NeuralGF: Unsupervised Point Normal Estimation by Learning Neural Gradient Function
Authors: Qing Li
Huifang Feng
Kanle Shi
Yue Gao
Yi Fang
Yu-Shen Liu
Zhizhong Han
Venue: NeurIPS 2023 poster (2023)
Keywords: Point Clouds
Normal Estimation
Neural Gradient

Abstract: Normal estimation for 3D point clouds is a fundamental task in 3D geometry processing. The state-of-the-art methods rely on priors of fitting local surfaces learned from normal supervision. However, normal supervision in benchmarks comes from synthetic shapes and is usually not available from real scans, thereby limiting the learned priors of these methods. In addition, normal orientation consistency across shapes remains difficult to achieve without a separate post-processing procedure. To resolve these issues, we propose a novel method for estimating oriented normals directly from point clouds without using ground truth normals as supervision. We achieve this by introducing a new paradigm for learning neural gradient functions, which encourages the neural network to fit the input point clouds and yield unit-norm gradients at the points. Specifically, we introduce loss functions to facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning a global surface representation of the data. Meanwhile, we incorporate gradients into the surface approximation to measure the minimum signed deviation of queries, resulting in a consistent gradient field associated with the surface. These techniques lead to our deep unsupervised oriented normal estimator that is robust to noise, outliers and density variations. Our excellent results on widely used benchmarks demonstrate that our method can learn more accurate normals for both unoriented and oriented normal estimation tasks than the latest methods. The source code and pre-trained model are publicly available.

:e Unsupervised Point Normal Estimation by Learning Neural Gradient Function
Qing Li!
Huifang Feng² Kanle Shi Yue Gao
Yi Fang"
Yu-Shen Liu Zhizhong Han School of Software, Tsinghua University, Beijing, China
2 School of Informaties, Xiamen University, Xiamen, China
"Kuaishou Technology. Beijing, China
"Center for Artificial Intelligence and Robotics, New York University Abu Dhabi, Abu Dhabi, UAE
DDepartment of Computer Science, Wayne State University, Detroit, USA www.s/tguu.eeu.....m. edu. fenghuifang@ste xmu edu. cn yfang@nyu. edu h312h@wayne.
{leoqli, gaoyue, Stttighhua
Abstract
Normal estimation for 3D point clouds is a fundamental task in 3D geometry processing. The state-of-the-an th-tth- methods rely on priors of fitting local surfaces learned from normal supervision. However, normal supervision in benchmarks comes from synthetic shapes and is usually not available from real scans, thereby limiting the learned priors of these methods. In addition, normal orientation consistency across shapes remains difficult to achieve without a separate post- processing procedure. resolve these issues, we propose a novel method for estimating oriented normals directly from point clouds without using ground truth normals as supervision. achieve this by introducing a new paradigm for learning neural gradient functions, which encourages the neural network to fit the input point clouds and yield unit-norm gradients at the points. Specifically, we introduce loss functions to facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning global surface representation of the data. Meanwhile, approximation to measure the minimum signed deviation of queries, resulting in a consistent gradient field associated with the surface. These techniques lead deep unsupervised oriented normal estimator that is robust to noise, outliers and density variations. Our excellent results widely used benchmarks demonstrate that our method can learn more accurate normals for both unoriented and oriented normal estimation tasks than the latest methods. The source code and pre-trained model are available at httpps:/ggthhb..omm/ Loo.oo/ooooo/LLO0/////
incorporate gradients into the surface
Introduction
Normal vectors are one of the local descriptors of point clouds and can offer additional geometric information for many downstream applications, such as denoising 770
segmentation [63] 651 and registration 662 68). Moreover, some computer vision tasks require the normals to have
consistent orientations, i.e., oriented normals, such as graphics rendering [0 21 61 and surface reconstruction [26 31 Therefore, point cloud normal estimation has been important research topic for long time. However, progress in this area has plateaued
some time, until the recent intro- duction of deep learning. Currently, growing body of work shows that performance improvements can be achieved using data-driven approaches [24 40 39], and these learning-basse based methods often
Corresponning author
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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(a)
(b)
Figure 1: (a) Existing surface fitting-base based normal estimation methods can only produce unoriented normals (arrows) with inconsistent orientations. (b) Our method can estimate oriented normals (arrows) from unevenly sampled point clouds (black dots) with varying noise levels. The color of the arrow indicates the angle error compared to the ground truth.
give better results than traditional data- iata-indeeendent methods. However, they suffer from shortcomings in terms of robustness to various data and computational efficiency [36]. Besides, they usually rely on a large amount of labeled training data and network parameters to learn high-dimensional features. More importantly. the presence of varying noise levels, uneven sampling densities, and various complex geometries poses challenges for accurately estimating oriented normals from point clouds.
Most existing point cloud normal estimation methods aim to locally fit geometric surfaces and compute normals from the fitted surfaces. Recent surface fitting-based methods, such as DeepFit (8), AdaFit [86] and GraphFit [38] generalize the n-jet surface model [ to learning- -based frameworks that predict pointwise weights using deep neural networks. They obtain surface normals by solving the weighted least squares polynomial fitting problem. However, as shown Fig. [aa, their methods can not guarantee consistent orientations very well, since n-jet only fits a local surface, which is ambiguous for determining orientation. Furthermore, their methods need to construct local patches based on each query point for surface fitting. This results in the algorithm needing to traverse all patches in the point cloud, which time-consuming. Most importantly, they rely on the priors learned from normal supervision to fit local surfaces, while such supervision is not always available.
we introduce neural gradient functions (NeuralGF) to approximate the global surface representation and constrain the gradient during the approximation. Our method includes two basic principles: (1) We formulate the neural gradient learning as iteratively learning an implicit surface representation directly from the entire point cloud for global surface fitting. (2) We constrain the derived surface gradient vector accurately describe the local distribution and make it consistent across iterations. Accordingly. we implement these two principles using the designed loss functions, resulting model that can provide accurate normals with consistent orientations from the entire point cloud (see Fig. (b)) through a single forward pass, and does not require ground truth labels or post-processing Our method improves the state-of-the-ae results while using much fewer parameters. To summarize, main contributions include:
Inspired by neural implicit representations 6 46 49 16,,
Introducing a new paradigm of learning neural gradient functions. building on implicit geometric representations, for computing consistent gradients directly from raw data without ground truth.
Implementing the paradigm with the losses designed to train neural networks to fit global surfaces and estimate oriented normals with consistent orientations in an end-to-end manner.
Reporting the state-of-the-an performance for both unoriented and oriented normal estimation on point clouds with noise, density variations, and complex geometries.
2 Related Work
Unoriented Normal Estimation. Point cloud normal estimation has been extensively studied over the past decades and widely used many downstream applications. In general, most of these studies focus on estimating unoriented normals, i.e., finding more accurate perpendiculars of local surfaces.
The Principle Component Analysis (PCA) (6] is the most commonly used algorithm that has been widely applied to various geometrie processing tasks. Later, PCA variants
[] 59 54 35 27). Voronoi

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diagrams 1 [1] and Hough transform [1] have been proposed to improve the performance by preserving the shape details or sharp features. Moreover, some methods [37 [5
23 36
are designed to fit complex surfaces by containing more neighboring points, such as moving least
squares (37, truncated Taylor expansion (n-jet) [TS] and spherical surface fitting 23 These methods usually suffer from the parameters tuning, noise and outliers. To achieve more robust capability, they usually choose a larger neighborhood size but over-smooth the details. In recent years, deep learning-based normal estimation methods have been proposed and achieve great success. The regression-bast based methods predict normals from structured data or raw point clouds. Specifically, some methods [22 666 43 transform the 3D points into structured data, such 2D grid representations.
39 focus on the direct regression for unstructured point clouds. In contrast to regressing the normal vectors directly, the surface fitting-based methods aim to train a neural network to predict pointwise weights, then the normals solved through weighted plane fitting (36 14 or weighted polynomial surface fitting 8 86 84 80, (8) on local neighborhoods. Although these deep learning- based methods perform better than traditional data-independent methods, they require ground truth normals as supervision. In this work, we propose to use networks to estimate normals in an unsupervised manner.
Inspired by PointNet [69], other methods [24] 83 25
9 81 82
40, 41
19
Consistent Normal Orientation. Estimating normals with consistent orientations is also important and has been widely studied. The most popular and simplest way for normal orientation is based on local propagation, where the orientation of a seed point is diffused to the adjacent points by a Minimum Spanning Tree (MST). Most existing normal orientation approaches are based on this strategy, such the pioneering work (26) and its improved methods [34 69 72 67 76 29). More recent work [53] integrates a neural network to learn oriented normals within a single local patch and introduces a dipole propagation strategy across the partitioned patches achieve global consistency. These propagation- based approaches usually assume smoothness and suffer from local error propagation and the choice of neighborhood size. Since local consistency is usually not enough to achieve robust orientations across different inputs, some other methods determine the normal orientation by applying various volumetric representation techniques, such as signed distance functions [50] 551. variational formulations [71 2 28 visibility [30 17). active contours [75), isovalue constraints (74) and manding-number field
In addition to the above approaches, the most recent researches (24 25 773 399 aim to learn oriented normals directly from point clouds in a data-driven manner. Similar to the unoriented normal estimation task, these deep learning- based methods perform better than traditional methods, but they require expensive ground truth as supervision. contrast, our method achieves better performance in an unsupervised manner.
Neural Implicit Representation. Recent works 152
57 propose to learn neural implicit representa- tion from point clouds. Later, one class of methods 6 7
47 48 focuses on training neural networks to overfit a single point cloud by introducing newly designed priors. SAL 66] uses unsigned regression to find signed local minima, thereby producing useful implicit representations directly from raw 3D data. IGR (22) proposes an implicit geometric regularization to learn smooth zero level set surfaces. Neural-Pull [46] uses the signed distance to move a point along the gradient for finding its nearest path to the surface. SAP [60] proposes a differentiable Poisson solver to represent the surface as an oriented point cloud at the zero level set of an indicator function. These methods focus on accurately locating the position of the zero iso-surface of an implicit function to extract the shape surface. However, they ignore the constraints on the gradient of the function during optimizing the neural network. We know that the gradient determines the direction of function convergence, and the gradient of the iso-surface can be used as the normal of the extracted surface. If the gradient can be guided reasonably, the convergence process can be more robust and efficient, avoiding local extremum caused by noise or outliers. Motivated by this, we try to incorporate neural gradients into implicit function learning to achieve accurate oriented normals.
46
60
85
3 Method
Preliminary. Neural implicit representations are widely used in representing 3D geometry and have
achieved promising performance in surface reconstruction. Benefiting from the adaptability and approximation capabilities of deep neural networks (5), recent approaches (57 22 learn signed distance fields as a mapping from 3D coordinates æ to signed distances. These approaches
46 48 use them to
represent surfaces as zero level sets of implicit functions/ f. i.e., S = (a E R² [[ 0) 0}. where
f: 123 + is network with parameter 0, which is usually implemented as multi layer perceptron
3

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Optimization
Moving Operation
Normal Estimation
12,
f(z
0)
f
x
V
10.
Input
Output
(imsidle)
(inside)
(imidee)
fax;0)
Surface
Surface
Surface
Vf
Moving Operation
1000
Operation
Maving
Step
Step
(P n)
(outside)
(outside)
(outside)
Overview of our method. The optimization is formulated as an iterative moving operation of the positions of point set GQ, which is sampled from the raw data P. while the normal estimation is a single forward differentiation of the learned function f(z; f) using P as input. We move the input set {Q, G} through multiple steps using signed distance fü and gradient f,, learned from f2; 0) to obtain the output set {Q', G'). }. The gradient initialized to point outside the surface.
Figure 2:
(MLP). If the function f is continuous and differentiable, the normal of a point p on the surface can be represented as np = V fp)) f f())l, where ||-1| denotes the euclidean -nom and V f indicates the gradient at p. which can be obtained in the differentiation process of f. Specifically, there are some important properties of the gradient: (1) The maximum rate of change of the function defined by the magnitude of the gradient [| and given by the direction of Vf. (2) The gradient vector (p) is perpendicular to the level surface fp)
We aim to train neural network with parameter 0 from input points æ to obtain the gradient f during the optimization of f. The optimization encourages / to fit æ and the gradient f to approach the normal of the implicit surface. We sample query point set Q = {q: ما and a surface point set G= {وy No from raw point cloud P. Specifically, Q is sampled through some probability distribution from P 13 (46.. To explicitly indicate the underlying geometry, we sample the point set G CC P that may be distorted by noise, and verify that it can facilitate the learning of f (see Sec. 4.2). As shown in Fig. 2 our optimization pipeline is formulated as an iterative moving operation of the union of points Q and G, more specifically, with I steps of position movement. During normal estimation, the entire point cloud P is fed into the network to derive the gradient embedded in the learned function. To learn the neural gradient function based on the implicit surface representation of f. we consider a loss of the form
Overview.
((0) = EEND (z; V(x)) + (F(((() F(x)))
(I)
where T: RR R is a differentiable similarity function, and adopt the standard 1.2 (Fuclidean) distance. Vx:; 0) and F(x;;) 0) represent the approximated gradient vector and global surface representation of the underlying geometry P. respectively. V(x) and F(x) denote the corresponding measures, respectively. We will introduce them in the following two sections. À >> 0 is a weighting factor. The expectation E is made for certain probability distribution a D in 3D space.
3.1 Learning Global Surface
The nejet surface model represents a polynomial function that maps points (z,z) ER? to their height that is not in the tangent space over a local surface [8] 86 38). In contrast, we aim to learn an implicit global surface representation of the point cloud. The gradient indicates the direction in 3D space in which the signed distance from the surface increases the fastest, moving a point along the gradient will find its nearest path to the surface. According to this property, we adopt a moving operation introduced in [46] to project a query point to q' and o' q l(q: 0) ng. n, fg: /) [g: 0)ll- We also use f to denote the normalized gradient for simplification. Q, we expect the function ff could provide correct signed distance f(g; 0) and gradient V 0) that can be used move g the nearest point q over the surface, and we minimize the error || - dll. In this way. the function ff can be used as an implicit representation of the underlying surface and the zero level set of will be valid manifold describing the point cloud. In
For the point

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Ca(0)= = Be.BBED ( -00 C - (Q',
this work, the moving operation for input point set {Q, G} is formulated as
Σ F. V.. (2)
where {Q', G'} is the position of point set {Q, G} after moving in multiple steps I (see Fig. 2 and {2, G} is the target position. Si represents the signed distance of points during the i-th position movement step. In practice, we find the nearest point of Q and G from the raw point cloud P the target position. Different from previous works 48). we introduce multi-step movement strategy to optimize the point set {Q', to cover the surface instead of single-step movement. The insight of our design is that it is always difficult to directly reach the target in one step if some points are distributed far from the surface or distorted by noise. At this time, the optimization of the moving operation may go through many twists and turns to converge.
We optimize La) in an end-to-end manner based the moved point set {Q', G'). For the points G C P in the first step, we expect them to be on the underlying surface. For the points {Q', G'} moved after the first step, we expect them be close to the underlying surface possible. Hence, we leverage a regression loss term on the predicted distances f. and f.f of point set {Q, G2, i.e.
(r) (19)2 + ESi) and fi 199
2=2
3.2 Learning Consistent Gradient
standard way for estimating unoriented point normals is to a plane to the local neighborhood N(p, P) of each point p [77 Given a neighborhood size K, we let V(p) E R denote the set of centered coordinates of points that neighborhood. Then the plane fitting described as finding the least squares solution of
K Σ
K Σ
1
arg min n::
I--- V(p)xl? and V(p); = Ph-
K Pt. Pk. E P).
(4)
P
However, the inner product in the formula allows the distribution of normals on both sides of the surface (positive or negative orientations) to satisfy the optimization. Therefore, this approach cannot determine the normal orientation with respect to the surface, and a consistent orientation needs to be solved by post-processing, such as MST-based orientation propagation. In this work, we directly estimate oriented normals by incorporating neural gradients into implicit function learning 7..
(0) = E. T
(TV (V( 0), V(=)) [1 0) n V(=))l+
where n the unit gradient (normal) of each point on the surface. The above formula aims to measure the minimum deviation of each point from the underlying surface. Specifically, the gradient is perpendicular to the surface and the distance from the surface increases the fastest along the direction of the gradient. Therefore, we measure the deviation from P to Q by finding the optimal gradients. Generally, raw point clouds may contain noise and outliers that can severely degrade the accuracy of the surface approximation. To reduce the impact of inaccuracy in P. we introduce multi-scale neighborhood size (K, J.n for V(x) based statistical manner, that is
K. Σ Dh: Ds E (N, P), == 1, .K.
V.(4)
Kg
where Nk, (q. P) denotes the K₂ nearest points of q Q in P. V(q) is built as vector from the averaged position a z= , NK. ()/P) K, on surface to query q. Then, the objective of Eq. 5 is tumndd into the aggregation of the error each neighborhood size scale, i.e.,
Nx
V(x)) = Σ 1 v. -v(9)
(. = Ea
where ;= ... I is the number of steps. In practice, we only use Si of the first step and its V to measure the deviation, since we want the optimization to converge as fast as possible, which also implies that the position Q' after the first move step in Eq. (2) is as close to the surface as possible.
5

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PCA+MST
PCA+ODP
PCA-SNO
SHS-Net
AdaFit-MST
AdaFit-ODP
Sda
PCPNet
HSurf-NetMMST
HSurf-Ne-ODD
Ours
Figure 3:
Visual comparison of oriented normals. We only show a partial cross-section of closed shape for better visualization. The color of the arrow indicates RMSE, where purple means the same orientation the ground truth and red means the opposite.
Table 1:
RMSE of oriented normals on datasets PCPNet and FamousShape. Our method achieves SOTA performance even when compared to supervised baselines. means the code is uncompleted.
PCPNet Dataset
FamousShape Dataset
Category
Noise
Density Gradient
Naise
Density
Nome
0.12%
0.6%
1.2%
Stripe
Average
Nume
0.12%
0.6%
1.2%
Stripe
Gradient
Average
PCA
26
MMST
35
19.05
30.20
31.76
39.64
27.11
23
28.52
35.88
41.67
38.09
60.16
31.69
35.40
40.48
PCA
SNNO
18.55
25.46
26.52
32.25
39.39
61.91
PCA
26
+DDP
33
28.96
86
34.91
51.52
28.70
23.00
32.
30.47
31.29
41.65
84.00
39.41
30.72
42.92
LRR
MMST
43,48
47.58
38.58
44.08
48.45
46.77
44.82
56.24
57.38
45.73
64.63
66.35
56.65
$7.83
LRR
+NN
44.87
43.45
33.46
45.40
46.96
37.73
41.98
59.78
60.18
45.02
71.37
62.78
59.90
59.84
LRR
79
HODP
53
28.65
25.83
36.11
53.89
26.41
23.72
32.44
39.97
42.17
48.29
88,
+4.92
47.56
51.93
AdaFit
+MST
27.67
43.69
54.39
36
40.46
41.87
43.12
39.33
62.28
60.27
45.57
42.00
AdaFir
56
+SNO
67
26.41
24.17
40.31
48.76
27.74
31 11.56
33. 16.16
27.55
37.60
69.56
62.77
27.86
29.19
42.42
53
26.37
39.19
72.91
HSurf-Net
40
+MST 26
29.82
44.49
50.47
55.47
40.54
43.15
43.99
54,02
42.67
68.37
65.91
52.52
53.96
56.24
HSurf-Net
+SNO
30.34
33,46
40.49
38.
62.04
HSurf-Net
40
+0DP
13
26.91
24.85
35.87
51.75
26.91
20.16
31.07
43.77
43.74
46.91
72.70
45.09
43.98
49.37
PCPNet
24
33.34
34.22
40.54
44.46
37.95
35.44
37.66
40.51
41.09
46.67
54.36
40.54
44.26
44.57
DPGO
73
23.79
25.19
35.66
43.89
28,99
29.33
31.14
SHS-Not
30
10.28
13.23
25
25.40
35.51
16.40
17.92
19.79
21.63
25.96
41.14
52.67
26.39
28.97
32.79
Ours
10.60
24.76
33.45
18.70
19.28
36.22
50.27
17.23
17.38
In addition, we pursue better gradient consistency across iterations, i.e., make the gradient directions in each move step parallel. To achieve this, we align the gradient Vfs after the first step with the gradient V of the first step. Meanwhile, we also consider the confidence of each point with respect to the underlying surface. Thus, we introduce confidence- -eelddeeee-weghted cosine distance to evaluate the gradient consistency at points {Q, G), with the form of
Leon (0) = -= w. (1- (V,, VfA), and V (VS.VV.)
(8)
where (.) denotes cosine distance vectors. w =xexp(--> lfill) is an adaptive weight that indicates the significance of each imput point based on the predicted distance, making the model pay more attention to points near the surface. In summary, our method combines an implicit representation and gradient approximation with respect the underlying geometry of P. We not only consider the gradient consistency at q during iterations, but also unify the adjacent regions of q to compensate for the inaccuracy of P. This is important because the input point cloud may be noisy and some points are not on the underlying surface.
Loss Function. We optimize global surface approximation and consistent gradient learning in an end-to-end manner by adjusting the parameter of f using the following loss function
((0) (0) + Cc + (z) + >zCrey (
where X; 0.01, 0.1 and A 10 balance weights and are fixed across all experiments.
Neural Gradient Computation. The above losses require incorporating the gradient (a 0) in differentiable manner. As in 222 construct V (x; 0) as neural network in conjunction with

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PCA-MST
PCA+ODP
PCA+SNO
SHS-Net
PCA+AIST
PCA+ODP
PCA+SNO
SHS-NNt
AdaFieMMST
AdaFit-OPP
AdaFic-SNO
AdaFit-ODP
AdaFit-SNO
HSurt-NetMMST
HSurf-NoHOPP
HSu
Ours
HSurf-NeHMSST
HSurfNNeHOP
HSurGNorSNO
Ours
Figure 4:
Visual comparison of oriented normals point clouds with density variations and complex geometries. Point color indicates normal RMSE and red (180%) indicates reversed orientation.
Table 2:
RMSE of unoriented normal on datasets PCPNet and FamousShape. Our method achieves SOTA performance compared to unsupervised baseline methods.
PCPNet Dataset
FamousShape Dataset
Category
Noise
Density Stripe Gradient
Noise
Density Stripe Gradient
None 0.12%
0.6%
1.2%
Average
None
0.12%
0.6%
1.2%
Average
Supervised
PCPNet
9.64
22.84
11.73
14.58
24.95
Nesti-Net 9
7.06
10.24
17.77
22.31
8.64
8.95
12.49
11.60
16.80
31.61
39.22
12.33
11.77
20.55
21.96
7.73
11.84
DeepFit 8
6.51
9.21
16.73
23.12
7.92
7.31
11.80
11.21
16.39
29.84
39.95
11.84
10.54
19.96
7hang al.
5.65
9.19
16.78
22,93
6.68
6.29
11.25
9.83
16.13
29.81
39.81
9.72
9.19
19.08
AdaFit
5.19
9.05
16.45
21.94
6.01
5.90
10.76
15.78
29.78
38.74
8.52
8.57
18.41
GraphFit 58
5.21
8.96
16.12
21.71
6,30
5.86
10.69
8.91
15.73
29.37
38.67
9.10
8.62
18:40
NeAF
21.74
4.89
15.67
7.47
HSurf-Net
40
4.17
8.78
16.25
21.61
4.98
4.86
10.11
7.59
15.64
29.43
38.54
7.63
7.40
17.70
Du el
19
3.85
8.67
16.11
21.75
4.78
9.96
6.92
15.05
29.49
38.73
7.19
6.92
17.38
SI1S Nei
39
3.95
8.55
16.13
21.53
4.91
4.67
9.96
7.41
15.34
29.33
38.56
7.74
7.28
17.61
Unsupervised
Boulch of af. [
11.80
11.68
22.42
35.15
13.71
12.38
17.86
19.00
19.60
36.71
50.41
20.20
17.84
27.29
PCV
16.76
27.02
Jet
[1
12.35
12.84
18.33
27.68
13.39
13.13
16.29
20.11
20.57
31.34
45.19
18.82
18.69
25.79
PCA
27.52
LRR
9.63
11.31
20.53
32.53
10.42
10.02
15.74
17.68
19.32
33.89
49.84
16.73
16.33
25.63
Ours
7.89
9.85
18.62
24.89
9.21
9.29
13.29
13.74
16.51
31.05
40.68
13.95
13.17
21.52
fx;; 0). Specifically, each layer of the network has the form -=+==00000
==:==<WWW <WW ff b). where 0 0= (W, b)
is the learnable parameter of l layer with output y and Φ R + R is a non-linear derivable activation function. According to chain-rule, the gradient is given by 22
V +1 diag (o' (W of +6) WVy',
(10)
where ø' is the derivative of é, and diag(z) arrange its input vector T the diagonal of square matrix mm practice, we use the automatic differentiation module of PyTorch [88] to implement the computation of the gradient V fæ;; 0) in the forward pass.
4 Experiments
Architecture. To learn the neural gradient function, we use a simple neural network similar to 6 46 22]. It is composed of eight linear layers and a skip connection from the input to the intermediate layer. Except for the last layer, each linear layer contains 512 hidden units and is equipped with ReLU activation function. The last layer outputs the signed distance for each point. The parameters of the linear layer are initialized using the geometric initialization 6). After optimization, we use the network to derive pointwise gradients from the data P (see Fig.

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No Noise
Low Noise
Medium Noise
Distribution D. We make the distribution D concentrate in the neigh- borhood of in 3D space. Specifically, it is set by uniform sam- pling points p from P and placing an isotropic Gaussian Npp o²) cen- tered at each p. with stan- dard deviation parame- ter a is adaptively set to the distance from the 1- th nearest point to [ 71. We set I to 25 for the PCPNet dataset and 50 for the FamousShape dataset.
0.
POP
0.7
High
Deusity: Strige
Density: Gradient
1.0
PGP
PCA NS7
0.5
1.0
PM
Implementation. We choose the neighbor- hood scale set {KK)) Nk {1, K/2, K), K = 8, and set the parameter p of adaptive weight to 60. We select No 5000 points from the distribution DD and NG ftc=2000 2500 points from P to form the input [Q, G). The number of steps is set to 2 for all evaluations. As for metric, we use normal angle Root Mean Squared Error (RMSE) to evaluate the estimated normals and Percentage of Good Points (PGP) to show the normal error distribution 40 39
0.6
Oku
(brocholSddegree)
(ddgme)l
(irereoldddegrre)
Figure The PGP curves of oriented normal on the FamousShape dataset. is plotted by the percentage of good point normals (PGP) for a given angle threshold. Our method has the best value at large thresholds.
Point Cloud
Neural-Pull
CAP-UDF
OSP
PCP
Ours
Low
Medium
High
Figure 6: Comparison with surface reconstruction methods using point movement strategy, e-g., Neural-Pull [46), CAP-UDF [85), OSP [47] and PCP [48). The input point clouds have different levels of noise (low, medium and high).
4.1 Performance Evaluation
Evaluation of Oriented Normal. In this evaluation, we not only compare with one-stage base- line methods, including PCPNet (24]. DPGO [3] and SHS-Net [39], but also compare with two- stage baseline methods that combine representative algorithms based on different design philoso- phies. Specifically, we use the various combinations of unoriented normal estimation methods (PCA (26), AdaFit [86] and HSurf-Net (40) and normal orientation methods (MST [26]. SNO [7] and ODP (53). Note that PCPNet, DPGO, SHS-Net, AdaFit and HSurf-Net rely on supervised training with ground truth normals. We report quantitative evaluation results on datasets PCPNet and FamousShape (39 in T Our method has better performance under the vast majority of
data categories (noise levels and density variations), and achieves the best average results. It is worth
8

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Point Cloud
MST
ODP
SNO
PCPNet
StES-Net
GCNO
Ours
The reconstructed surfaces on wireframe point clouds with sparse and non-uniform sampling. There are less than 1000 points in each input point cloud. The initial normals of MST, ODP and SNO are provided by PCA. GCNO 777 performs similarly to our method.
Figure 7:
mentioning that method achieves a large performance improvement the FamousShape dataset, which contains different shapes with more complex geometries than the PCPNet dataset. We provide visual comparison of the estimated normals in Fig- which shows normal vectors of a column section with a gear-like structure. In Fig. we show the normal error map on point clouds with uneven sampling and mulli--banchh structure. The normal error distribution shown in Fig. S It is clear that our unsupervised method has advantages over baselines in all categories when the angle threshold larger than 50%. The evaluation results demonstrate that our method can consistently provide good oriented normals in the presence of various complex structures and density variations.
Evaluation of Unoriented Normal. In general, estimating unoriented normals is easier than estimating oriented normals as it does not need to explore more information to determine the orientation, but only focuses on finding the perpendicular of local plane surface according to the input patch. Moreover, existing deep learning- methods for unoriented normal estimation all rely on supervised training with ground truth normals. In contrast, our method is designed for estimating oriented normal in unsupervised manner. To evaluate the unoriented normals, we compare with the baselines using our oriented normals and ignore the orientation of normals. In Table 2 we report the quantitative evaluation results of supervised and unsupervised methods on datasets PCPNet and FamousShape. We can see that, compared unsupervised methods, our method achieves significant performance gains on almost all data categories of these two datasets and has the best average results. The supervised methods learn from ground truth normals, and their supervision provides perfect surface perpendiculars, leading to superior results. Our better results than PCPNet highlight the strong learning ability of our method without labels.
Surface Reconstruction. For surface reconstruction from point clouds, we compare our method with the latest methods in two different ways. (1) We determine the zero level set of the learned implicit function f via signed distances, and use the marching cubes algo- rithm to extract the global surface representation of the point cloud. A visual comparison of the extracted surfaces on point clouds with different noise levels is shown in Fig. 6 where our surfaces are much cleaner and more complete under the interference of noise. (2)
Table 3: Comparison of oriented normal estimation. We use implicit representation methods to estimate oriented normals.
Noise Density
Average
IGR
22
54.77
75.90
65.33
SAP 60
57.56
41.32
49.44
SAL 6
46.69
43.78
45.24
Neural-Pull 44
48.48
26,22
37.35
Ours
36.92
26.08
31.50
Based on the estimated oriented normals, we employ
the Poisson reconstruction algorithm [33] to generate surfaces from wireframe point clouds, and the
results in Fig. 7 show that our method can handle extremely sparse and uneven data. The reconstructed surfaces unsupervised methods on LiDAR point clouds of the KITTI dataset (20] are shown in
Fig. 8 where our undistorted surface exhibits more realistic road scene of real-world data.
Moreover, we repurpose some implicit representation methods, which are designed for surface reconstruction, to estimate oriented normals from point clouds. The comparison of normal RMSE on some selected point clouds in the datasets PCPNet and FamousShape reported Table 3 A visual comparison of the reconstructed surfaces on noisy point clouds is shown in the supplementary material. We can see that our method has clear advantages, especially on noisy point clouds. We provide more evaluation results different data in the supplementary material.

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Table 4: Ablation studies for unoriented and oriented normal estimation on the FamousShape dataset.
Unoriented Normal
Oriented Normal
Category
Noise
Density
Noise
Density Stipe
None
0.12%
0.6%
1.2%
Stripe
Gradient
Average
None
0.12%
0.6%
1.2%
Gradient
Average
w/o Coeg
13.76
16.45
31.32
42.56
13.72
12.87
21.78
16.63
19.15
36,77
55.22
16.89
16.35
85.84
w/o La
13.98
16.66
30.88
40.70
14.01
13.30
21.59
16.74
19.30
35.72
49.98
17.21
17.39
26.06
w/o Loom
13.93
17.13
33.81
45.95
13.95
13.02
22.97
23.97
26.70
40.49
69.06
23.26
17.87
33.56
(8)
win Co
28.41
31.34
49.04
49.65
30.60
26.47
35.92
46.97
43,00
73.64
75.61
43.10
35.33
52.94
w/o m
13.76
16.56
31.45
41.61
14.00
12.83
21.70
16.81
19.40
37.06
52.97
18.12
16.11
26.74
13.44
16.14
31.35
41.80
13.71
12.90
21.56
16.20
18.99
36.66
51.47
17.61
18.60
26.59
(b)
w/o G
13.86
16.73
31.10
40.73
14.02
13,30
21.62
16.99
21 26.66
36,70
50.30
17.37
20.35
27.23
I
14.03
17.05
45.97 33.42
13.94
13.36
22.96
23.39
26.34
40.10 63.89
19.47
27.89
33.31
(c)
7.
13.94
26.07
(1)
14.11
17.25
31.62
41.36
13.72
12.85
21.82
23.87
23.14
36.47
53.32
17.82
16.35
28.50
[
14.31
16.67
30.43
40.12
14.45
13.37
21.56
17.26
19.36
35.34
49.13
18.33
16.88
26.05
(d)
(1,K)
13.72
16.62
30.77
40.28
13.92
13.06
21.39
16.54
19.40
35.65
51.98
17.02
16.55
26.19
13.60
17.34
30,90
40.43
13.70
12.56
21.42
16.66
23.60
36.17
49.44
16.86
14.72
26.24
K =
16
14.32
16.69
30.86
41.25
14.40
13.40
21.82
17.26
19.42
35.85
50.98
17.43
17.17
26.35
32
15.02
17.18
30.61
41.15
15.44
14.26
22.28
17.82
19.79
35.70
50.64
19.28
18.22
26.91
Final
13.74
31.05
40.08
13.95
13.17
21.52
16.57
19.28
36.22
30.27
17.23
17.38
26.16
4.2
Ablation Studies
We aim to achieve the best performance on average in both oriented and unoriented nor- mal estimation. We provide the ablation re- sults the FamousShape dataset in Table (d), which are discussed as follows.
PCA-ODP
PCA+SNO
(a) Loss. We do not use one of the four loss terms Eq. (9) and the weight w in Eq. (8). respectively. We also try incorporating the weight w into (0) Eq. 7 These ab- lations do not provide better performance in both oriented and unoriented normal estima- tion. We observe that the performance drops a lot without using L. (0). and that Ceon performance. Note that (,) plays a more important role on the PCPNet dataset than the FamousShape dataset our experiments.
PCA-MST
Ours
has larger impact
Figure 8:
The reconstructed surfaces on the KITTI dataset using estimated normals of unsupervised meth- ods. Points are colored with their heights.
(b) Input. We do not sample subset G from P. and take Q as the input instead of {Q, G}. The results verify the positive effect G on explicitly indicating the surface and improving performance.
(c) Iteration. We set the number of steps I to and 3, respectively. More iterations bring better performance, but we still choose I= after weighing the running time and memory usage.
(d) Scale Set. The scale set [K, NK in Eq. (6) is chosen to be {1. K/2, K) K= = in implemen- tation. Here we change the scale set to different sizes, e-g., {1}, [K} and (1, K), K=8, or different values of K, egg., 16 and 32. Some of these settings have advantages in single evaluation, but do not give better results for both oriented and unoriented normal estimation, such as {K} and K).
5 Conclusion
In this work, we propose to learn neural gradient functions from point clouds to estimate oriented normals without requiring ground truth normals. Specifically, we introduce loss functions facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning global surface representation of the data. Meanwhile, we incorporate gradients into the surface approximation to measure the minimum signed deviation of queries, resulting in consistent gradient field associated with the surface. Extensive evaluation and ablation experiments are provided, and our excellent results in both unoriented and oriented normal estimation demonstrate the effectiveness of innovations our design. Future work includes adding more local constraints to further improve the accuracy of normals.
10

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6 Acknowledgement
This work was supported by National Key R&D Program of China (2022YF33800000, the National Natural Science Foundation of China (62272263, 62072268), and in part by Saingua--Kaishor Institute of Future Media Data.
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[81] Haoran Zhou, Honghua Chen, Yidan Feng. Qiong Wang. Jing Qin, Haoran Xie. Fu Lee Wang. Mingqiang Wei, and Jun Wang. Geometry and learning co-supported normal estimation unstructured point cloud. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pages 19238-13247, 2020.
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[84] Jun Zhou, Wei Jin, Mingjie Wang. Xiuping Liu, Zhiyang Li. and Zhaobin Liu. Improvement of normal estimation point clouds via simplifying surface fitting. Computer-Aided Design, page 103533, 2023, [85] Junsheng Zhou, Baorui Ma, Yu-Shen Liu, Yi Fang, and Zhizhong Han. Learning consistency-awarr unsigned distance functions progressively from raw point clouds. Advances Neural Information Processing Systems, 35: 11441-1169944 2022.
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AdaFit: Rethinking learning-
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Reviews:
[{"review_id": "D10oXt1W5C", "reviewer": "Reviewer_HD1H", "summary": "The paper proposes a method that predicts oriented surface normals given a 3D point cloud. The method is based on SDF surface reconstruction. Given a potentially noisy point cloud, a SDF is fitted using an approach similar to Neural-Pull [45]. The surface normal at each point can then be computed based on the gradient of the SDF. Since a global SDF of the shape is recovered, the orientation of the predicted surface normal is also globally consistent. Additionally, the paper proposed various improvements on top of Neural-Pull to improve normal prediction quality: a multi-step movement strategy during optimization, and a multi-scale neighborhood size strategy.", "questions": "* How does the proposed method compare with previous works in terms of speed?\n* It is possible to directly repurpose surface reconstruction methods for normal estimation? How would they perform?", "limitations": "The limitation and societal impact of the paper is adequately addressed in the supplemental material.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The proposed method is unsupervised -- It is optimized on a per-shape basis and unlike [23,8,39,38], it does not need to be trained on a large collection of shapes, which also avoids any training-evaluation domain gap.\n* Unlike local fitting methods [8,82,37,15], the proposed method produces globally consistent surface orientation thanks to the use of a global SDF.\n* The proposed method achieved state-of-the-art performance on both oriented and unoriented surface normal estimation. It worked well especially for noisy point clouds.\n* The paper has included very comprehensive ablations to show the effects of various design decisions as well as the new components such as multi-step supervision and multi-scale neighbor selection.", "weaknesses": "* The proposed method resembles many existing works on point cloud 3D reconstruction, such as SAL [6], Neural-Pull [45], Shape As Points (Peng et al.), NDF (Chibane et al.). In fact, recovering the normal can be considered as the side effect of surface reconstruction -- once the oriented surface is obtained, the oriented normal can be obtained naturally. As there is no comparison on surface normal quality with such methods in the paper, it is not clear if the proposed method has any significant benefit over these methods from  surface reconstruction community.\n* Compared to feed-forward methods such as PCPNet [23], the proposed method requires per-shape optimization, which can be computationally expensive.\n* The paper will be easier to follow if it can devote some paragraphs to the connections between state-of-the-art surface reconstruction and normal estimation literature.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### 1. Comparing with other implicit representation methods and repurposing surface reconstruction methods for normal estimation.\n\nWe use some implicit representation methods to estimate oriented normals.\nThe comparison of normal RMSE on point clouds in the datasets PCPNet and FamousShape is reported in the following table and the visual comparison is shown in *Rebuttal PDF Fig.6*.\nWe can see that our method has clear advantages.\n\nWe also have compared with some other surface reconstruction methods in Fig.6 of the paper and Fig.1 of the supplementary material.\nOur method can reconstruct better surfaces, especially on noisy point clouds.\nMoreover, our method can handle point clouds with uneven sampling and open surface structure, such as the KITTI dataset in Fig.7.\n\n|Method  |SAP [1]     |IGR [2]     |SAL [3]     |Neural-Pull [4] |Ours       |\n|:-:     |:-:         |:-:         |:-:         |:-:             |:-:        |\n|Noise   |57.56       |54.77       |46.69       |48.48           |***36.92***|\n|Density |41.32       |75.90       |43.78       |26.22           |***26.08***|\n|Average |49.44       |65.33       |45.24       |37.35           |***31.50***|\n\n[1] Peng et al., Shape As Points: A Differentiable Poisson Solver, NeurIPS 2021. \n\n[2] Gropp et al., Implicit Geometric Regularization for Learning Shapes, ICML 2020. \n\n[3] Atzmon et al., SAL: Sign Agnostic Learning of Shapes from Raw Data, CVPR 2020. \n\n[4] Ma et al., Neural-Pull: Learning signed distance functions from point clouds by learning to pull space onto surfaces. ICML 2021.\n\n\n### 2. The per-shape optimization is computationally expensive compared to PCPNet.\n\nExisting learning-based normal estimation methods, such as PCPNet, require ground truth normals as supervision for training, and there is still room to improve, especially in some challenging cases.\nAlthough PCPNet can predict point normal in a forward process with pre-trained parameters, which is much faster than ours, it does not generalize well to unseen cases. \nWe resolve this issue using an overfitting strategy, which significantly improves the performance on unseen cases.\nOur method directly learns normals from raw data without using ground truth and achieves better performance for various inputs.\nIn addition, our method can also provide better surface reconstruction results.\nWe do not intend to replace methods such as PCPNet that can directly use pre-trained models on different data, but rather as a new exploration that can provide another option for the 3D computer vision community to easily obtain more accurate normals and surfaces from point clouds.\n\n\n### 3. Add a paragraph to the connection between SOTA surface reconstruction and normal estimation.\n\nIn recent years, researchers have paid more attention to the global consistency of normal orientations, such as iterative Poisson Surface Reconstruction (iPSR) [Hou et al. 2022], Parametric Gauss Reconstruction (PGR) [Lin et al. 2022] and Stochastic Poisson Surface Reconstruction (SPSR) [Sellán and Jacobson 2022].\nFor example, iPSR runs Poisson reconstruction in an iterative manner and updates normals using the generated surface of the last iteration.\nPGR regards normals and surface elements in the Gauss formula as unknown parameters and optimizes the parametric function space.\nIn addition to traditional approaches, deep neural networks have been applied to gather information for orientation and reconstruction.\nSome works learn the implicit function directly from the input point cloud and eliminate the need for training data, such as SAL [Atzmon et al. 2020], Neural-Pull [Ma et al. 2021], IGR [Gropp et al. 2020] and SAP [Peng et al. 2021].\nFor example, SAP proposes a differentiable Poisson solver to represent shape surfaces as oriented point clouds, and the point positions and normals are updated during the optimization of surface.\nNeural-Pull predicts the signed distance field to move a point along or against the gradient for finding its nearest path to the surface, and its gradient is equivalent to normal.\nAlthough these surface reconstruction methods do not aim to estimate normals, they will add normals to the constraint conditions during the surface optimization process to assist surface reconstruction.\nThe experimental results of these methods also validate the benefits of using normals in surface reconstruction.\n\nWe will add more discussions in the revised version.\n\n\n### 4. Comparison of running speed.\n\nIn Section 2 and Table 1 of the **supplementary material**, we have compared the network parameters, and running time of the learning-based methods."}, {"review_id": "xHDIkRnPOG", "reviewer": "Reviewer_vh4j", "summary": "This work proposes to learn neural gradient functions from point clouds to estimate oriented normal in an unsupervised manner. Specifically, this method introduces several loss functions to constrain query points to iteratively fit the underlying surface, which is defined by the sampled points. Meanwhile, the local gradients are incorporated into the surface approximation to measure the minimum signed deviation of queries, resulting in a consistent normal field associated with the surface. Lastly, some evaluations demonstrate the superior performance of the proposed method over existing approaches. ", "questions": "1. I believe that the designed loss functions can facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning a global surface representation of the data. However, I am doubting that if the incorporate gradients can achieve a consistent normal field, especially handling some challenging cases. Is it possible to give more details?\n2. Both point sets of Q and G are sampled from input point cloud, so in the test stage, how to compute the normal for all points?\n", "limitations": "1. I do not see any failure cases in the paper. It is essential to show some failure cases for the reader to investigate the failure model to benefit future research.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "1. This paper is well organized and nicely written. The presentation of motivation is clear and smooth.\n2. The work is well motivated. The idea of learning neural gradients for normal estimation is inspiring. \n3. Some visual and quantitative evaluation are promising. \n", "weaknesses": "1. I am not sure why we need a distribution D. \n2. I do not see any running efficiency statistics, which would be important for a fair assessment.\n3. I think there is a missing comparison with [73]. Also, some challenging cases from [73] should be included.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### 1. Why need a distribution D?\n\nThe optimization of our method is formulated as an iterative moving operation of the points to the target positions.\nWe use this distribution to generate the training data, i.e., query points, which are pulled onto the surface during optimization.\nAs in Line 117, the input point set Q is sampled from the raw point cloud via distribution D.\nLine 206 describes how to construct the distribution.\n\n\n### 2. Running efficiency.\n\nIn Section 2 and Table 1 of the **supplementary material**, we have compared the network parameters, and running time of the learning-based methods.\n\n\n### 3. Comparison with GCNO and using challenging cases of GCNO [73].\n\nWe conduct an evaluation on a dataset that has the same shapes as the FamousShape dataset but each shape in this dataset contains only 5000 points.\nAs shown in the following table, we report quantitative comparison results of oriented normal estimation on this dataset with sparse point clouds.\nThe traditional baseline algorithms, including GCNO and PCA+MST, are implemented in C++ on the Windows platform and run on an Intel i9-11900K CPU.\nOther learning-based methods are implemented in Pytorch on the Linux platform and run on an NVIDIA 2080 Ti GPU.\nWe can see that our method has the best RMSE result.\nComparisons on the challenging cases of GCNO [73] and other cases are shown in ***Rebuttal PDF***.\nThese results demonstrate the good performance of our method on sparse point clouds.\n\n|Method |HSurf-Net+ODP |PCA+MST |PCPNet  |SHS-Net  |GCNO   |Ours     |\n|:-:    |:-:           |:-:     |:-:     |:-:      |:-:    |:-:      |\n|RMSE   |62.51         |45.40   |48.48   |32.64    |45.14  |**24.35**|\n\n[73] Xu et al., Globally consistent normal orientation for point clouds by regularizing the winding-number field. ACM TOG 2023.\n\n\n### 4. Doubt about incorporating gradients to achieve consistent normals.\n\nDuring the learning of global surface representation, the gradient of the signed distance field determines the convergence direction of the zero iso-surface, and the consistency of the gradient affects the quality of the final result.\nTo ensure a continuous and smooth surface, adding constraints to the gradient for optimization can improve the robustness of the algorithm for surface representation and avoid local extremum caused by noise or outliers.\n\nIn this work, we introduce the multi-step moving strategy along the gradient in Eq.(2), the gradient uniformity of multi-scale neighborhood size in Eq.(7), and the gradient consistency of multi-step moving in Eq.(8).\nThe quantization results in Table 1 and Figure 5, the noisy data in Figure 6, and the uneven data with open structure of the KITTI dataset in Figure 7 indicate that our method can obtain consistent normal under various geometric structures, noise, and non-uniform densities.\nOverall, the shape in the FamousShape dataset has a more complex geometry and topology than the PCPNet dataset, while the point clouds in the KITTI dataset are extremely uneven, sparse, and has an open surface structure.\nThe challenging cases provided in Question 3 and more cases in ***Rebuttal PDF*** show the good performance of our method.\nAt the same time, we also provided a detailed analysis of the limitation and failure cases of our method in the supplementary material.\n\n\n### 5. How to compute the normals of all points?\n\nDuring training, the point sets Q and G sampled from raw point clouds are used as input.\nAs in Line 122, during testing, the entire point cloud is fed into the trained network to derive the gradient at each point, and the solved gradient is used as the normal.\n\n\n### 6. Failure cases.\n\nThe failure cases are already provided in Section 4 of the **supplementary material**.\nOur method fails on noisy point clouds of a thin sheet with a hollow structure.\nThe integration of noisy points from the upper and lower planes leads to the blurring of the internal and external structures, and the algorithm assumes that all point clouds belong to the same plane."}, {"review_id": "PQlPVm08sM", "reviewer": "Reviewer_wJx4", "summary": "In this work the authors present an unsupervised framework for predicting globally consistent, accurate normals given a point cloud as input. The crux of the method is to predict an implicit surface representation (a signed distance function to be exact) from the input point cloud by leveraging the fact that the gradient of the Neural SDF of surface points gives the surface normal at that point. The authors distinguish their method from existing similar approaches by overcoming drawbacks like lack of global consistency by designing a loss function that penalizes incorrect signed predictions and also considers multi-scale neighborhoods. Furthermore, they consider a multi-step iterative approach for refining their surface normal estimations. They show impressive results in terms of RMSE of unoriented/oriented normals across challenging datasets across different levels of sampling densities and noise.", "questions": "1. The density-based experiments are unclear to me. Could the authors please explain what the “gradient” and “stripe” settings exactly imply? The supplementary shows a few qualitative examples but can you explain how they were generated? It also seems that supervised methods perform comparably w.r.t. the proposed method. Could the authors give an intuition for why their method suffers from density variation but not on noisy point clouds (see Table 2 and Fig. 5, especially at low threshold angles)? Which hyperparameters can be tuned to address noise vs. density variation and what is the trade-off for the same? This would help with a better understanding of the authors’ contributions.\n\n2. In the analysis of un-oriented point clouds, we can see that LRR is the second-best performing unsupervised normal estimation method. Yet, we do not see an application of LRR with techniques for introducing orientation like MST in the study on the estimation of oriented normals. Adding LRR + (some orientation method) type of methods to Table 2 and Fig. 5 would make the results a bit more stronger. Basically this can help address the question of why don't we just pick a SoTA unoriented unsupervised point cloud method and do some simple orientation method?\n", "limitations": "Limitation section in the supplementary. Potential other limitations can include shapes with open surface and shapes without volume.\n", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method achieves SoTA performance compared to other unsupervised methods on the PCPNet and FamousShape dataset. \n2. In the presence of noisy input points, their method is quite competitive even against supervised baselines which is a big plus.\n3. Accurately oriented point clouds are hugely sought-after in downstream applications like surface reconstruction and their results on single object surface reconstruction demonstrates the value of their method in this important application.\n", "weaknesses": "1. Comparison with other implicit representations. At its core, the proposed method learns a Neural SDF for the shape represented by the input point cloud and the estimated normals are simply the gradients computed from this Neural SDF using automatic differentiation. As such I believe there should be more comparison drawn to similar methods like Neural-Pull [1], SIREN [2], SAL [3], and Neuralangelo [4]. In particular, [4] uses a finite-difference-based approach to estimate the surface normal during training and I would be curious to see if a similar technique can get similar results in lesser resources for normal estimation.\n\n\n2. Hyperparameter tuning. Since the method is optimization based, one potential risk is mishandling the hyper-parameter tuning, which turns out to be very important. For example, how does the method choose the hyper-parameter for each shape? Does a different input point set require different hyper-parameters or one hyper-parameter can be used for a wide range of shapes? I think tuning hyper-parameter of a neural field fitting procedure can drastically change the performance, and arguably with proper hyper-parameter tuning, one can probably find a way to curate smooth solutions to different input instances. As a result, it's very essential for this kind of test-time optimization-based method to have a rigorous hyper-parameter tuning procedure reported to eliminate the risk of accidentally adding human judgment into producing the results. \n\nReferences:\n\n[1] Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces. https://arxiv.org/abs/2011.13495\n[2] SIREN. https://arxiv.org/abs/2006.09661\n[3] SAL. https://arxiv.org/abs/1911.10414 \n[4] Neuralangelo: High-Fidelity Neural Surface Reconstruction. https://research.nvidia.com/labs/dir/neuralangelo/\n[5] NKSR. https://research.nvidia.com/labs/toronto-ai/NKSR/ \n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### 1. Comparison with other implicit representations.\n\nWe use some implicit representation methods to estimate oriented normals.\nThe comparison of normal RMSE on point clouds in the datasets PCPNet and FamousShape is reported in the following table and the visual comparison is shown in *Rebuttal PDF Fig.6*.\nWe can see that our method has clear advantages.\n\nWe also have compared with some other surface reconstruction methods in Fig.6 of the paper and Fig.1 of the supplementary material.\nOur method can reconstruct better surfaces, especially on noisy point clouds.\nMoreover, our method can handle point clouds with uneven sampling and open surface structure, such as the KITTI dataset in Fig.7.\n\nWe will add references to the mentioned methods in the revised version.\nNeuralangelo [5] aims to recover dense 3D surfaces via image-based neural rendering and only provides the source code of a Blender addon.\nRewriting the algorithm to estimate normals from point clouds is difficult during the short rebuttal period.\n\n|Method  |SAP [1]     |IGR [2]     |SAL [3]     |Neural-Pull [4] |Ours       |\n|:-:     |:-:         |:-:         |:-:         |:-:             |:-:        |\n|Noise   |57.56       |54.77       |46.69       |48.48           |***36.92***|\n|Density |41.32       |75.90       |43.78       |26.22           |***26.08***|\n|Average |49.44       |65.33       |45.24       |37.35           |***31.50***|\n\n[1] Peng et al., Shape As Points: A Differentiable Poisson Solver, NeurIPS 2021.\n\n[2] Gropp et al., Implicit Geometric Regularization for Learning Shapes, ICML 2020.\n\n[3] Atzmon et al., SAL: Sign Agnostic Learning of Shapes from Raw Data, CVPR 2020.\n\n[4] Ma et al., Neural-Pull: Learning signed distance functions from point clouds by learning to pull space onto surfaces. ICML 2021.\n\n[5] Li et al., Neuralangelo: High-Fidelity Neural Surface Reconstruction, CVPR 2023.\n\n\n### 2. Hyperparameter tuning.\n\nIn all experiments, we use the same network structure and loss weight factors.\nWe did not perform hyperparameter tuning for each shape.\nWe use the same hyperparameters for all shapes in a benchmark dataset, and only use different parameters to adopt different shapes in different datasets.\nSo, a different input point set does not require different hyperparameters in a dataset and one hyperparameter can be used for a wide range of shapes.\nOur method has a hyperparameter needed to be tuned, i.e., determining the standard deviation of distribution D for different datasets.\nThis hyperparameter is predefined in the code we provide, and we tend to choose a larger value for the dataset with sparse sampling or high curvature.\n\n\n### 3. Explanation of density-based experiments.\n\nThe 'gradient' simulates data collected using a 3D scanner, where nearby points are dense while far points are sparse.\nTo achieve this, we give higher weight to points that are closer to the simulated scanner in the probability-based sampling process.\nThe 'stripe' simulates the occlusion situation during the data collection, making the points in the occluded area sparse or disappear.\nTo achieve that, we divide the shape into multiple areas and sample the points in specific areas with extremely low weights.\n\n\n### 4. Suffer from density variation at low angle thresholds (Fig.5), but not noise?\n\nThe optimization of our method is formulated as an iterative moving operation of the input point position.\nThis strategy and the constraints for gradient uniformity of multi-scale neighborhood size make the model can handle noisy data.\nDuring optimization, the input point set Q is generated from the raw point cloud through a probability distribution D, which is built based on the neighborhood of the query points.\nTherefore, the density variation will affect the input data, and the generated points in sparse areas may be far away from the surface, increasing the difficulty of optimization in the point moving operation.\nAs stated in Question 2, we use the same parameters in all categories (clean, noise, and density variation) of a dataset for a fair comparison with other methods.\nA solution may be to choose different distributions for clean, noisy and uneven point clouds of a dataset, respectively.\n\n\n### 5. Experiment on LRR+(some orientation method).\n\nThe RMSE results of LRR+MST/QPBO/ODP on datasets PCPNet and FamousShape are shown in the following table and the curve of PGP is shown in *Rebuttal PDF Fig.4*.\nFor oriented normal estimation methods based on two-stage paradigms, the initial unoriented normals will affect the performance of normal orientation.\nFrom this table, we observe that higher-precision unoriented normals do not result in more accurate oriented normals using a normal orientation algorithm based on propagation strategy, such as PCA+MST vs. LRR+MST.\nThis means that even if we use better unoriented normal estimation algorithms (unsupervised or supervised), utilizing existing normal orientation algorithms does not necessarily lead to better orientation results.\n\n|Method      | PCA+MST | PCA+QPBO | PCA+ODP | LRR+MST | LRR+QPBO | LRR+ODP | Ours      |\n|:-:         |:-:      |:-:       |:-:      |:-:      |:-:       |:-:      |:-:        |\n|PCPNet      | 28.52   | 26.52    | 32.16   | 44.82   | 41.98    | 32.44   | **18.70** |\n|FamousShape | 40.48   | 41.31    | 42.92   | 57.83   | 59.84    | 51.93   | **26.16** |"}, {"review_id": "fkqG2rHsSh", "reviewer": "Reviewer_1E4M", "summary": "The paper introduces a method for estimating oriented normals from a given point cloud by utilizing a neural networkto model a sign distance field (SDF). The proposed approach involves training the SDF representation, which allows for easy querying of gradients at the positions of the point cloud. The method employs a set of loss functions that simulate an iterative process of moving query points to match target points based on the neural gradients. Experiments are conducted on unsupervised oriented normals estimation from input point clouds, which may contain noise, outliers, and density variations.", "questions": "Please see concerns and questions in weaknesses. Few further questions:\n\n- Are the loss coefficients tuned for different noise levels, or are they fixed throughout the experiments?\n- Can the presented method also yield the complete negative solution, i.e., $-\\boldsymbol{n}$, which is equivalent up to a global sign? Does this depend on the initialization or other factors?", "limitations": "The authors discussed limitation, however the limitation discussion and figure is shown soley in the appendix.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The authors clearly highlight the limitations of previous works, providing a good motivation for their proposed method. The promising results showcased in Figure 1 further support this motivation.\n- The introduction and related work sections are comprehensive, providing the necessary background for understanding the normal estimation task. This makes the paper self-contained.\n- The method is presented in a well-structured manner, starting with a high-level overview before delving into the details. This organization helps to keep the reader engaged and informed throughout the paper.", "weaknesses": "- The mathematical definitions and notations in the method section need to be revisited. Some crucial definitions, such as $f_i$ in Equation 2, are mentioned without being properly defined, making Sections 3.1 and 3.2 harder to understand. Additionally, the notation $ \\{ Q, G\\} $ is confusing as the operation involving this set containing two sets of points is not clearly defined or highlighted. The notation $f_i^G$ also requires clarification. Furthermore, the unit gradient $\\boldsymbol{n}$ mentioned in Line 164 lacks clear information about its 3D positions.\n- The authors refer to the post-processing step of computing the gradients of the learned SDF as \"inference\". However, \"inference\" typically refers to the process of applying a trained model on unseen inputs for generalization. This usage of terminology can be misleading and confusing.\n- Figure 2 does not fully capture the presented method as it introduces notations that are only defined in the text. It is unclear from the figure alone, even with the caption, what the input to the model is.\n- A comparison to other neural SDF methods that learn an SDF from an input point cloud, such as SAL (https://arxiv.org/abs/1911.10414) or IGR (https://arxiv.org/abs/2002.10099), is missing. These methods could be relevant for extracting normals easily, as in the suggested method.\n- The paper lacks a reference to SAP (https://pengsongyou.github.io/sap), which provides a scenario of reconstructing normals and a surface from an input point cloud.\n- It is recommended to include illustrations for the different losses mentioned, similar to the visualization in Figure 2 (i) at the bottom.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### 1. Some definitions, notations, and wordings need to be revised.\n\n$f_i$ in Eq.(2) represents the signed distance of each point during the i-th position movement process.\n$f_i^G$ in Eq.(3) represents the signed distance of each point in point set G during the i-th position movement process.\n\nAs in Line 117, Q and G are two point sets sampled from the raw point cloud in different ways, and they are used as the input of the network during optimization.\n\nThe unit gradient n in Line 164 is the normal of each point on the surface.\n\nThe wording 'inference' will be changed to 'normal estimation'.\n\nWe will revise the paper to further polish the definitions, notations, and wordings.\n\n\n### 2. Comments on Fig.2, loss and model input.\n\nWe will revise Fig.2 to clearly illustrate the method and add an illustration for the designed loss.\n\nDuring the optimization, the model input is two point sets Q and G sampled from the raw point cloud.\nDuring the normal estimation, the model input is the entire raw point cloud.\n\n\n### 3. Comparison with other neural SDF methods.\n\nWe will add references to the mentioned methods, namely SAL, IGR and SAP, in the revised version.\n\nWe use some implicit representation methods to estimate oriented normals.\nThe comparison of normal RMSE on point clouds in the datasets PCPNet and FamousShape is reported in the following table and the visual comparison is shown in *Rebuttal PDF Fig.6*.\nWe can see that our method has clear advantages.\n\nWe also have compared with some other surface reconstruction methods in Fig.6 of the paper and Fig.1 of the supplementary material.\nOur method can reconstruct better surfaces, especially on noisy point clouds.\nMoreover, our method can handle point clouds with uneven sampling and open surface structure, such as the KITTI dataset in Fig.7.\n\n|Method  |SAP [1]     |IGR [2]     |SAL [3]     |Neural-Pull [4] |Ours       |\n|:-:     |:-:         |:-:         |:-:         |:-:             |:-:        |\n|Noise   |57.56       |54.77       |46.69       |48.48           |***36.92***|\n|Density |41.32       |75.90       |43.78       |26.22           |***26.08***|\n|Average |49.44       |65.33       |45.24       |37.35           |***31.50***|\n\n[1] Peng et al., Shape As Points: A Differentiable Poisson Solver, NeurIPS 2021. \n\n[2] Gropp et al., Implicit Geometric Regularization for Learning Shapes, ICML 2020. \n\n[3] Atzmon et al., SAL: Sign Agnostic Learning of Shapes from Raw Data, CVPR 2020. \n\n[4] Ma et al., Neural-Pull: Learning signed distance functions from point clouds by learning to pull space onto surfaces. ICML 2021.\n\n\n### 4. More questions.\n\nThe coefficients for each loss are fixed across all experiments.\n\nThe oriented normal, i.e., gradient, is derived from the learned neural gradient function, and its sign depends on the initialization of the network.\nWe do not observe that our method yields the complete negative solution of the sign.\nIn the evaluation, if all normals have negative orientations with respect to the ground truth, we can simply reverse their orientation."}, {"review_id": "hEjZa56TtL", "reviewer": "Reviewer_6zu3", "summary": "The paper proposes a method to learn normals for point clouds using a neural representation. The idea is to perform a multi-step point moving and compute a series of losses to constrain the gradient of the field to describe shape locality with consistency. The paper proposes a large series of comparisons with the existing methods, showing promising results, although the training time is particularly slow.\n\n== POST-REBUTTAL ==\n\nAfter the discussion, the authors addressed my concerns. From other reviews, I also see a general consensus for acceptance, and the general recommendations are about clarifying some minor aspects. The only negative reviewer seems unwilling to defend the rejection position and does not point to significant weaknesses. Hence, I decided to increase my score and lean toward acceptance. Authors already agreed to incorporate the suggested changes, which are indeed essential; especially, I would stress that the two figures that show partial sections of complete shapes should be clearly described so as to not be deceptive.\n\nI wish authors best of luck with their work.", "questions": "A) From 3.2 I cannot completely understand how the method obtains the correct normal orientation between the two possible directions. Is it obtained thanks to the multi-scale neighbourhood size?\nB) From supplementary material, the training time seems dramatically slower than other methods. What is the main cause of that?", "limitations": "Limitations are sufficiently discussed in the supplementary material", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1) The paper obtained good reconstruction results; the shown shapes have varying numbers of different geometries, and the method seems to outperform the competitors.\n2) While the method part is not straightforward to understand, the attached code and the implementation details should provide enough details for replicability.", "weaknesses": "1) The paper does not convey a clear and precise message about its contribution. I am not an expert in this specific field, and I have quite a hard time understanding what makes this work different from previous ones. Accordingly to the conclusion, the main contribution seems in introducing a new loss/optimization schema. However, from the ablation study (which is also not easy to inspect, given the number of experiments and the lack of a best in bold), the \"full\" method is not always the clear winner, and it makes unclear the contribution of the losses. From my understanding, the main advancement is in providing multi-scale neighbour consistency, which helps orient all the normals in the same direction. I suggest clearly stating the main insights and adding more structure to 3.2 (e.g., with paragraph titles). \n\n2) Some works seem missing: [A] proposes a smoothness regularization for implicit representations; [B] involves a differentiable Poisson Shape reconstruction. I think a proper discussion would be useful, and especially fostering this kind of discussion in the related works. For example, at the moment, the two paragraph end highlighting that the method is unsupervised, and achieves better performance than the previous works. But how is this obtained? What is the key aspect that enables such advancement and is not considered in the previous works (if any)?\n\n\n\n[A]: Implicit Geometric Regularization for Learning Shapes, Implicit Geometric Regularization for Learning Shapes, Gropp et al., ICML 2020\n[B]: Shape As Points: A Differentiable Poisson Solver, Peng et al., NeurIPS 2021", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### 1. Discussion of contributions, and improvements to previous works.\nAs we can see from Table 2, existing supervised methods can achieve high-precision unoriented normal.\nHowever, from Table 1, we can observe that higher-precision unoriented normal do not result in more accurate oriented normal using a normal orientation algorithm based on propagation strategies, such as PCA+MST vs. AdaFit+MST and PCA+QPBO vs. HSurf-Net+QPBO.\nThis means that even if we develop better unoriented normal estimation algorithms, utilizing existing normal orientation algorithms will not necessarily lead to better orientation results.\nIn brief, the bottleneck of oriented normal estimation is correctly determining the orientation.\nThe supervised methods require expensive ground truth as supervision and have been extensively studied, while unsupervised learning of normals is still an unexplored field.\nBased on the above observations, we focus on how to use an unsupervised manner to directly learn oriented normals with higher orientation accuracy, instead of learning unoriented normals.\n\nBoth IGR [ICML 2020] and SAP [NeurIPS 2021] learn neural shape representation from point clouds without oriented normal.\nIGR proposes an implicit geometric regularization to favor a smooth zero level set of an implicit function f, i.e., f(x)=0.\nSAP proposes a differentiable poisson solver to represent the shape surface as an oriented point cloud at the zero level set of an indicator function f, i.e., f(x)=0.\nThese methods focus on accurately locating the position of the zero iso-surface of f(x) to extract the shape surface.\nHowever, they ignore the constraints on the gradient of the function during optimizing the neural network.\nWe know that the gradient determines the direction of function convergence, and the gradient of the iso-surface can be used as the normal of the surface.\nIf the gradient can be guided reasonably, the convergence process will be more robust and efficient, avoiding local extremum caused by noise or outliers.\nMotivated by this, we try to incorporate neural gradients into implicit function learning to achieve oriented normals.\n\nIn this work, we introduce a neural gradient function, consisting of the multi-step moving strategy along the gradient in Eq.(2), the gradient uniformity of multi-scale neighborhood size in Eq.(7), and the gradient consistency of multi-step moving in Eq.(8).\nFrom the ablation studies in Table 3(a), we can see that the performance of the algorithm drops a lot if we do not use these strategies, especially the losses in Eq.(7) and Eq.(8).\nIn Fig.6 of the paper and Fig.1 of the supplementary material, thanks to the losses in Eq.(2) and Eq.(7), our method is more robust against noise compared to surface reconstruction methods.\nThe multi-scale neighbor consistency mentioned by the reviewer, i.e., Eq.(7), is one of the important advancements of this work.\n\n### 2. Clarification of ablation study.\nFor the ablation studies in Table 3, we report results for both tasks of unoriented and oriented normal estimation.\nThe first two categories (a) and (b) in the table are obtained by removing modules (losses and inputs).\nIt can be seen that the performance of the algorithm drops relative to the complete method 'Full', especially the losses in Eq.(7) and Eq.(8).\nThese studies demonstrate the effectiveness of our novel designs and the contribution of the losses to improving the performance of the algorithm.\nThe latter two categories (c) and (d) are experiments on parameters, and they are based on using all losses.\nSome parameter settings provide better results in a single task, but do not give better results for two tasks.\nTo handle different cases of more datasets, such as KITTI, we choose the parameters that perform the best on average in both tasks.\nWe cannot conclude that the contribution of the losses is unclear because the 'Full' is not always the winner.\n\n### 3. How to obtain the correct normal orientation.\nThe normal orientation is achieved by the gradient of the learned implicit function, and our proposed neural gradient function learns an implicit global surface representation from data.\nThe implicit representation approaches, such as signed distance fields (SDF), represent the surface as zero level set of an implicit function f, i.e., f(x)=0.\nTherefore, we can train a neural network to regress singed distances, where SDF<0 for inside, SDF>0 for outside, and SDF=0 for surface, so that the SDF increases from inside to outside of the surface.\nThen, the gradient vector field of the SDF is obtained, and the gradient on the iso-surface should have a uniform orientation.\n\nIn our method, the optimization is formulated as an iterative moving operation of points to the target surface.\nAccording to Neural-Pull [ICML 2021], the gradient indicates the direction in 3D space in which the signed distance from the surface increases the fastest, so moving a point along or against (decided by SDF) the gradient will find its nearest path to the surface.\nWe can obtain the gradient at each point using the learned SDF, and the gradient is perpendicular to the surface and points to inside or outside based on the initialization of the network.\n\n### 4. The training time is slower than other methods.\nNote that we only reported the running time of various methods for predicting normal from point cloud (testing), and Table 1 in supplementary material does not include the training time of other methods.\nThe other supervised methods can be trained in advance using ground truth, and then their trained model is used for testing.\nIn contrast, our unsupervised method does not require training data and ground truth to train the model, but needs to be optimized based on test data to obtain its learned function, so we provide our optimization (training) time in the table.\nAs for training time, our method has a similar time cost (about 40 hours) to the SOTA method SHS-Net on the entire PCPNet dataset using an NVIDIA 2080 Ti GPU."}]

Author rebuttal:
We thank the reviewers for their valuable comments, and our responses to all reviewers are as follows. More visual comparison results are shown in the provided PDF.

Meta-review: The revievers appreciated the strong results, clear presentation, and comprehensive ablations.
A major concern shared by reviewers was the relation to SDF-based surface reconstruction methods, which they found insufficiently discussed. Moreover, experimental comparisons to surface-reconstruction-based approaches and on sparse point clouds were missing. The rebuttal addressed these concerns and experimental results confirmed the superior performance of the proposed method over baselines. All reviewers except for unresponsive vh4j found their concerns addressed in the rebuttal and raised their score. The AC agrees with the majority of the reviewers and recommends acceptance.
Decision: Accept (poster)