameters may significantly exceed the number of sample points. Consequently, specifying only fo can result in ghost artifacts in the reconstructed surface, such as bulging effects, redundant parts, or excessive surface variations. For instance, consider the last row of Fig. The reconstructed surfaces may differ significantly from the desired surface, even when V foll to seek more effective control over the gradient direction.
any surface. || fo
holds almost everywhere. This observation motivates us
Why not smoothness energy. In previous literature, various forms smoothness energy, such Dirichlet Energy (28). Hessian energy p 54], and Laplacian energy (6]. have been employed. Generally, the inclusion of smoothness terms can encourage smoothness in the SDF, reducing variations, and thereby mitigating the emergence of ghost geometries.
However, for these smoothing-bard geometry, they often sacrifice the ability to accurately represent geometric details. In contrast, regularization term enforces the alignment between the gradient and the Hessian, which significantly differs from enforcing simple smoothness. As shown in Fig. 2 created 'L' shape with 100 boundary points, following the same experimental setup as DiGS [6]. to illustrate the contrast between various approaches. Smoothing- based approaches tend not to reduce the smoothness energy zero, even at termination. In contrast, our regularization term can approach zero more closely. Moreover, aligning the gradient and the Hessian follows an inherent property of the SDF, independent of the specific case. This sets our approach apart from smoothing-base methods that require case-by-case tuning of smoothing weights. As result, our approach exhibits superior feature-preserving ability.
approaches, while they can reduce the occurrence of ghost
We conducted two ablation studies in Section 4.3 which clearly demonstrate that our alignment loss term not only more effectively suppresses the occurrence of ghost geometry but also yields more faithful reconstruction results than traditional smoothing- based approaches.
3.4
Loss function
Based on the aforementioned discussion, our neural function fo has be least C2 continuous. In this paper, we consider two types of activation functions. The first activation function is Sine, which facilitates learning high-frequency information as pointed out in SIREN 40). Obviously, Sine is Crinf continuous and satisfies our requirements. Additionally, Neural-Pul.
5 leverages the SoftPlus
activation function, which is smooth version of ReLU and can also serve our purpose.
To this end, we establish the entire loss function by integrating the alignment loss into the existing loss configuration such that the network optimization can be formulated as
arg min { Eold (fo) + EaEiin (fe))
(6)
5

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Normal C.
Chamfer
F-Score
JOU
meant
std.4
mean]
std.
meant
std.j
std.j
SPSR
22
90.58
3.30
4.66
4.64
75.28
25.76
90.58
6.77
NSP*
10
90.74
5.48
8.85
6.96
52.42
28.55
79.08
12.41
SAL 1
86.69
9.66
29.98
31.86
25.76
22.43
60.62
18.61
IGR 19
80.85
11.88
62.54
48.44
26.28
35.91
33.33
24.51
SIREN 20
83.79
10.20
34.19
46.77
32.34
30.13
41.14
15.82
DIGS 0
95.82
4.44
4.59
4.94
78.87
27.34
89.54
8.83
OSP
94,73
59.12
25.82
50.84
12.56
iPSR 20
93.22
5.26
5.95
5.97
68.42
26.36
85.01
9.34
PGR
27
91.90
4.93
7.34
4.81
51.44
23.12
78.34
11.17
POCO
96.41
3.53
3.62
4.21
85,42
23.13
94,40
5.55
Ours (SIREN)
96.52
3,31
3.38
3.64
88.71
18.28
89.80
6,47
Table 2: Quantitative comparison of surface reconstruction under ShapeNet (13) with 3K points. The methods marked with *** require normals, and the methods marked with ++ are supervision based.
where Eatign (fo) is given by
Σ LBg Latign (g),
qe
and Eold (fo) is the original loss function of existing works (the details will discussed in the experiments). We leverage SIREN [40] without the normal version and Neural-Puis Puull [5] in our experiment. Since the alignment effect between gradients and Hessian should be more emphasized near the underlying surface, we leverage adaptive weighting scheme inspired by [15]
Bg exp (-5 (f((g)),
(8)
we set 8 = 10 by default. At the same time, we set the default value of a to More details can be checked in our supplementary material.
4 Experiments
Metrics. The indicators for comparison include normal consistency, chamfer distances, and F-Score, where normal consistency (%, abbreviated as 'Normal C.) reflects the degree which the normals of the reconstructed surface agree with the normals of the ground-truth surface, chamfer distance (scaled by 10S, using Lı-norm) measures the fitting tightness between the two surfaces, and F-Score (%) indicates the harmonic mean of precision and recall (completeness). We set the default threshold of F-Score 0.005. All meshes are uniformly scaled [-0.5, 0.5), and 100K points are sampled from each mesh for evaluation.
4.1 Optimization-bn -ased Surface Reconstruction
Surface Reconstruction Benchmark (SRB). The Surface Reconstruction Benchmark (SRB)
49
contains five shapes, each of which has challenging features, e.g., missing parts and rich details. The approaches for comparison include screened Poisson surface reconstruction (SPSR) (23), SIREN 40
Shape as points (SAP) (38), iPSR (20), Predictive Context Priors (PCP) [3 CAP-UDF (55] and DIGS [6. Note that SPSR leverages normals given by the input scans. As shown in Tab. our
method outperforms the existing methods, as evidenced by superior results in both Chamfer distance and F-score metrics. In particular, the visual comparison presented in Fig. demonstrates that our method is capable of accurately recovering the hole feature of the Anchor model, despite the absence of points on the inner wall. Furthermore, successfully preserves the adjacent gaps of the Lord Quas model. In addition, we compared our approach with DiGS using SRB, following the evaluation settings of DIGS. Detailed performance statistics can be found in the supplementary material.
ShapeNet. The ShapeNet dataset 13 comprises diverse collection of CAD models. We follow
the splitting of [50] for the 13 categories of shapes with total 260 shapes. Our comparison
performed using 3K points. The baseline methods include Screened Poisson Surface Reconstruction
(SPSR) 23). NSP [50, SAL (T. IGR [19], SIREN 40), DiGS (6, OSP 4. iPSR 20) and PGR (27).
Note that SPSR and NSP require normal inputs. To ensure the validity of our experiments, provide

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ShapeNer
ABC
Things 10K
Input
SPSR*
SIREN
DIGS
PGR
POCO*
Ours
GT mesh
Figure Comparison of our method other methods under ShapeNet Thingi10K [57]
ABC
and
ground-truth -ruuh normals to those that require normal information. Additionally, we include the learnable baselines Shape as Points (SAP) [38] and POCO (8 for comparison and retrain them from scratch on ShapeNet with 3K points. Based on the quantitative results presented in Tab, 2 and the visual comparison Fig. Our method effectively suppresses unnecessary surface variations and adapts the implicit representation the inherent complexity encoded by the point cloud. Furthermore, reconstruction accuracy is comparable to supervised methods. Specifically, our method outperforms POCO in Normal Correctness, Chamfer, and F-Score, but lags slightly behind POCO in terms of IOU.
ABC and Thingi10K. The ABC dataset 24 contains variety of CAD meshes, while Thingi10K [57] comprises shapes with rich geometric details. follow [6] to perform split- ting for each dataset and randomly sample 10K points from each mesh. Each dataset contains 100 shapes. The bascline methods include Screened Poisson Surface Reconstruction (SPSR) (23] SAL M IGR [19], SIREN 44)), Neural-Puus [5]. Shape Points (SAP) (38] DiGS 6 iPSR 20)
and PGR 27 Note that we found that the supervised version SAP does not generalize well (ShapeNet) when using a global PointNet-basedd encoder. Therefore, we compare our approach against the unsupervised version of SAP. We also include the supervised method POCO [8] for comparison. To assess the generalization ability of supervised methods, we retrain them using 10K points on ShapeNet. The quantitative comparison results are presented in Tab. 3 Furthermore, the visual comparison on ABC (24) and Thingil0K [57] (see Fig. (4) demonstrates that our method can effectively recover CAD features such as small holes and thin plates and has ability accurately recover high-fidelity geometric details.
shapes not present in the training
3D Scene. Finally, we conducted tests on a 3D scene dataset [56] by sampling 20K points for each model. The quantitative comparison results presented in Tab, 4 and visual comparison in
Normal
Chandir
F-Score
moin]
sd._
7.38
Ours (STREN)
58.37
Neural-Puo Pull Ours (Neural-Pull)
$2.05
5.34
33.96
17.14
20.9
25.46 28.86
Fig. [5) Our method, when combined with either SIREN 40 or Neural-Pull (5). yields significant improvements even with only 20K points for the scene-level reconstruction.
$7.32
5.1
12.04
9.79
3530
Table 4: Scene [66]
Quantitative comparison on 3D
Notably, our method effectively captures both the details and overall shape of these challenging scenes.
4.2
Shape Space Learning
The D-Faust 7 dataset contains high-resolution raw scans (triangle soups) 10 humans with various poses. We follow DualOctreeGNN [46] to perform splitting that 6K scans used for training and
7

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ABC
Thingi10K
Normal C.
Chamfer
F-Score
Normal C.
Chamfer
F-Score
meant
std.4
mean]
std..
meant
std.4
meant
std.4
mean]
std.j
meant
std.j
SPSR
74.54
26.65
97.15
2.95
3.93
1.79
77.03
23.71
SAL
86.25
8.39
17:30
14.82
29.60
18.04
92.85
5.01
13.46
7.97
27.56
14.64
KR
82.14
16.12
36.51
40.68
43.47
40.06
90.20
10.61
27.80
34.8
54.28
39.99
SIREN
40
82.26
9.24
17.56
15.25
30.95
22.23
88.30
6.53
17.69
13.47
26.20
19.74
Neural-B
94.23
4.57
6.73
5.15
42.67
10.75
96.15
2.80
5.89
1.12
46.44
8.53
SAP 88
81.59
10.61
15.18
16.60
45.88
33.67
92.60
7.03
10.61
13.84
53.32
31.91
DiGS
94.48
6.12
6.91
6,94
66.22
32.01
97.25
3.30
5.36
5.59
74.45
27.11
iPSR 20
93.15
7.47
4.84
4.06
71.59
24.96
96.46
3.57
4.41
2,94
74.88
22.72
94.11
4.52
96.80
3.25
2.01
72.86
POCO+
8
92.90
7.00
6.05
6.80
68.29
26.05
95.16
5.00
5.61
9.42
73.92
25.79
Ours (SIREN)
96.50
3.95
3.72
2.48
83,50
19.28
97.92
2.90
3.34
3.14
89.68
Table Quantitative comparison on ABC points.
and Thingi10K [57].
Each raw point cloud has 10K
Nenual-uull
Ours (Nerual-Pall Pulli
SEREN
Ours (SIREN)
Figure 5: Visual comparison of our method to other methods under 3D Scene
56
Normal C.
Chamler
F-Seore
meant
std.]
metin
sid
std.]
IGK 1
3.34
29.01
33.61
73.32
14.05
SAL
96.77
0.81
2.82
4.67
91.35
9.15
SALD X
97.01
0.92
3.06
1.32
$8.56
12.73
JGR*
SAL'
SALD"
DualOctreeGNN*
IGR MV DIGS 0
$7.93
3.40
48.56
2.35
6.54
0.11
87.60
1.91
11.87
37.77
7.11
3.50
Ours (SIREN)
Table 5: Quantitative comparison DFAUST (7].
All methods marked with *** means leverages with normals in the training stage and marked with + means supervised.
DIGS
IGR
DualOetreeGNN
Ours
Figure 6: Visual comparison on DFAUST
2K scans for testing. In the training phase, we adopt the encoder following Convolutional Occupancy Networks [T). Besides, we adopt the FiLM conditioning [I]] that applies an affine transformation to the network's intermediate features as SIREN is weak in handling high-dimensional inputs [I]] 33
The baseline approaches include IGR 19. SAL M. SALD (2). DualOctreeGNN (46) and DiGS 6
Also, we further demonstrate the results of IGR, as well as the results of Dual OctrreGGN trained without normals. As shown in Fig. 6 although IGR can generate well details, yields spurious planes away from the input. SAL, supervised with unsigned distance, can only produce smooth results. Additionally, SALD, with the support of normal supervision, can generate more details, but its reconstruction accuracy is even worse than SAL since it suffers from large systematic misalignment that does not respect input poses.
By contrast, DualOctreeGNN gets the most impressive results since the well-designed oc- tree network can capture local prior for details.
Normal C.
Chanfe
F-Score
mncan"
std...
mean]
std.).
meant
std.j
91.02
4.34
4.55
4.10
29.41
However, the performance either IGR Du- alOctreeGNN is compromised without normals.
0.6
92.65
3.95
4.17
4.20
80.44
23.46
96.52
3.38
3.64
60
93.04
1.03
2.19
0.73
82.91
24.73
Table 6: Effect of weights of Latign

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DiGS cannot yield reliable results though it also based on SIREN. To summarize, our method can learn shape space without requiring input normals or additional supervision but still produces faithful shapes. It's important point clouds without input normals are not watertight, despite the small Chamfer distances.
that the resulting surfaces of DualOctreeGNN for
4.3 Ablation Study
Our ablation studies were conducted on the ShapeNet dataset 13 shapes, with 20 shapes per category for a total of 260 shapes, each represented by 3K points. We use SIREN the activation function for baseline.
which comprises 13 categories of
Comparison with smoothness energies In Figure 7 we increased the weighting coefficient of the regularization term to 1 10 x 108, and 1 x 10", respectively, for both our approach and those based on smoothing. It's evident that even with a large weight, our regularization term consistently produces high-fidelity surfaces, in contrast to the smoothing-basee approaches, whereas their reconstruction results may become overly smooth or even fail (Note that double layers for Dirichlet energy).
A similar trend is observed in the 3D context. As illustrated in Figure 9 our method excels in reconstructing faithful and high-fidelity shapes from input data with 100K points, while the other approaches tend to produce over-smoothed results.
We also conducted ablation study compare the quantitative performance of our energy term with that of Dirichlet Energy [28] and Hessian Energy [6]. The results presented in Tab. demonstrate that our energy
Ealign (.ee significantly improves the performance. It's worth noting that the comparison between DiGS [6] (with a Laplacian energy) and ours has been made in previous subsections, which shows that ours has a superior performance. Therefore, in this subsection, we do not include Laplacian energy for comparison.
T 10a
T= 10
== 10
Difihlet
Resien
n)
Figure 7: Ablation study the weighting coefficient of the regularization term.
Effeet of weights of Talien- We conducted an ablation study to evaluate the effect of the weight a of Lalign- By selecting different values, i.c., {0.06, 0.6, 60), we keep the statistics in Tab. which demonstrates that the best performance was achieved when the weight was set to 6.
Effect of adaptive per points weights. We demonstrate the effectiveness of the adaptive weight Bg for each query in Tab. 7 We also compare the performance of different decay parameters ó used to compute 3g Eq. (8) and find that 8 yields the best results.
Normal C. std.j
Chanfer
F-Score std.j
mean].
std.
8=
91.85
7.22
4.73
3.91
79.95
22.38
$=
7.23
22.38
8= 10
96.52
3.31
3.38
3.64
88.71
18.28
Table 7: Effect of adaative per points weights.

---

Comparison to LSA. Recently, LSA [3 pro-
posed enforcing alignment between the gradient of each level set and the gradient of the O-level set to account for the direction of the gradient. However, it is important to note that the gradi- ent the O-level set may not align with the true surface normals for its results. As such, using the gradient the O-level set as a reference may not be particularly effective. We present results for SIREN [40] using both our loss and LSA under six shapes from the Stanford Scanning dataset with million points same as the settings of LSA in its main paper. As shown in Fig. & our method effectively eliminates ghost geometry in empty areas due to incorrect gradient direction, while LSA does not. Tab. 9 presents the comparison statistics for three different approaches: 1) our methods combined with SIREN 40), 2) original SIREN and 3) LSA.
meant
sd...
mean)
sid.1
mean
sid..
Thrichlet Energy
92.91
7.48
4.30
3.75
77.67
23.83
Hession energy
93.28
7.21
4.38
385
81.12
25.41
Ours
96.52
3.31
3.38
364
88.71
18.28
Table 8: Quantitative comparison to Smooth En- ergy.
Normal C.
Charffer
F-Score
std.j
mean].
std.3
std.j
SIREN
93.37
5.58
7.19
7.57
74.82
29.21
LSA (SIREN)
96.40
1,04
4.10
1.19
86.38
7.37
Ourx (SIREN)
98.25
1.03
2.19
0.73
97.66
4.15
Table 9: Quantitative comparison to LSA
31
(SIREN)
Ours (SIREN)
Figure 8: Visual comparison with SIREN 40 and
LSA 31 on Stanford Scanning dataset.
Inpux
DIGS
Our
Ingut
DXGS
Out
Dirichlet energy (SIREN wab n)
Hessian energy
Laplacian energy DDIS
(SIREN
(SIREN
Figure 10: Our method encountering challenges while handling sparse in- puts.
Figure 9: Comparison with smoothness energies in 3D con- text.
Conclusion
In this paper, we propose novel approach for surface reconstruction from unoriented point clouds by aligning the gradient and the Hessian of the Signed Distance Function (SDF). Unlike existing smoothness energy formulations that minimize SDF volatility, our approach offers more effective control over gradient direction, allowing the implicit function adapt to the inherent complexity of input point cloud. Comprehensive experimental results demonstrate that our approach effectively suppresses ghost geometry and recovers high-fidelity geometric details, surpassing state-of-thed methods in terms of reconstruction quality.
Our current algorithm still has a few trawbacks. Firstly, it struggles with processing super sparse inputs, such as sketch point clouds LiDAR data (as shown in Figure TOp. The inherent challenge arises from completing extensive missing parts and closing the gaps between the input stripes. Secondly, faces difficulties when dealing with large-scale scenes, like those from Matterport3D [12). mainly due to the neural networks' catastrophic forgetting issue. Representing large-scale scenes within a single network becomes exceedingly challenging. our future work, we plan to address these challenges. To handle sparse inputs more effectively, we are considering implementing supervision mechanism. Additionally, we aim to adopt a sliding window strategy, similar approaches like DeepLS [10] and BlockNeRE 43 to mitigate the catastrophic forgetting issue.
10

---

6
Acknowledgement
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This work is supported by the National Key R&D Program of China (2022YFB3003200), the National Natural Science Foundation of China (62002190, 62272277, 62072284), and the Key Research and Development Plan of Shandong Province of China (2020ZLYS0)).
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14


Reviews:
[{"review_id": "Oz6RCReyHc", "reviewer": "Reviewer_XXzk", "summary": "This paper introduces a novel approach to learning the SDF directly from point clouds without the use of surface normals. The key insight behind this is that aligning the gradient of the SDF with its Hessian allows for better training of the distance field. Through extensive experiments, the paper demonstrates the effectiveness of the proposed approach in accurately recovering the underlying shape and effectively reducing geometry artifacts.\n\n", "questions": "Can the proposed method guarantee the reconstruction of watertight surfaces? For large-scale scenes, are the reconstructed surfaces watertight?", "limitations": "The paper lacks a discussion of potential limitations and the negative social impact of the proposed method.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper exhibits a good writing style, effectively conveying the ideas and concepts to the readers. The content is clear, concise, and well-structured, allowing for easy understanding of the proposed method.\n\n- The paper introduces a novel method for learning the signed distance function directly from unoriented point clouds. \n\n- The paper presents extensive experimental results that showcase the effectiveness of the proposed approach. \n\n", "weaknesses": "- The paper does not explicitly discuss potential limitations or challenges of the proposed method. Addressing and acknowledging these limitations would provide a more comprehensive understanding of the method's applicability and potential areas for improvement.\n\n- The paper lacks a time analysis of the proposed method. While the experimental results demonstrate the effectiveness of the approach, the paper does not provide insights into the computational efficiency or time complexity of the method. Understanding the time requirements would be valuable for assessing its practicality and potential for real-time applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewers for their appreciation and invaluable comments. We will incorporate all feedback into the revised version.\n\n**Q1: Limitations and negative social impact**\n\nWe shall point out the following limitations in this revision: (1) it is weak in handling sparse input such as sketch point clouds or LiDAR. In particular, most LiDAR data in the KITTI dataset are partial scans. This presents significant challenges in closing the gaps between the stripes and completing large missing parts. We enclosed the relevant experimental results in the **PDF file** (**Fig. 4**) and will include them in the supplementary. (2) Its potential negative social impact includes unauthorized replication of mechanical designs and intentional content creation as other reconstruction methods.\n\n**Q2: Time analysis of the method**\n\nWe made a detailed comparison with IGR, SIREN, and DiGS on the timing cost and the number of parameters for optimizing for a single point cloud in **Sec.B.2** and **Tab. 3** in the supplementary material. See the table below for more information.\n\n|            |$\\text{IGR}$|$\\text {SIREN}$|$\\text{DiGS}$|$\\text {Ours}$|\n|:------------|---------|:---------:|:---------:|:---------:|\n|#$\\text{parameters}$|$\\text{1.86M}$|$\\text{264.4K}$|$\\text{264.4K}$|  $\\text{264.4K}$|\n|$\\text {time[ms]}$|$\\text{50.73}$|$\\text{11.52}$|$\\text{36.28}$|$\\text {40.10}$|\n\nThe optimization time statistics are made on a single GTX 3090 GPU. It can be seen that the timing costs of DiGS and ours are higher than the conventional SIREN since DiGS and ours need a second-order optimization. However, ours is more computationally efficient than IGR.\n\n**Q3: Guarantee about the watertight reconstruction surface?**\n\nSince our approach fits the underlying SDF, which naturally guarantees that the reconstruction surface (the 0-isosurface) is watertight. But it must be pointed out that when the points are too sparse or with a lot of missing parts, the produced surface may consist of many separate connected components. See **Fig.4 in the PDF file**. Furthermore, for large-scale scenes (such as from Matterport3D), it is extremely hard to represent large-scale scenes within a single network due to the catastrophic forgetting issue of the neural networks. The sliding window strategy proposed by recent works (e.g. DeepLS [ECCV 2020] and BlockNeRF [CVPR 2022]) seems helpful. We shall list it as an interesting future work. Thanks."}, {"review_id": "7WDLmffqgG", "reviewer": "Reviewer_krE5", "summary": "The paper considers the task of surface reconstruction from point clouds without normals and uses neural signed distance functions. They consider previous smoothness losses, and come up with their own by taking the derivative of the Eikonal equation, thus guaranteeing that their loss constrains a fundamental property of SDFs. They motivate this loss in various other ways (which to me are fairly unclear). They also show good results on many datasets, and compare to many different types of approaches.", "questions": "- Is there a reason you are using most of DiGS' datasets but not using their version of the metrics?\n- Your explanation of the implication is Rodigues' formula is not clear. I couldn't find the formula name in your citation [34], and looking else online it seems that the formula states that one of the eigenvalue-eigenvector pairs of the Hessian should be a principle curvature, principle direction pair, not all eigenvalue-eigenvector pairs? Why does it imply equation (3)?\n- What is Figure 3 trying to show us?", "limitations": "Not discussed.", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "- Using second order information derived from the Eikonal equation makes sense\n- Lots of experiments and comparisons to many types of methods", "weaknesses": "- One of the benefits of the split of the ShapeNet dataset is that you use is that you can compute IoU with it, which is a very different but important measure to IoU.\n- Your explanation for why L_{align} is different from the Eikonal term is essentially that it works better for finitely many points, which is not an explanation. You should explain that with finitely many points constraining second order information reduces the number of possible solutions, and maybe allude to the toy problem in DiGS (or even construct the same toy problem with your loss term).\n- Your explanation for why not smoothness energy is not very compelling either. Your main argument is that other energies are not 0 at the optimum, hence it is hard to choose how regularisation to apply. A much better argument is your earlier point that the loss is derived from a fundamental property of SDFs, while the others are general smoothing losses that are not specifically guiding towards a proper SDF and will be smoothing in unwanted ways.\n\nOverall great idea and results, not great presentation/soundness. Your explanation for your loss needs to be more clear, keeping it to it is a fundamental property of an SDF works just fine. A toy problem to back this up could be to see what happens when you regularise the 2D shapes with high weight with the different smoothness energies, mostly likely other methods would stop obeying the points and go for a smoother surface while yours would still obey the input points.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for constructive feedback. In the following, we will address the main concerns carefully and seriously.\n\n**Q1: Compute IoU under ShapeNet**\n\nFollowing your advice, we report the IoU statistics under ShapeNet, see **Tab. 1 in the global response box**. At the same time, we will add the statistics to the paper.\n\nIt can be seen that our method is a bit inferior to the supervised method POCO (note that ours outperforms POCO in terms of Normal Consistence, Chamfer distance, and F-Score, as shown in the main paper). However, our method is consistently better than the other optimization-based methods.\n\n**Q2: Compared to the Eikonal term and other smoothness energy terms**\n\nThanks for pointing out this issue. As you observed, when one enforces the Eikonal term on finite and discrete points, still a large number of invalid solutions (with ghost geometry) cannot be filtered out. By contrast, second-order information can effectively limit the solution space.\n\nSecondly, the parameters of the network are typically larger than input points, leading to over-parameterized neural networks with an extensive array of parameters, which can further convolute the optimization process, as described by Empirical Analysis of the Hessian of Over-Parametrized Neural Networks [Sagun et al., 2017].\n\nAs for various smoothness energy terms (i.e., Laplacian energy used by DiGS), they have a negative effect (missing geometric details) although they can reduce the number of possible solutions. In other words, the ability to accurately represent geometric details is greatly diminished. By contrast, our regularization term is based on the fundamental geometry property of SDF, which doesn’t enforce unwanted smoothness. We construct a similar toy problem as DiGS with the shape 'L' with 100 points to demonstrate the effect of our term. We set the weighting coefficient of all regularization terms to 100 for fairness (default coefficient used by DiGS), shown in **Fig.1 in the PDF file**. The experimental results validate our observation. (Note that DiGS leverages 15K points in its main paper for the toy example.)\n\nTo further validate the effect of our term, we increase the weighting coefficient of the regularization term to 1e3, 1e5, and 1e7, respectively, as shown in **Fig. 2 in the PDF file**. It can be seen that even with large weight, our regularization term consistently yields high-fidelity surfaces without introducing over-smooth and ghost geometry. However, smoothness energy terms easily degenerate.\n\nIn essence, other general smoothness energies, which are not always to zero, tend to force the derivative to 0 and will be smoothing in unwanted ways. Instead, our regularization term is more relaxed to avoid over-smooth results, which enforces the gradient to become one of the eigenvectors of the Hessian matrix, with the corresponding eigenvalue being 0 that only the second derivative in the gradient direction being zero satisfies the fundamental property of SDF. To illustrate the feature-preserving ability of our method, we conducted experiments under a lion shape with 100K points as input. As shown in **Fig. 3 in the PDF file**, our method not only suppresses ghost geometry but also recovers high-fidelity geometric details. However, smoothness energy terms tend to produce over-smoothed results. \n\n**Q3: Different metrics compared to DiGS**\n\nTo our knowledge, there are two commonly used evaluation settings under SRB, one used by DiGS [CVPR 2022] and the other proposed by Shape as Points [NIPS 2021].\n\nThe version of DiGS operates in the resolution of $512^3$ and uses Chamfer distance and Hausdorff distance at the original scale, while the version of Shape as Points operates in the resolution of $256^3$ and uses Chamfer distance, F-Score, and Normal Consistency, but it does not give the number of evaluation points. Therefore, we sample 100K evaluation points and follow the settings of Shape as Points in this paper. Following your advice, we also show the results under the settings of DiGS, see **Tab. 2 in the global response box**.\n\nIt can be seen that our method outperforms DiGS in terms of Hausdorff distance. To summarize, our method consistently outperforms DiGS, in terms of whether Hausdorff distance or for F-Score (indicated in the main paper).\n\n**Q4: Misleading of Rodrigues’ formula**\n\nWe are sorry for this misleading, it is actually not related to Rodrigues’ formula. A more suitable explanation and reference is as follows ref. [30]: See Chapter 2. When the situation reduces to 2D, it coincides with your observation - one of the eigenvalue-eigenvector pairs of the Hessian should be a principle curvature, principle direction pair. We will fix this in the revision.\n\n**Q5: The objective of Figure 3**\n\nThanks for pointing out the misleading figure. We shall replace it with a more informative figure (**see Fig. 1 in the PDF file**).\n"}, {"review_id": "DFcB3SeDTK", "reviewer": "Reviewer_JRpq", "summary": "The authors provide a novel regularizing term for training of implicit neural SDF representations, motivating the function gradient to be a zero eigenvector of the Hessian near the zero level set (the surface). They demonstrate that it can be used to reduce the appearance of ghost geometry and increase quality of the representations.", "questions": "See above weaknesses.", "limitations": "The authors adequately address the limitations of the methods, which are mostly that there is still room for further improvement in their output.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- A principled novel regularization term that seems to significantly improve the learned representations.\n- Is tackling an important application domain, as neural geometry representations are broadening the application domain of deep learning techniques.", "weaknesses": "- It seems unclear to me how the method is specifically designed to handle unoriented point clouds. If there is a claim that the regularizer somehow motivates consistent alignment of resulting normals, it'd be nice to have some commentary on this.\n- The evaluation metrics are a little strange to me:\n   - It's unclear how they are comparing normals, as there would need to be a correspondence between the reconstructed surface and the ground truth surface.\n   - F-score seems a bit of an odd choice. I initially assumed that they are using the interior of the isosurface to specify points classified as inside the volume, but the default threshold 0.005 seems quite low for this. So perhaps they are asking for the network output to classify points on the shape, but I am not sure of how they do this either, as nearly all points will evaluate to nonzero values. \n  - In my opinion, Hausdorff would be good to include, as it would give a sense of the maximum error.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate that the reviewer finds our paper novel. We address additional comments below.\n\n**Q1: How to handle unoriented point clouds?**\n\nYes, our alignment term encourages consistent orientations of normal vectors. Based on our $C^{\\infty}$ activation function (sine and softplus), when we request the Hessian matrix to degenerate along the eigenvector corresponding to the zero eigenvalue, that is the second derivative in the gradient (normal) direction is zero, the degeneration direction of the Hessian matrix in the neighborhood also tends to be consistent, leading to consistent normals.\n\n\n**Q2: Confusion about the evaluation metric**\n\nIn the paper, we used the commonly used metrics for evaluating reconstruction methods including Chamfer distance, F-Score, and Normal consistency proposed by OccupancyNet [CVPR 2019] and Convolutional Occupancy Networks [ECCV 2020]. We give their definitions in **Sec. C in the supplementary material**.\n\nFor the purpose of comparing normals, we sample the same number of points from the reconstructed surface and the ground-truth surface, respectively. For each sample point on the reconstructed surface, we find the nearest point in the other point set and keep the dot-product of the two normals. Note that the computation is bi-directional.\n\nThe original F-Score is defined as the harmonic mean between precision and recall. To define the F-Score in our scenario, we sample the same number of points from the reconstructed surface and the ground-truth surface, as mentioned above. For recall, it counts how many points on the GT mesh lie within a certain distance (threshold t) to the reconstruction surface. For precision, it counts the percentage of points on the reconstructed mesh that lie within a certain distance (threshold t) to the GT. The F-Score is then defined as the harmonic mean between precision and recall, see the definitions of precision and recall in **Sec. C in the supplementary material**\n\nFurthermore, we agree that Hausdorff distance is a good choice. We compared ours and DiGS using Chamfer distance and Hausdorff distance under SRB, see **Tab. 2 in the global response box**.\n\nWe will add it in the revision."}, {"review_id": "WVwdKju4qy", "reviewer": "Reviewer_csEg", "summary": "The paper under review proposes a method for signed distance function (SDF) reconstruction from point cloud data without the use of normal data at the points.  The paper proposes a new term as a regularization to complement the EIkonal loss, which is a necessary condition for an SDF.  The purpose of this regularization is to reduce \"ghosting\" effects.  The term is obtained by differentiating the Eikonal constraint, resulting in a second-order constraint, namely that the Hessian times the gradient is zero.  The paper shows experimental comparison to SOA on multiple benchmarks with favorable results.", "questions": "Address the weaknesses.", "limitations": "Not discussed.", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- New regularization term for neural SDFs; original to best of my knowledge; neural SDFs are a very relevant research topic of interest in vision/graphics\n- Paper is mostly clear\n- Experiments on many benchmark datasets, with favorable results to the authors' approach", "weaknesses": "- Experimental results on SRB and ShapeNet use different metrics than recent papers in area, e.g., DiGS[6] - not clear why they have been changed.  Please report results with previous protocols.\n- A new sampling strategy is reported in supplementary for the alignment loss.  Not clear what the effect of this is, and why it is needed.\n- Theoretical justification is weak, e.g., line 156-174; there is no clear understanding of this new term - no clear explanation of why the ghosting is reduced with this term.\n--The authors attribute ghosting to in practice only applying the Eikonal constraint to a sampling of points; unclear if this is true due to random sampling or this is only part of the problem.  The Eikonal equation does not have a unique solution which is in part why there is ghosting.\n--Unclear why the smoothness term not being min zero means it's difficult to regulate. \n--It is stated that the norm and direction of the gradient is de-coupled.  However, a minimization of the loss is being considered - the optimizing flow for both the Eikonal and alignment will both change the direction & norm of the gradient during minimization.\n- Line186-187: the constraint is true a.e. so why is the adaptive weighting needed?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the valuable comments. We will incorporate all feedback into the revised version.\n\n**Q1: Different metric compared to DiGS**\n\nTo our knowledge, there are two commonly used evaluation settings under SRB, one used by DiGS [CVPR 2022] and the other proposed by Shape as Points [NIPS 2021].\n\nThe version of DiGS operates in the resolution of $512^3$ and uses Chamfer distance and Hausdorff distance at the original scale, while the version of Shape as Points operates in the resolution of $256^3$ and uses Chamfer distance, F-Score, and Normal Consistency, but it does not give the number of evaluation points. Therefore, we sample 100K evaluation points and follow the settings of Shape as Points in this paper. Following your advice, we also show the results under the settings of DiGS, see **Tab. 3 in the global response box**.\n\nIt can be seen that our method outperforms DiGS in terms of Hausdorff distance. To summarize, our method consistently outperforms DiGS, in terms of whether Hausdorff distance or for F-Score (indicated in the main paper).\n\n\n\n**Q2: Effect of sampling strategy for alignment**\n\nThe sampling strategy is not new. Actually, we use the sampling strategy following IGR [ICLR 2020] and NeuralPull [ICML 2021], to sample more points near the input points rather than uniformly the bounding box.\n\nThe rationale behind this lies in that the SDF only guarantees to be differentiable in a narrow region nearby the surface (considering faraway surface points which have two or more same minimal distance that cannot be differentiable). Besides, for the reconstruction methods, the 0-isosurface is of more interest than other level sets. We compared different sampling strategies as **Tab. 4 in the global response box** (We disable adaptive weight here to validate sampling strategies only). It can be seen that sampled points near the surface are better than samples randomly on the bounding box.\n\n**Q3: Understanding of ghost geometry and our term**\n\nThanks for pointing out this issue. As you observed, the Eikonal term has many invalid solutions (with ghost geometry) that cannot be filtered out on finite and discrete points, as discussed in DiGS. By contrast, second-order information can effectively limit the solution space.\n\nSecondly, the parameters of the network are typically larger than input points, leading to over-parameterized neural networks with an extensive array of parameters, which can further convolute the optimization process, as described by Empirical Analysis of the Hessian of Over-Parametrized Neural Networks [Sagun et al., 2017].\n\nOur regularization term is based on the fundamental geometry property of SDF to limit the solution space and reduces the number of possible solutions. The occurrence of ghost geometry contradicts the fundamental property of SDF. We construct a similar toy problem as DiGS with the shape 'L' with 100 points to demonstrate the effect of our term. We set the weighting coefficient of all regularization terms to 100 for fairness (the default coefficient used by DiGS) shown in **Fig. 1 in the PDF file**. The experimental results validate our observation. (Note that DiGS leverages 15K points in its main paper for the toy example.)\n\nTo further validate the effect of our term, we increase the weighting coefficient of the regularization term to 1e3, 1e5, and 1e7, respectively, as shown in **Fig. 2 in the PDF file**. It can be seen that even with large weight, our regularization term consistently yields high-fidelity surfaces without introducing over-smooth and ghost geometry. However, smoothness energy terms easily degenerate.\n\nIn essence, other general smoothness energies, which are not always to zero, tend to force the derivative to 0 and will be smoothing in unwanted ways. Instead, our regularization term is more relaxed to avoid over-smooth results, which enforces the gradient to become one of the eigenvectors of the Hessian matrix, with the corresponding eigenvalue being 0 that only the second derivative in the gradient direction being zero satisfies the fundamental property of SDF. To illustrate the feature-preserving ability of our method, we conducted experiments under a lion shape with 100K points as input. As shown in **Fig. 3 in the PDF file**, our method not only suppresses ghost geometry but also recovers high-fidelity geometric details. However, smoothness energy terms tend to produce over-smoothed results. Finally, we agree that the norm and direction of the gradient cannot be totally de-coupled. We shall make it more rigorous in this revision.\n\n\n\n**Q4: Effect of adaptive weighting**\n\nAs discussed in #Q2, it is better to emphasize the points nearby the underlying surface. We conducted an ablation study about how adaptive weighting influences the reconstruction result; **Tab. 7 in the main paper** demonstrates that an adaptive weighting scheme ($\\delta = 0$ means disabling it with the adaptive weighting constantly is 1) a better strategy. "}, {"review_id": "nP3ICN3zmV", "reviewer": "Reviewer_ApDg", "summary": "The authors propose a new smoothing loss to better regularize the estimated Signed Distance Field (SDF) from unoriented point clouds. By aligning the SDF's gradient and its Hessian, the quality of reconstructed surfaces improves significantly. Compared with commonly used Eikonal loss, the proposed loss is more effective. Experiments validate the effectiveness of the proposed method as well as not so sensitive to input noise and hyper-parameters. ", "questions": "1.\tThe first paragraph in Sec 3.3 requires figure illustration. It’s difficult to understand why Equ. 4 intuitively works. What is Equ. 4 enforcing? What do you mean by \"alignment\"? A few figures are necessary. It is not clear at all where Equ. 4 comes from.\n2.\tIn Figure 2, The 100 points inputs are different for three methods? Please show the same inputs’ results as comparisons; otherwise, it is  not convincing.\n3.\tIn Figure 3, it is NOT evidence that the alignment loss can suppress the ghost geometry as mentioned in Line 173. The authors need clearly show which region in Figure 3 is problematic, and why the last column is good. It seems the alignment loss works similar to Dirichlet loss, and I do not see any benefit. Are the black lines zero-isosurface? Then the last column’s result seems quite bad.\n4.\tThe big Q in Equ. 7 is not defined.\n5.\tTable 5’s caption needs explanations of each row, especially the difference between same name’s rows.\n", "limitations": "The current experiments only show surface reconstruction from point clouds. But I think the loss could benefit other scenarios, like incorporated into Neus, but the authors do not show its effectiveness. Also, whether the loss works for unsigned distance field (UDF) also? This could be an interesting future topic. The current point cloud scenario is somewhat limited.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The proposed loss is novel and theoretically reasonable.\n2. The loss could be easily worked with different methods as a plugin.\n3. The experiments are almost complete and big improvements validate the effectiveness.", "weaknesses": "The paper is poorly written. The authors write the paper more like from a mathematic point of view rather than computer vision point of view. Many details are missing, and some said \"obviously\" and \"evidence\" are not clear at all. For a computer vision paper, Figures are much more important for illustration and detailed explanation of Figures and Equations, especially where to look at in the figure and why an equation intuitively works.\n\nHowever, if the authors could provide details in the rebuttal process and the final version, I believe that the paper could be accepted. If not, it should be rejected, as this is not a math paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for considering our method novel and theoretically reasonable.\n\n**Q1: Explanation of Sec 3.3 and Eq. 4.**\n\nWe are sorry for the unclear writing. We will further improve the readability in the revision. Recall that one of the basic properties of SDF is the Eikonal condition (line. 116) $\\lVert\\nabla f_\\theta\\rVert_2^2=1$, we can differentiate both sides of the identity and we can get the eigenfunction of the Hessian matrix (Eq. 2): $\\mathbf{H}\\_{f\\_\\theta} \\nabla f\\_\\theta \\(\\boldsymbol{q}\\)=\\mathbf{0}$, which implies the gradient of SDF is one of the eigenvectors of the Hessian matrix of SDF $f$ and its corresponding eigenvalue is 0. The more detailed proof is available in ref. [30]. The property implies that the gradient must align with one eigenvector of the Hessian matrix with 0 eigenvalue. Our regularization term (Eq. 4) is exactly based on this fundamental property. Therefore, we use \"alignment\" to enforce the gradient to become one of the eigenvectors of the Hessian matrix, with the corresponding eigenvalue being 0 that is the second derivative in the gradient direction is zero. Experimental results show that the alignment requirement is effective, i.e., not only effectively suppresses the ghost geometry but also preserves geometric details.\n\n\n\n**Q2: The issue of Fig. 2**\n\nThanks for pointing this out!\n\nWe follow the 2D toy problem settings as DiGS which samples 100 points from analytical equations at each epoch. So it seems the points leveraged by different methods are different. We agree that it is misleading and we conducted experiments with fixed points in **Fig. 1 in the PDF file**, which is used to replace Fig. 2. \n\n**Q3: The issue of Fig. 3**\n\nWe are sorry for this issue. We will replace it with our new figure (**Fig. 2 in the PDF file**) to demonstrate our benefits.\n\n**Q4: The definition of Q**\n\nQ is the set of sampled points and we defined it in **Sec. A in the supplementary material**.\n\nWe follow IGR [ICLR 2020] to sample point clouds and directly leverage the sampled points produced by NeuS on Ray for multi-image input.\n\nSpecifically, suppose that $p_i$ is a point in the input $P$, we define the Gaussian function rooted at $p_i$ and take the distance to its $k$-th nearest neighbor ($k = 50$ by default follow IGR) as the standard deviation. Then we sample points from each distribution. \n\n**Q5: Explanation of Table 5**\n\nThanks for the question. In the paper, all methods marked with '*' means it leverages with normals in the training stage, and those marked with '+' means this is a supervised one. We have compared different optimization-based methods under DFAUST and the different symbols mean its training settings.\n\nIn all, our method can learn shape space without requiring input normals or additional supervision but still produces more faithful shapes than others. \n\nWe will revise it following your advice.\n\n**Q6: Extension of our method**\n\nThanks for your constructive comments! It is interesting to explore our method with multi-image input and fit UDF.\n\nWe have incorporated our loss into NeuS and conducted the experiments under the DTU dataset in **Sec.D.5 in the supplementary material** (we enclosed the experimental results in the submission phase). It can be seen that our method supports multi-image inputs and effectively improves the quality of results compared to the original NeuS.\n\nFor UDF, unfortunately, it does not satisfy the Eikonal term since the norm of gradient is 0 on the 0-isosurface. Therefore, the key alignment property does not hold anymore and thus our algorithm may not work well for fitting UDF."}]

Author rebuttal:
We sincerely appreciate all valuable comments and suggestions, which helped us to improve the quality of the article. We have carefully considered all your advice and addressed the questions raised, see details in the separate response and the attached PDF files. Specifically:

- We clarify the motivation and insight of the method compared to the Eikonal term and other general smoothness energy terms. Moreover, we conducted 2D experiments compared to them (the black bold line represents 0-isosurface), see the figures in the attached PDF files.

- We report the IoU under ShapeNet:


|                      |                         | $\text{SPSR}^*$ | $\text{NSP}^*$ | $\text{SAL}$ | $\text{IGR}$ | $\text{SIREN}$ | $\text{DiGS}$ | $\text{OSP}$ | $\text{iPSR}$ | $\text{PGR}$ | $\text{POCO}^{+}$ | $\textbf{Ours (SIREN)}$ |
|:--------------------:|:-----------------------:|:---------------:|:--------------:|:------------:|:------------:|:--------------:|:-------------:|:------------:|:-------------:|:------------:|:-----------------:|:-----------------------:|
| $\text{IOU}$ |  $\text{mean}\uparrow$  |     $81.01$     |     $67.34$    |    $48.89$   |    $30.28$   |     $35.65$    |    $78.44$    |    $57.86$   |    $75.53$    |    $69.26$   |  $\textbf{84.25}$ |         $81.19$         |
|                      | $\text{std.}\downarrow$ |     $15.10$     |     $21.85$    |    $30.93$   |    $33.46$   |     $25.96$    |    $18.72$    |    $27.28$   |    $19.78$    |    $20.22$   |  $\textbf{13.62}$ |         $16.37$         |

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ Table 1: IoU under ShapeNet

- We compare with DiGS under SRB using the evaluation settings of DiGS:

||$\text {Mean}$||$\text {Anchor}$||||$\text{Daratech}$||||$\text{DC}$||||
|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|
||$\text{GT}$||$\text{GT}$||$\text{Scans}$||$\text{GT}$||$\text { Scans }$||$\text{GT}$||$\text{Scans}$||
|$\text{Method}$|$d_C$|$d_C$|$d_C$|$d_H$|$d_{\vec{C}}$|$d_{\vec{H}}$|$d_C$|$d_H$|$d_{\vec{C}}$|$d_{\vec{H}}$|$d_C$|$d_H$|$d_{\vec{C}}$|$d_{\vec{H}}$|
|$\text{DiGS}$|$\textbf{0.19}$|$3.52$|$0.29$|$7.19$|$0.11$|$1.17$|$\textbf{0.20}$|$3.72$|$0.09$|$1.80$|$0.15$|$\textbf{1.70}$|$0.07$|$2.75$|
|$\textbf{Ours}$|$\textbf{0.19}$|$\textbf{2.98}$|$\textbf{0.28}$|$\textbf{4.79}$|$0.24$|$1.78$|$\textbf{0.20}$|$\textbf{2.52}$|$0.13$|$1.84$|$\textbf{0.14}$|$1.88$|$0.10$|$2.77$|

||$\text {Gargoyle}$ ||||$\text {Lord Quas}$ ||||
|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|
||$\text {GT}$||$\text{Scans}$||$\text{GT}$||$\text { Scans }$||
|$\text{Method}$|$d_C$|$d_H$|$d_{\vec{C}}$|$d_{\vec{H}}$|$d_C$|$d_H$|$d_{\vec{C}}$|$d_{\vec{H}}$|
|$\text{DiGS}$|$\textbf{0.17}$|$\textbf{4.10}$|$0.09$|$0.92$|$\textbf{0.12}$|$\textbf{0.91}$|$0.06$|$0.70$|
|$\textbf{Ours}$|$0.19$|$4.56$|$0.15$|$1.82$|$0.14$|$1.13$|$0.09$|$0.95$|

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ Table 2: Comparison under Surface Reconstruction Benchmark.

- We explore the effects of different sampling strategies under SRB and the evaluation settings of DiGS. Note that Bbox-sampling means we sample randomly in the bounding box ([-1, 1]) of the input and then utilize them for our alignment term, and $k$ means the $k-th$ of the nearest sampling following IGR[ICLR 2020]:

||$\textbf{Mean}$||
|:---|:---:|:---:|
||$\textbf{GT}$||
|$\text{Method}$|$d_C$|$d_H$|
|$\text{Bbox-sampling}$|$\text{0.32}$|$\text{4.51}$|
|$\text{k=1}$|$\text{0.32}$|$\text{4.90}$|
|$\text{k=25}$|$\text{0.28}$|$5.87$|
|$\mathbf{k=50 (ours)}$| $\textbf{0.19}$|$\textbf{2.98}$|
|$\text{k=75}$|$\text{0.26}$|$\text{4.31}$|
|$\text{k=100}$|$\text{0.28}$|$\text{5.34}$|     

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ Table 3: Effects of different sampling strategies.

---
Please let us know if you have any other questions. We would be pleased to engage in further dialogue with you.

Thanks for your time,

The Authors

Meta-review: This submission received 5 positive recommendations. Initially, the reviewers were concerned about the evaluations and the details. The authors addressed most of the concerns in the rebuttal and the responses to reviewers’ additional comments. The reviewers reached a consensus of acceptance after the discussion period. The AC read through the submission, the review, the rebuttal and the discussions. The AC agrees that the authors presented an interesting idea of constraining gradients in the learning of implicit representations, and also addressed the reviewers’ concerns by additional responses and convincing results. Per this, the AC supports the reviewers’ decision, and accepts this submission. The decision was discussed with and approved by the SAC. Please follow the reviewers’ comments to improve the manuscript.
Decision: Accept (poster)