MultiPull: Detailing Signed Distance Functions by Pulling Multi-Level Queries at Multi-Step
Authors: Takeshi Noda
Chao Chen
Weiqi Zhang
Xinhai Liu
Yu-Shen Liu
Zhizhong Han
Venue: NeurIPS 2024 poster (2024)
Keywords: surface reconstruction
point cloud
SDF
Primary area: machine_vision

Abstract: Reconstructing a continuous surface from a raw 3D point cloud is a challenging task. Latest methods employ supervised learning or pretrained priors to learn a signed distance function (SDF). However, neural networks tend to smooth local details due to the lack of ground truth signed distnaces or normals, which limits the performance of learning-based methods in reconstruction tasks. To resolve this issue, we propose a novel method, named MultiPull, to learn multi-scale implicit fields from raw point clouds to optimize accurate SDFs from coarse to fine. We achieve this by mapping 3D query points into a set of frequency features, which makes it possible to leverage multi-level features during optimization. Meanwhile, we introduce optimization constraints from the perspective of spatial distance and normal consistency, which play a key role in point cloud reconstruction based on multi-scale optimization strategies. Our experiments on widely used object and scene benchmarks demonstrate that our method outperforms the state-of-the-art methods in surface reconstruction.

MultiPull: Detailing Signed Distance Functions by Pulling Multi-Level Queries at Multi-Step
Takeshi Nodal*
Chao Chen
Weiqi Zhang'
Xinhai Liu²
Yu-Shen Liu
Zhizhong Han
School of Software, Tsinghua University, Beijing, China Tencent, Hunyuan, Beijing, China
Department of Computer Science, Wayne State University, Detroit, USA
yeth21 @mails tsinghua. edu. chenchao19@tsinnghua. org cn
zwq23@mails tsinghua. edu. adlerzzhliu@tencent com
Liuyushenttsing inghua. edu. cn h312h@wayne.edd
Abstract
Reconstructing continuous surface from raw 3D point cloud is a challenging task. Recent methods usually train neural networks overfit on single point clouds infer signed distance functions (SDFs). However, neural networks tend to smooth local details due to the lack ground truth signed distances or normals, which limits the performance of overfitting-based methods in reconstruction tasks. To resolve this issue, we propose a novel method, named MultiPull, to learn multi-scale implicit fields from raw point clouds by optimizing accurate SDFs from coarse to fine. We achieve this by mapping 3D query points into a set of frequency features, which makes it possible to leverage multi-level features during optimization. Mcanwhile, we introduce optimization constraints from the perspective of spatial distance and normal consistency, which play key role in point cloud reconstruction based on multi-scale optimization strategies. Our experiments on widely used object and scene benchmarks demonstrate that our method outperforms the state-of-the methods in surface reconstruction. Project page: htttps: httess///taeeeieeiiiihhttu. wit.tgithhh........... io/MultPPull
Introduction
Reconstructing surfaces from 3D point clouds is an important task in computer vision. It widely used in various real-world scenarios such autonomous driving, 3D scanning and other downstream applications. Recently, using neural networks to learn signed distance functions from 3D point clouds has made huge progress set of 3D continuous field, and the surface can be further extracted using the marching cubes algorithm [. supervised methods [TO
3 4
g SDF represents surface as the zero-level
12 3 a continuous field is learned using signed distance supervision. Some methods employ multi-level representations [14 15 such as Fourier layers and level of detail (LOD) 16. 17], learn detailed geometry. However, these methods require 3D supervision, including ground truth signed distances or point normals, calculated on a watertight manifold. address this issue, several unsupervised methods [I8] [9 220 21 22) were proposed to directly infer an SDF by overfitting neural networks single point cloud without requiring ground truth signed distances and point normals. They usually need various strategies, such as geometric constraints [18 19 20 and consistency constraints 222 23), for smoother and more completed signed distance field. However, the raw point cloud a highly discrete approximation of the surface, learning
"Equal contribution. 'The corresponding author is Yu-Shen Liu.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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(a) Step 1
(b) Step
(c) Step
(d) Reference
Figure 1: Visualization of the 31) shape reconstruction. In (a), (b) and (c), SDFs learned from point cloud by optimizing multi-level query points at multi-step. At each step, we optimize query points at one level with frequency features at this specific level conditions. This enables the network to progressively recover coarse-to-fine geometry details.
SDFs directly from the point cloud is often inaccurate and highly ambiguous. This makes hard for the network to learn accurate SDFs on local details, resulting in over-smooth reconstruction.
To address this issue, we propose MultiPull, to learn an accurate SDF with multi-scale frequency features. It enables network to predict SDF from coarse to fine, significantly enhancing the accuracy of the predictions. Furthermore, to optimize the SDF at different scales simultaneously, we introduce constraints the pulling process. Specifically, given query points sampled around 3D space input, we use a Fourier transform network to represent them as set of Fourier features. Next, we design network that can leverage multi-scale Fourier features to learn SDF fields from coarse to fine. To optimize the signed distance fields with multi-scale features, we introduce a loss function based gradient consistency and distance awareness. Compared with Level of Detail (LOD) methods we can optimize the signed distance fields effectively without a need of signed distance supervision, recovering more accurate geometric details. Evaluations widely used benchmarks show that our method outperforms the state-of-the-art methods. Our contribution can be summarized follows.
16 17 24),
We propose a novel framework that can directly learn SDFs with details from raw point clouds, progressing from coarse to fine. This provides new perspective for recovering 3D geometry details.
We introduce multi-level loss function based on gradient consistency and distance aware- ness, enabling the network to geometry details.
Our method outperforms state-of-the-art methods in surface reconstruction in terms of accuracy under widely used benchmarks.
Related Work
have attempted to analyze the geometric modeling of objects, which do not require large-scale datasets. With the advent of extensive and
Classic methods for geometric modeling 25
26
27
28
intricate 3D datasets like Net (29] and ABC 30 learning-based methods have achieved
significant advancements [18] 12 31 32 22 33
34
35|
36
37
38 These approaches learn implicit
representations from various inputs, including multi-view images 39 40
42 43 44), point clouds
45 46 47
and voxels 48 49 50]
Learning Implicit Functions with Supervision. Supervised methods have made significant progress in recent years. These methods leverage deep learning networks learn priors from datasets or use real data for supervision [10
53 [44] to improve surface reconstruction performance. Some supervised approaches use signed distances and point normals as supervision, or leverage occupancy labels to guide the network's 's learning process. In order to improve the generalization ability of neural networks and learn more geometric details, some studies learn geometry prior of shapes through supervised learning.
[1 51
52

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Learning Implicit Functions with Conditions. To alleviate the dependence on supervised infor- mation, recent studies focus on unsupervised implicit reconstruction methods. These methods do not require pretrained priors during optimization. For example, NeuralPull (NP) 21 learns SDF by pulling query points in nearby space onto the underlying surface, which relies on the gradient field of the network. CAP (55] further complements this by forming dense surface by additionally sampling dense query points. GridPull [23] generalizes this learning method to the grid, by pulling the query point using interpolated signed distances on the grid. In addition, some studies explore surface reconstruction more deeply and propose innovative methods, such as utilizing differentiable Poisson solutions [56], or learning signed [37 588 19 51 or unsigned functions [59] with priors. However, inferring implicit functions without 3D supervision requires a lengthy convergence process, which limits the performance of unsupervised methods on large-scale point cloud datasets. To address this, we propose a fitting-based frequency feature learning strategy that efficiently learns implicit fields without the need for additional supervision.
Learning Implicit Functions with LOD. Level-Of-Detal (LOD) models [6 17 24 are used to simplify code complexity and refine surface details through the architecture of multi-level outputs. Previous studies have explored multi-scale architectures in various reconstruction tasks. For example, NGLOD [16] uses octree-based feature volumes to represent implicit surfaces, which can adapt to shapes with multiple discrete levels of detail and enable continuous level-of-detall switching through SDF interpolation. MFLOD [17] applies Fourier layers to LOD, which can offer better feasibility in Fourier analysis. However, it difficult to optimize multi-level features simultancously to learn 3D shapes. To address this issue, propose novel strategy to optimize multi- nevei-level frequency features, allowing the network to progressively learn geometric details from coarse to fine.
ho (Qs)
9/3
Ereecn
h; (0s)
3/1
LSNN
kttus (Qa)
YN
LSNN
MNN-2 (20)
W
UML-2
Linst
Query Points
AN--1 (0w)
LSNN
Fressn
Qo
frequency features
(a) Frequency Feature Transformation (FFT) Module
(b) Multi-Step Pulling (MSP) Module
Hadamard Product
Ququary points in y. freeuuncyy featuro
SDF predictions
layers
LSNN
Linear Sequence
i-th pulling
Neural Network
Figure 2: Overview of our method: (a) Frequency Feature Transformation (FFT) module and (b)
Multi-Step Pulling (MSP) module. In (a), we learn Fourier bases (Q) from query points Q using the Fourier layer and obtain multi-level frequency features Ys through Hadamard product. In (b), using multi-level frequency features from (a) and a linear network LSNN with shared parameters, calculate the distance(D) of Q; to its corresponding surface target point Qt to predict a more accurate surface. We visualize the predicted SDF distribution map corresponding the frequency features in (a) and the reconstruction from each step of SDF predictions on the right side of (b).
3 Method
Overview. The overview of MultiPull is shown in Fig. 2 We design neural network to learn an implicit function if from a single 3D point cloud by progressively pulling a set of query points Qo onto the underlying surface, where Qo is randomly sampled around the raw point cloud S. Our network mainly consists of two parts as follows.
(1) The Frequency Feature Transformation (FFT) Module ( Fig. a())) aims to convert the query points Qu into introducing frequency features lies in flexible control of the degree of details. (2) The Multi-Step Pulling (MSP) Module (Fig. ((b)) is designed to predict f with coarse-to-fine details under the guidance of frequency features Y. At the i-th step, we pull Q: to Qi+1 using the predicted signed distances 8; f(2s: fi) and the gradients at Q., according to its feature y/s. To this end, we constrain query points be as close their nearest neighbor point S.
multi-level frequency features Y = {y,, E 0. Nz. -1]]. The key insight for
3

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3.1 Frequency Feature Transformation (FFT) Module
We introduce a neural network learn frequency features Y from point clouds. The network manipulates input Qu through several linear layers to obtain an initial input 2y and set of Fourier basis h; (Qo), 2 [0, I]. formulated as follows.
{
(hi(Qu) sin(w; + di),
(I)
20 ho ho(Qo),
where and @; are the parameters of the network, and N,. is the number of layers of the network.
To effectively represent the expression of the raw input in the frequency space, we choose the sine function as the activation function and employ the Hadamard product to compute the intermediate frequency feature output. Since the Hadamard product allows the representation of frequency components as the product of two feature inputs, denoted as a and b, which can be formulated as:
1 sin(a) sin(b) デ (sin(a + b m 2) sin(a b+ -)
Through Eq. 22. we can calculate the frequency component 2; of h4 (Qo), and then obtaín the output of the i-th layer, formulated as:
2j h: (Q,))) @ (Wiz-1+ + [1, 1],
y/ Wi2 bi,
where @ indicates the Hadamard product, W,, b, parameters of the network.
Frequency networks based on the Multiplication Filter Network (MFN) 14] typically employ uniform or tixgh-weight initialization for network parameters in practice. This approach overlooks the issue of gradient vanishing in deep network layers during the training process, Icading to underfitting and making the network overly sensitive to hyperparameter changes. To address this challenge, propose new initialization scheme that thoroughly considers the impact network propagation, aiming ensuring uniform distribution of initial parameters. Specifically, we dynamically adjust initial weights, which can be formulated as:
V Vn x sin(it/NL), E [1,Nz - 1],
where N₁. and 1 are the number of layers and the parameters of the network, respectively. We leverage the standard deviation as the initialization range to ensure that the parameters Eq. (3) are within reasonable range. As shown in Fig. 3 we compared the parameter distributions of different linear layers. The initialization scheme based on MFN results gradient vanishing and small activations in deeper linear layers. In contrast, our initialization scheme ensures that the parameters of each linear layer follow standard normal distribution.
14
LB
311
MFN- Boad
9ux
Qpm
01
9xx
015
1k
13
Owr
415
0.25
412
02
412
022
412
Figure 3: Comparison of parameter distributions of different linear layers especially in
(L2, L1, Le, Lg). We show the different initialization strategies on the results of the reconstruc- tion task and the visualization effects in Appendix B
3.2 Multi-Step Pulling (MSP) Module
In Fig. ()),, we demonstrate our idea of learning an accurate implicit function with multiple frequency features. Given set of frequency feature Y, use frequency features y in Y as the

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input along with query points Qi for the MSP module. We follow NP[2T] to construct initial query points and calculate the stride and direction of Q, i-th step for pulling it to the target surface point. Furthermore, we use the direction of the gradient as V VS(Q:. y/) and signed distance s(Q Vi) for the pulling, where V.(Q.. 1/6) represents the fastest increase in signed distance 3D space, pointing away from the surface direction. Therefore, Q; = Q f(Q;-1, Bi-1) V VI(Q;-I.
V f( ((Q)--:)i-1)) l12. For each step of pulling the query points Qi. corresponds to a nearest point the surface, and the distance between query points and surface points can be described as di D = =1Q - qi l.. Based on this, we initiate the optimization by pulling query points Qi the target points 9; progressively. Therefore, can obtain the combined loss Lpull under optimal conditions:
Cpull = Σ EL,1,
(5)
1
where / is the step of moving operation. However, optimizing all query points accurately through this equation alone is challenging when merely constraining surface points. This is because the query points Qi may be located across multiple spatial scales with inconsistent gradient directions, indicating that simultaneous optimization becomes challenging. Consequently, some outlier points may not be effectively optimized. Additionally, for sampling points near target points, some surface constraints are required to enable the network to accurately predict their corresponding zero level- set to avoid optimization errors. Therefore, we will further advance Eq. (5) from the perspectives
of distance constraints, gradient consistency, surface constraints in Sec. 3.3 to enhance network performance.
3.3 Loss Function
Distance-Aware Constraint. Inspired by FOCAL 660 we design a novel constraint with distance- attention weights ensure that the network pays more attention to the optimization of underfitting query points in space and optimizes the SDFs simultaneously. This allows query points different distances from the surface to be optimized properly, and assigns higher attention weights for outlier and underlying points:
{ ai, 02 so cootmax((Di, D2),
(6)
Lrecom an D1 + a D2 + D3.
where on and 02 are calculated from D1, D2 by the softmax activation, y a scaling coefficient we set to 2 by default. Here, we only consider 3 steps, which is trade-off between performance and efficiency.
Consistent Gradients. We additionally introduce consistency constraints in the gradient direction. This loss encourages neighboring level sets keep parallel, which reduces the artifacts off the surface and smooths the surface. We add a cosine gradient consistency loss function to encourage the gradient direction at the query points to keep consistent with the gradient direction at its target point on the surface, which aims to improve the continuity of the gradient during the multi-step pulling, We use Q1, Q2 and Q3 to represent the query points that have been continuously optimized by the multiple steps. We take the one with the lowest similarity score to measure the overall similarity.
{ (vi( cos( V f(Q,, 3:), V f(Qo, yo)), Egrat = - min(L~v(1)) (Q1),
(Qa));
where (Qi) represents the loss of cosine similarity between query points Q and target surface points 9.
Surface Constraint. We introduce the surface constraint for the implicit function f. aiming to assist the network in approaching the zero level-set on the surface at final step. Hence, we constrain the f(Q1, uI) approaches zero at the final step:
Lanf =I1 s(Qr. wr) ||
(8)
Joint Loss Function. Overall, we learn the SDFs by minimizing the following loss function L.
L Lrecon + BC + 8C
where and are balance weights. the subsequent ablation experiments, we validated the effectiveness of different loss functions.
5

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Experiments
In this section, we evaluate the performance of MultiPull surface reconstruction by conducting numerical and visual comparisons with state-of-the-ar methods on both synthetic and real-scan datasets. Specifically, in Sec. 4.1 experiment synthetic shape datasets with diverse topological structures. Furthermore, in Sec. 4.2 we report our results across various scales on real large-scale scale scene datasets. Meanwhile, consider FAMOUS as the verification dataset the ablation studies to compare the effectiveness of each module in MultiPull in Sec. 4.3
4.1 Surface Reconstruction for Shapes
Datasets and Metrics. For the single shape surface reconstruction task, perform evaluation on multiple datasets including ShapeNet [29], FAMOUS [10], Surface Reconstruction Benchmark (SRB) 45 Thingi10K [6]] and D-FAUST [62] We conduct validation experiments on subcate- gories within the ShapeNet dataset, while the remaining datasets are experimented on the complete dataset. For metric comparison, leverage LI and L2 Chamfer Distance CD and L2, Normal Consistency (NC), and F-Score as evaluation metrics.
ShapeNet. We evaluate approach the SetpeNet 29 according to the experimental settings of
GP
We compared our methods with methods including ATLAS [3]. DSDF (51), NP [21), PCP shown in Tab. We report CD12, NC and F-Score metries for ShapeNet, where we randomly sample 10,000 points on the reconstructed object surface for evaluation. MultiPull outperforms the state-of-the- -ht---r methods. Compare to previous gradient-based methods in Fig. 4 our method performs better by revealing more local details of these complex structures. We provide detailed results in Appendix
23
64
GP 23],
Table 1: Reconstruction accuracy on ShapeNet in terms of CD NC and F-Score with thresholds of 0.002 and 0.004.
Methods
CD22 100
NC
F-Score 0.002
F-Score 0:001
ATLAS
1.368
0.695
0.062
0.158
DSDF
0.766
0.884
0.212
0.717
NP
0.038
0.939
0.961
0.976
PCP
0.0136
0.9590
0.9871
0.9899
GP
0.0086
0.9723
0.9896
0.9923
Ours
0.0075
0.9737
0.9906
0,9932
NP
PCP
GP
Ours
GT
Figure 4: Visual comparison of reconstructions on ShapeNet.
FAMOUS. We evaluate the performance of our method on the FAMOUS dataset according to the experimental settings of PCP [64] and NP 21.. Our method demonstrates superiority over recent

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approaches, including GP 233 PCP 664, GenSDF 665 FGC [66] NP 21, and IGR [57]. As shown in Tab. 2 we compared the recent methods using CD12 1. and NC metrics, and our method exhibits outstanding performance. To demonstrate the effectiveness of our method in reconstruction accuracy, we visualize the error-map for comparison in Fig- | Compare to the the state-of-art methods, our method has better overall reconstruction accuracy (bluer).
GaSDDF
NP
GP
OKt
GT
Table 2: Reconstruction accuracy on FA-
MOUS in terms of CD L2 and NC.
Methods CDr x 100
NC
IGR
1.65
0.911
High
GenSDF
0.668
0.909
NP
0.220
0.914
FGC
0.055
0.933
PCP
0.044
0.933
Low
GP
0.040
0.945
Ours
0.035
0.953
Figure MOUS.
Visual comparison of error maps on FA-
Noiay Input
PIM
SAP
Table 3: Reconstruction accuracy on SRB in terms of CD21 and F-Score with a threshold of 0.01.
Methods CD x 100
NC
P2M
0.116
64.8
SAP
0.076
83.0
NP
0.106
79.7
BACON
0.089
82.7
CAP
0.073
84.5
GP
0.070
85.1
Ours
0.068
85.7
Figure 6: Visual comparison SRB.
SRB. We validate our method on the real scanned dataset SRB, following the experimental settings of VisCo [67] and GP [23] In Tab. we compared approach with recent methods including P2M (68) SAP [56], NP [21), BACON 15), CAP (55] GP 23). We use CD,,1 and F-Score to evaluate performance and surpass all others in terms of these metrics. As depicted in Fig. 6 our method excels in reconstructing more complete and smoother surfaces.
D-FAUST. We evaluate method the D-FAUST dataset with SAP (56) settings. As indicated in Tab. 4 we compared our approach with recent methods including IGR [57]. SAP [56), GP (23] Our method excels in CD F-Score and NC. As illustrated in Fig. 7 compared to other methods, our approach demonstrates superior accuracy in recovering human body shapes.
Thingi10K. We assess the performance of our approach on the Thingil0K dataset, following the experimental setup of SAP
[66]. We compared our approach with recent methods including IGR (57).
SAP (56). BACON [5 across CD F-Score and NC metrics. illustrated Fig. g our method can reconstruct surfaces with more accurate details.
GP
2 As indicated in Tab. 5 our method surpasses existing methods

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Table 5: Reconstruction accuracy under Thingi 10K terms of CD11 and F-Score with a thresh-
Table 4: Reconstruction accuracy under D- FAUST in terms of CD11 and F-Score with
old of 0.01.
threshold of 0.01.
Methods
CDri
F-Score
Methods
CDA1
F-Score
NC
IGR
0.440
0.505
0.692
IGR
0.235
0,805
0.911
SAP
0.054
0.940
0.947
SAP 0.043
0.966
0.959
BACON
0.053
0.946
0.961
GP
0.015
0.975
0.978
GP
0.051
0.948
0.965
Ours
0.009
0.986
0.988
Ours
0.048
0.953
0.968
Input
IGR
SAP
GP
Ours
Figure 7: Visual comparison on D-FAUST.
Figure 8: Visual comparison on Thingi 10K.
4.2 Surface Reconstruction for Real-Scan Scenes
Datasets and Metrics. For the scene reconstruction task, we validate our method on the 3DScene 669 and KITTI 770 datasets assess the performance on large-scale datasets. We keep the same evaluation metrics as those used for shape reconstruction in Sec. 4.1
compared approach with recent methods including ConvOcc [39], NP [21], PCP 64 and GP 23 We report the evaluation results of CD L1. CD L2 and NC, and compared our method with the latest approaches listed in Tab,
3DScene. In accordance with the experimental settings of PCP
64
As illustrated in Fig. our method outperforms prior-based methods and overfitting based methods.
ConvOcc
NP
PCP
GP
Ours
GT
High
Figure 9: Visual comparison of CD error maps on 3DScene.

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Table 6: Reconstruction accuracy on 3DScene in terms of CD CD and NC.
Methods
CD X 100
CD
NC
ConvOcc
13.32
0.049
0.752
NP
8.350
0.0194
0.713
PCP
0.11
0,007
0.886
GP
0.10
0.006
0.903
Ours
0.094
0.006
0.918
Ours
GP
Figure 10: Visual comparison on KITTI.
We validate our method on the large-scale scanned point cloud dataset KITTI [70], which contains 13.8 million points. As shown in Fig. 10 our approach is capable of reconstructing more complete and accurate surfaces compared to the GP method [33 GP struggles to reconstruct contin- uous surfaces such walls and streets, whereas our method achieves more detailed reconstruction of objects various scales in real scanned scenes. It demonstrates that method is robust when handling point cloud with various scales.
KITTI.
4.3 Ablation Experiments
Effect of Frequency Layers. We denote the j-th layer of the frequency network as Lj. a specific combination of frequency feature layers can be formulated as [L4, L.j. T.)., where ji.j.kt k) [1, - 1). We evaluate the effectiveness of the frequency transformer layers Tab. with CD and NC, replacing the frequency network with linear layers results in decrease in the performance of the CD,2 and NC metrics. The performance of using only one layer(L.) surpasses linear layers. With the increase of the frequency layers, {Lg produces best results.
Table 7: Effect of frequency features.
Layer CD x 100
NC
Linear
0.042
0.920
LA
0,040
0.926
LAs Lo
0.037
0.933
La, Los Ls.
0.036
0.948

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Effect of MSP Module. We report comparisons with different features in Tab. The 'Layer' column denotes the combination frequency features obtained by the FFT module. For instance. (T4. La. [.g] } represent the frequency features from the 4th 6t and 8th layers guiding the pulling of the query point the MSP network, respectively. We find that the accuracy of the network increases with the number of steps. After considering both performance metrics and time efficiency, we have set Step=3 by default.
Table 8: Effect of MSP Module.
Step
CDt2 x 100
NC
0.040
0.926
2
0.037
0.933
0.036
0.948
4
0.036
0,942
0.0357
0.955
Effect of Loss Functions. We compared CD metric under different loss strategies in Tab. 9 As shown in the table, Weighting query points at different scales effectively enhances reconstruction accuracy and the reconstruction loss allows the network to obtain a complete shape with local details. Furthermore, The gradient loss improves the surface continuity of the object. And the surface supervision loss facilitates the learning of more precise zero-level sets, which also improves the accuracy.
Table 9: Effect of loss functions.
Loss
CDr. x 100
NC
Equlll
0.0443
0.937
Lrecom
0.0383
0.946
Lrecon
Loii
0.0367
0.948
Lrccon + Laim + Lear
0.0352
0.954
Effect of Different Levels of Noise. We evaluate the reconstruction performance of our method on the Famous dataset under two levels of noise: Mid-Level and Max-Level noise. As shown Tab, IO our method outperforms the majority of approaches even in the presence of noisy inputs.
Table 10: Effect of different levels of noise.
Noise level
NP
PCP
GP
Ours
Mid-Noise
0.280
0.071
0.044
0.037
Max-Noise
0.310
0.298
0.060
0.058
5 Conclusion
We propose a novel method to learn detailed SDFs by pulling queries onto the surface at multi-step. We leverage the multi-level features to predict signed distances, which recovers high frequency details. Through optimization, our method is able to gradually restore the coarse-to-fine structure of reconstructed objects, thereby revealing more geometry details. Visual and numerical comparisons show that our approach demonstrates competitive performance over the state-of-the-aa methods.
Acknowledgment
This work was supported by National Key R&D Program of China (2022YFC3800600) and the National Natural Science Foundation of China (62272263, 62072268), and in part by Tsinghua- Kuaishou Institute of Future Media Data.
10

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A Implementation Details.
Our network consists of two main parts: frequency feature transformation and multi-step pulling modules in Fig 2 (a) and (b), respectively. For frequency feature transformation, we transform the raw point cloud into frequency features M, where M is initialized to 256. Same for the multi-step pulling module, we train a linear sequence neural network (LSNN) with shared parameters and we fix intermediate layer output dimension to 512. In the construction of query points, we establish the corresponding pairs between query points and their nearest points on surfaces. Specifically, we follow NP [21] to construct 40 queries for each point of the point cloud, the construction these query points follows Gaussian distribution. During the reconstruction process, we the Marching Cubes algorithm 27 to extract the mesh surface.
During the training process, we do optimization in 40,000 iterations, with an average time of 24 minutes for single-object reconstruction. We utilize single NVIDIA RTX-3090 GPU for both training and testing.
B Additional Experiments
Effect of Frequency Features. To further validate the superiority of frequency features, we exclude multi-step pulling, and only use single-layer frequency feature for performance verification against linear layers. We note the frequency feature in the ?-th layer as L,- We compared the CD and NC of specific layers(L2, L1, Lo, Ls) with the linear layers. As shown the Tab. [I the performance of frequency features at different layers is superior to the linear layers, and with an increase in the number of layers, higher-level frequency conditions enhance the network's performance.
Table 11: Effect of frequency features.
Layer CD12 100
NC
Linear 0.042
0,920
L1 0.040
0,926
LG 0.038
0,931
Ls 0.037
0.935
Effect of Initialization Strategies. We compared our initialization strategy with random initialization and MFN-based method (BACON (15) as example. We compared the metries of these initialization methods in Tab. 12 which shows that combining random BACON initialization with our approach does not yield satisfactory results. To further demonstrate the advantages our initialization method, visually compared SDF with random initialization and BACON initialization strategies. As shown in the Fig. 11 our method significantly outperforms other initialization methods terms of convergence speed. In addition, method can enable the network to learn more accurate signed distance field. We compared the results with the same iterations and the final results under the default settings for different methods (Final) in Fig. [2, these failed reconstructions based on MFN demonstrate the instability of parameter initialization.
reconstruction results also indicate that reasonable initialization
Table 12: Effect of initialization strategies.
Initialization CD 100
NC
Random 0.042
0.938
BACON
0.038
0.946
Ours 0.035
0.950
Effect of Parameters on Networks. We compared the parameter quantities the methods listed in Tab. 13 below. It shows that the parameter number of PCP[64] is the largest among all the three methods, while NP(2T] has the least parameters. networks with the similar amount of parameters, we increase the parameters of NP and MultiPull to match PCP. The comparison in the Tab. [4] indicates that both NP and MultiPull show the improved performance. This demonstrates that more parameters are beneficial to improve the performance, but
further investigate the performance of
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Iteration
100
200
500
1000
Final Result
Random
BACON
Ours
Figure 11: Comparison of signed distances in optimization
Iteration
2000
Final
BACON
Ours
BACON
Ours
Figure 12: Comparison of the initialization strategies in optimization.
performance comes from novel methods rather than parameters. Meanwhile,we we further evaluate inference time with NP and PCP.As shown in Tb. 15 the local based method (PCP) has the longer inference time than the global based method, with NP having the fastest inference time.
Table 13: Comparison at different parameter levels.
Method
Parameters
CD 1.2 x 100
NC
NP
1843708
0.220
0.914
PCP
7894022
0.044
0.933
Ours
2648094
0.035
0.953
Table Comparison at a uniform parameter levels.
Method
Parameters
CD 100
NC
NP
7907441
0.1966
0.917
PCP
7894022
0.044
0.933
Ours
7856620
0.0317
0.957
Table 15: Comparison of Inference Time.
PCP
NP
Ours
Inference Time(min)
0.17
0.138
0.141
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Design of Lgrad- We further discuss the design of Lgad and effectiveness performance. We use minimum (min) as the baseline and compare with using the average (avg). As shown in Tab. 16 using (gga to calculate the similarity of query points at different time steps results in slight increase CD error. contrast, (min) achieves similar level of similarity but better performance in CD metrics. Therefore, we calculate minimum of the gradient similarities as constraint to prevent significant deviations in the moving direction during training, making the network more sensitive changes in gradient direction.
Table 16: Comparison at a uniform parameter levels.
w/o Lerad
(avg)
Cera((mnn
CD22 x 100
0.0388
0.0383
0.0352
NC
0.945
0.950
0.954
Feature Comparison. We combine MSP with the linear layers and the traditional feature learning encoder PointMLP to explore the effectiveness of MSP. We present the results of combining different feature learning networks with MSP methods in Tab. 17 We denote the single moving operation in MSP Pull and multiple feature learning networks baselines. Due to the lack of FFT module, the same features are used for multiple offsets in linear+MSP and PointMLP + MSP. As shown in Tab. 17 the combination of different feature extraction networks and MSP achieved better results in terms of both CD and NC metrics. We further demonstrate the effectiveness of MSP in Fig. 13 and Fig. 14 in the PDF. PointMLP/ linear+MSP can generate finer local details compared to PointMLP linear Pull.
Table 17: Comparison of Reconstruction Accuracy in CD12 100
Iteration
10K
20K
40K
Linear+Puu Pull
0.078
0.061
0.055
PointMLP+Puil
0.064
0.045
0.037
Linear+MSP
0.073
0.058
0.041
PointMLP+MSP
0.053
0.042
0.037
Iterations
10K
20K
40K
Error-map(40K)
PointMLP+Pull
PoiNtMLP+MSP
Ours
Small
Large
Figure 13: Comparison of different feature encoders and MSP module on FAMOUS dataset.
Default setting of Step. We set Step to by default for two reasons: (1) MSP can bring partial
accuracy improvement when learning more steps, but more network parameters are also needed. We
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Iterations
10K
20K
40K
moor-map(40kk
Linear+Pull
Large
L LiS
Small
Ours
Figure 14: Comparison of different feature encoders and MSP module on FAMOUS dataset.
present comparisons under different steps in Tab. 8 (2) Although deeper frequency features can represent more comprehensive information, combining frequency features from different layers can achieve better results. We denote the combination of the frequency. We conduct the experiment according to the default settings, and the results are shown Tab. T8
Table 18: Comparison of CD222 12 100, NC under different feature combination.
L4, L4, La
L6 L6; Ls
Lgs Lgs Lg
L Los Li
CD12 x 100 NC
0.0417 0.922
0.0397 0.927
0.0360 0.938
0.0343 0.948
C Additional Results
Comparison Details for ShapeNet. The complete comparison under all the five scenes of the ShapeNet dataset. The results are shown in Tab. 19 to Tab 21 We use Chamfer Distance (CD11,CD12) and NC evaluation metrics.
Table 19: Reconstruction accuracy on ShapeNet in terms of CD12
Class
ATLAS
DSDF
NP
PCP
Ours
Display
1.094
0.317
0.039
0.0087
0,0082
0.0074
Lamp
1,988
0.955
0.080
0.0380
0.0347
0.0301
Airplane
1.011
1.043
0.008
0.0065
0.0007
0.0006
Cabinet
1.611
0.921
0.026
0.0153
0.0112
0.0105
Vessel
0.997
1,254
0.022
0.0079
0.0033
0.0028
Table
1.311
0.660
0.060
0.0131
0.0052
0.0047
Chair
1.575
0.483
0,054
0.0110
0.0043
0.0036
Sofa
1.307
0.496
0.012
0.0086
0.0015
0.0009
Mean
1.368
0.766
0.038
0.0136
0.0086
0.0075
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Table 20: Reconstruction accuracy on ShapeNet in terms of NC.
Class
ATLAS
DSDF
NP
PCP
GP
Ours
Display
0.828
0.932
0.964
0.9775
0.9847
0.9855
Lamp
0.593
0.864
0.930
0,9450
0.9693
0.9710
Airplane
0.737
0.872
0.947
0.9490
0,9614
0.9633
Cabinet
0.682
0.872
0.930
0.9600
0.9689
0.9693
Vessel
0.671
0.841
0.941
0.9546
0.9667
0.9671
Table
0.783
0,901
0.908
0.9595
0,9755
0.9758
Chair
0.638
0.886
0.937
0.9580
0,9733
0.9757
Sofa
0.633
0.906
0.951
0. 0.9680
0,9792
0.9822
Mean
0.695
0.884
0.939
0.9590
0.9723
0.9737
Table 21: Reconstruction accuracy on ShapeNet terms of Score with thresholds of 0.002 and
0.004.
1-Score
ATLAS
DSDF
NP
PCP
GP
Ours
ATLAS
DSDF
NP
PCP
GP
Ours
Display
0.071
0.632
0.989
0.9939
0.9963
0.9966
0.179
0.787
0.991
0.9958
0.9963
0.9980
Lamp
0.268
0.891
0,9382
0.9455
0.9473
0.077
0.478
0,924
0,9402
0.9538
0.9561
Airplane
0.070
0.350
0.996
0.9942
0.9976
0.9989
0.179
0.566
0.997
0.9972
0.9989
0.9991
Cabinct
0.077
0.573
0,980
0,9888
0.9901
0.9913
0.195
0.694
0.989
0.9939
0.9946
0.9969
Vessel
0.058
0.323
0.985
0.9935
0.9956
0,9962
0.153
0.509
0,990
0,9958
0,9972
0,9979
Table
0.080
0.577
0,922
0.9969
0.9977
0.9982
0.195
0.743
0.973
0,9958
0,9990
0,9988
Chair
0.050
0.447
0.954
0.9970
0.9979
0.9980
0.134
0.665
0.969
0,9991
0.9990
0.9994
Sofa
0.058
0.577
0.968
0.9943
0.9974
0,9981
0.153
0.734
0.974
0.9987
0,9992
0.9992
Mcan
0.062
0.212
0.961
0.9871
0.9896
0,9906
0.158
0,717
0.976
0.9899
0.9923
0,9932
Comparison Details for 3DScene. We also provide detailed metrics for single scenes in 3DSccnne dataset. We evaluate it by CD11, CD12 and NC. As shown in the Tab. 22 our approach achieves the best performance across all scenes.
Table 22: CD11. CD12 x 100 and NC comparison under 3DScene.
ConvOcc
NP
PCP
GP
Ours
Burghers
CD 100
26.69
1.76
0.267
0.246
0.228
CD
0,077
0,010
0.008
0.008
0.008
NC
0.865
0.883
0.914
0.926
0.934
Lounge
CD12 100
8.68
39.71
0.061
0.055
0.048
CDLi
0.042
0.059
0.006
0.005
0.005
NC
0.857
0.857
0.928
0.922
0.949
Copyroom
CDra x 100
10.99
0.051
0.076
0.069
0.064
CDL1
0.045
0.011
0.007
0.006
0.006
NC
0,848
0.884
0.918
0.929
0.923
Stonewall
CD x 100
19.12
0.063
0.061
0.058
0.043
CDLi
0.066
0,007
0.0065
0,006
0.005
NC
0.866
0.868
0.888
0.893
0.936
Totepole
CD12 100
1.16
0.19
0.10
0.093
0.087
CDr
0.016
0.010
0.008
0.007
0.007
NC
0.325
0.765
0.784
0.847
0.851
Computational Complexity. We report our computational complexity in Tab. 23 we present numerical comparisons with the latest overfitting- methods, including NP and PCP, using different point counts, such 20K and 40K. The benchmark rounds for both NP and PCP are set at 40K. NP does not require learning priors, resulting in the highest operational efficiency. PCP needs to learn priors, which requires additional time. To achieve more refined results, we dedicate extra time to learning the frequency features of point clouds and computing the sampling point strides. Consequently, our speed is slower compared to However, it is noteworthy that our method outperforms PCP and operates faster even without using local priors.
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Table 23: Comparison of computational complexity.
Time/GPU Memory
20K
40K
NP
12min/1.5G
15min/2.3G
PCP
14min/1.90
18min/2.7G
Ours 13min/1.8G
16min/2.5G
D Limitation
We propose method that approximates the accurate signed distance field through multi-step opti- mization, achieving more precise results. However, there is still room for further optimization in terms of time and computational efficiency as shown Tab. and Tab. [4 In future work, we will continue to explore how to integrate multi-resolution (such as NGLOD 16 and Instant-NGP [71) features effectively to balance computational efficiency and accuracy.
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NeurIPS Paper Checklist
1. Claims
Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer:[Yes)
Justification: Our main claims made in the abstract and introduction accurately reflect the paper's contributions and scope.
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Reviews:
[{"review_id": "9YXxqa2EwI", "reviewer": "Reviewer_uTM5", "summary": "This paper proposes to learn multi-scale implicit fields from 3D point clouds for accurate optimization of SDFs in a coarse-to-fine manner. The spatial query points are first mapped to frequency features through the FFT module. Then the MSP module is designed to exploit the multi-level frequency features to progressively pull the query points towards to target surface. Various experiments and ablation studies demonstrate the effectiveness of the proposed learning approach.", "questions": "In lines 82-83, the authors wrote “The creation of 3D shapes LOD usually depends on mesh decimation, which has difficulty in blending between LODs.” I notice that this particular sentence is directly copied from the paper of MFLOD [14]. Explanations are needed.", "limitations": "The presented limitations are vague and less constructive.", "rating": 3, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The adopted frequency-domain learning paradigm and the pulling-based SDF learning paradigm are effective and advantageous.\n2. The experimental comparison shows the performance superiority of the constructed learning pipeline.", "weaknesses": "The major weakness of this work is its limited novelty. In general, I think the proposed techniques are just an incremental combination of the pulling-based SDF learning approach [18] (Neural-Pull) and the multiplicative fourier learning approach [14] (MFLOD), although with some marginal modifications in order to combine them. The authors seem to have intentionally overlooked the introduction and detailed discussion of these two types of highly related works. Hence, reviewers that are not very familiar with the two lines of works may overestimate the value and technical novelty of the proposed method. Note that I am not saying that combining some existing techniques is not meaningful. What I mean is that, for this paper, the contribution cannot reach the bar of NeurIPS.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your acknowledgements in our method. In the following, we respond to the comments with respect to the weaknesses and questions. All experiments presented in the rebuttal are based on the FAMOUS dataset.\n\n##### **(1) Novelty.**\n\n> In general, the proposed techniques are just an incremental combination of the pulling-based SDF learning approach [18] (Neural-Pull) and the multiplicative fourier learning approach [14] (MFLOD), although with some marginal modifications in order to combine them. and particular sentence is from the paper of MFLOD.\n\nAlthough we use feature extraction in MFLOD and pulling operation in NeuralPull as basis, simply combining these two operations does not work well due to unstable optimization or poor performance. Hence, our key contributions include the initialization scheme and multi-step pulling (MSP) modules with multiple constraints, rather than a simple combination. First, our initialization scheme can produce more discriminative frequency features and make each Fourier layer more stable and robust under different frequencies, adapting to more complex geometries, while avoiding gradient vanishing. Second, our MSP module with multiple constraints can infer more accurate SDFs and more consistent gradients using frequency based features as conditions. Whereas NeuralPull infers the SDFs by one-step pulling, it is difficult to directly and accurately reach the target in one step for queries, espcially for the queries that are far from the surface. In addition, NeuralPull does not consider gradient consistency on different level sets, which drastically decreases the accuracy of SDFs. In contrast, MultiPull infers the SDFs by multi-step pulling, the distance-aware constraint pays more attention to queries far from the surface, together with the help of the surface constraint, allowing us to pull any queries to correct locations on the zero-level set. We will make this more clear in our revision.\n\n##### **(2) Missing Citation.**\nWe were supposed to add a citation for the sentence that we get from MFLOD, but we forgot to do that. We will add a citation to this sentence in our revision.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KzMuJwYa6A", "reviewer": "Reviewer_NaoM", "summary": "This paper proposes MultiPull, a method for reconstructing a surface model by SDF from a 3D point cloud containing only the coordinates of each point. The paper proposes a method for estimating the SDF using the Fourier transform of the surface model predicted from the point cloud only at multi-scale. It also introduces gradient consistency and distance awareness for multi-scale consistency. The key idea is to use multi-scale frequency features, for which we introduce an FFT module corresponding to an encoder and an MSP module corresponding to an implicit decoder. The proposed method shows high reconstruction performance compared to existing methods and is effective on different datasets.", "questions": "- l.77: \"However, inferring implicit functions without 3D supervision requires a very long convergence process, which limits the performance of unsupervised methods in large-scale point cloud data.\" I agree with this point from my experiments, but can the author provide a specific reference in the paper?\n- l.99: \"To this end, we constrain query points to be as close to their nearest neighbor on S.\" I understood sampling from the nearest neighbor, but did not understand how Q is constructed. Are non-nearest points just ignored?", "limitations": "- I agree with the limitations described in the paper. This is a promising direction for future work.\n- If there are areas of future work that you would like to see described in Weaknesses and Questions, these may be limitations.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The problem of reconstructing a surface from only 3D point clouds, accessible in a real environment by a 3D sensor, is a common but important problem and an important technology for real-world applications of 3D data. However, the reconstruction performance of existing methods is limited, and a better method is needed. The proposed method aims to fill this gap, and the significance of the research is highly appreciated.\n- The paper is well written, especially in its straightforward and accurate description of the difficulties in setting up the problem. The detailed description of the proposed method also makes the paper easy to understand.\n- The insight into the initialization of MFN is an interesting point, which I consider to be one of the contributions of the paper.\n- Experimental results on various datasets are presented, showing the effectiveness of the proposed method.", "weaknesses": "- l.138: \"\\cdot\" is better than \"\\times\" to avoid confusion with the cross product.\n- My understanding is that there seems to be no guarantee that the scale of the displacement of the query point at each frequency varies with the frequency of interest. Is this understanding correct? Is there some possible constraint on the magnitude of the slope that would be consistent with the frequency of interest? (Or is it that this concern does not exist, since the experimental separation by frequency has been verified?)\n- l.166: For the purpose of consistency, would it not be appropriate to use the average or correlation rather than argmin to select only one of Q1, Q2, or Q3? If you have any additional explanation, I would appreciate it.\n- In the introduction, it would be desirable to specify whether the proposed method is a test-only optimization (i.e., does not require prior training, such as NeuralPull or SALD) or a method that requires prior training (e.g., DeepSDF). Figure 3 in [*1] clearly shows the position of the proposed method in relation to related research. My understanding is that the paper plots to the same point as NGLOD, and it is preferable to explicitly state that only test time is considered. (If my understanding is incorrect, please correct me.)\n- l.243: After considering both performance metrics and time efficiency, we have set Step=3 by default.\", can the authors discuss this trade-off? Given the nature of the LOD and the trade-off with the 3D geometry compression ratio, there may be future applications related to mesh compression.\n\n[*1] Francis Williams, Zan Gojcic, Sameh Khamis, Denis Zorin, Joan Bruna, Sanja Fidler, Or Litany. Neural Fields As Learnable Kernels for 3D Reconstruction. CVPR2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your acknowledgements in our method and writing. In the following, we respond to the comments with respect to the weaknesses and questions in turn. All experiments presented in the rebuttal are based on the FAMOUS dataset.\n\n##### **(1) Concern about frequency feature consistency.**\n\n> There seems to be no guarantee that the scale of the displacement of the query point at each frequency varies with the frequency of interest. Is there some possible constraint on the magnitude of the slope that would be consistent with the frequency of interest? \n\n**RESPONSE**: Yes, your understanding is correct. There is no guarantee for that. But one thing we observed during experiments is that the movement is getting smaller and smaller for the last several pulling since the pulled queries are approaching to the zero-level set. Thus, we use high frequency features as conditions to make the network more sensitive, so that we can capturer high frequency geometry details. We extract different levels of frequency features for the query points in the FFT module and guide multi-step movements in the MSP. As the reviewer mentioned, we separate the frequency features at different scales, so our constraint on frequency feature consistency mainly lies in the direction consistency during multi-step movements. Throughout the continuous displacement process, we use $\\mathcal{L}_{grad}$ to constrain the movement direction of the query points at different steps, ensuring consistency across different frequency features.\n\n##### **(2) Design of $\\mathcal{L}_{grad}$.**\n\n> For the purpose of consistency, would it not be appropriate to use the average or correlation rather than argmin to select one of $Q_1$, $Q_2$, or $Q_3$? \n\n**RESPONSE**: Ideally, the movement directions of query points at different steps ($Q_1$, $Q_2$, and $Q_3$) should keep consistent. We ensure this consistency by calculating the gradient similarity loss $L_{grad}$. To prevent significant deviations in the moving direction during training, we use argmin as a constraint, making the network more sensitive to changes in gradient direction. To further illustrate this, we use argmin as the baseline and compare it with using the average in $\\mathcal{L}_{grad}$. As shown in Tab. A, when using the average to calculate the similarity of query points at different steps, the CD error slightly increases. Conversely, argmin achieves a similar level of similarity but leads to better performance in terms of the CD metrics.\n\n##### **(3) Suggestions of Introduction.**\n\n> In the introduction, it would be desirable to specify whether the proposed method is a test-only optimization. Figure 3 in [1] clearly shows the position of the proposed method in relation to related research. My understanding is that the paper plots to the same point as NGLOD, and it is preferable to explicitly state that only test time is considered.\n\n**RESPONSE**: As the reviewer mentioned, our method can learn signed distance functions without introducing extra priors, which is an overfitting based method. In our revision, we will further clarify the differences between the various types of deep learning networks mentioned in the introduction. Additionally, we will provide clear classification explanations based on the references you provided.\n\n##### **(4) Default setting of Step.**\n\n> After considering both performance metrics and time efficiency, we have set Step=3 by default.\", can the authors discuss this trade-off? \n\n**RESPONSE**: We set Step to 3 by default for two reasons: (1) MSP can bring partial accuracy improvement when learning more steps, but more network parameters are also needed. We present comparisons under different steps in Tab. B. (2) Although deeper frequency features can represent more comprehensive information, combining frequency features from different layers can achieve better results. We denote the combination of the frequency features as ${L_1, ..., L_n}$. We conduct the experiment according to the default settings, and the results are shown in Tab. C.\n\n##### **(5) Related works.**\n\n> \"However, inferring implicit functions without 3D supervision requires a very long convergence process, which limits the performance of unsupervised methods in large-scale point cloud data.\" Can the author provide a specific reference in the paper?\n\n**RESPONSE**: Learning accurate implicit functions with an unsupervised strategy is challenging. Here, we provide some related methods as references: CAP-UDF [2], NP[3], and PCP [4].\n\n##### **(6) Construction of query points.**\n\n> \"To this end, we constrain query points to be as close to their nearest neighbor on S.\" I understood sampling from the nearest neighbor, but did not understand how Q is constructed.\n\n**RESPONSE**: We construct query points based on the sampling methods of NP. For the input point cloud $s_i \\in S$, we sample 80 query points around $s_i$ based on the Gaussian distribution $\\mathcal{N}(s_i, \\sigma^2)$. We use $\\sigma^2$ to control the distance of the query points, we follow NP to set $\\sigma^2$ as the square distance between $s_i$ and its 50-th nearest neighbor. We set the default distance as $0.5 \\sigma^2$.\n\n[1] Francis Williams, Zan Gojcic, Sameh Khamis, Denis Zorin, Joan Bruna, Sanja Fidler, Or Litany. Neural Fields As Learnable Kernels for 3D Reconstruction. CVPR2022.\n\n[2] Zhou J, Ma B, Li S, et al. CAP-UDF: Learning Unsigned Distance Functions Progressively from Raw Point Clouds with Consistency-Aware Field Optimization[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024.\n\n[3] Ma B, Han Z, Liu Y S, et al. Neural-pull: Learning signed distance functions from point clouds by learning to pull space onto surfaces[J]. International Conference on Machine Learning, 2021.\n\n[4] Ma B, Liu Y S, Zwicker M, et al. Surface reconstruction from point clouds by learning predictive context priors[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2022: 6326-6337.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cGMfDH6cdZ", "reviewer": "Reviewer_ZVf1", "summary": "This paper presents a model for reconstructing SDF from point clouds. The proposed approach introduces components, including Frequency Feature Transformation and Multi-Step Pulling, to iteratively refine the reconstructed SDF. The experiments demonstrate that this method performs well across multiple 3D object and scene datasets.", "questions": "1. What’s the intuition for using frequency features and why using LSNN in MSP? \n2. From the visualization, the proposed method seems to produce results with fewer holes and smoother surfaces. Is this due to the model learning better features, or is there a specific loss design different from prior work?\n3. Could you compare the MSP using the same features from prior work to demonstrate its performance even without the proposed feature? This would help to convincingly show its effectiveness.", "limitations": "No negative social impacts have been identified.\nSee weakness.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The use of multiple levels of features and query is a valid and well-founded idea. The experiments are through and the visualization of the results looks good.", "weaknesses": "1. Some intuitions behind the methodology are not clearly explained. Specifically, the rationale for using frequency features to represent query points needs further clarification. While the use of multiple levels of query points is understandable, the choice of LSNN at each level and how query points are used in this model require more detailed explanation.\n2. Some of the notations are inconsistent. S is used for both raw point cloud in L91 and SDF surface on L99. D is used for both feature dimensions and distance.\n3. Minor: The number reported on the last row of Table 7 appears to be wrong.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your acknowledgements in our method and results. In the following, we respond to the comments with respect to the weaknesses and questions in turn. All experiments presented in the rebuttal are based on the FAMOUS dataset.\n\n##### **(1) Motivation of FFT.**\n\n> The rationale for using frequency features to represent query points needs further clarification. \n\n**RESPONSE**: Previous works fail to accurately represent the local geometry of objects by learning global features. While adopting local methods can alleviate this issue, they typically suffer from high parameter counts and extended training times. Inspired by LOD models, we transform complex features into a more intuitive and interpretable frequency domain representation for neural networks. By utilizing simple frequency domain feature transformations and linear layers (Equation (2) and (3) in the main text ), we can effectively represent features at different scales, it also aids in predicting implicit fields from coarse to fine, where high frequency features help produce more sensitive adjust in signed distance field, leading to high frequency geometry details.\n##### **(2) Details about LSNN.**\n\n> While the use of multiple levels of query points is understandable, the choice of LSNN at each level and how query points are used in this model require more detailed explanation.\n\n**RESPONSE**: LSNN is a neural network consisting of 8 fully connected layers with ReLU activation functions. The first 7 layers have a dimension of 512, while the last layer has a dimension of 1. We use it to predict the SDFs and gradient of query points at different steps. The FFT module encodes the query point $Q_0$ into frequency features $y$ at different scales. We select three of these frequency features as conditions for the multi-step offset of the MSP module's query point to predict the implicit field $f$ after the offset. We denote the $i$-th step query points and frequency feature as $Q_i$ and $y_i$, respectively. We concatenate $Q_i$ and $y_i$ and use the LSNN network with shared parameters to predict the signed distance $f(Q_i, y_i)$ and the gradient $\\nabla f(Q_i, y_i)$. In this way, we finally obtain the different levels of query points $Q_1$, $Q_2$, and $Q_3$.\n\n##### **(3) Analysis of Results.**\n\n> From the visualization, the proposed method seems to produce results with fewer holes and smoother surfaces. Is this due to the model learning better features, or is there a specific loss design different from prior work?\n\n**RESPONSE**: We analyze the effectiveness of different modules from two aspects: (1) We conduct ablation experiments on modules other than MSP in Tab. E. We first remove the FFT module and use linear layers to learn query point features for reconstruction, resulting in a significant increase in CD error, which proves the effectiveness of the FFT module. Next, we gradually remove different $L_{sim}$and $\\mathcal{L}_ {surf} $to explore the importance of these constraint terms. After that, we obtain slightly worse results after removing two losses. (2) We further explore the impact of MSP on accuracy. As shown in Tab. B, the CD error increases significantly when the steps are reduced, and we get the worst result when $Step=1 $. Overall, the offset steps and FFT module have the greatest impact for our method.\n\n##### **(4) Feature Comparison.**\n\n> Could you compare the MSP using the same features from prior work to demonstrate its performance even without the proposed feature? This would help to convincingly show its effectiveness.\n\n**RESPONSE**: We followed the reviewer's suggestion to combine MSP with the linear layers and the traditional feature learning encoder PointMLP [1] to explore the effectiveness of MSP. We present the results of combining different feature learning networks with MSP methods in Tab. F. We denote the single moving operation in MSP as Pull and use multiple feature learning networks as baselines. Due to the lack of an FFT module, the same features are used for multiple offsets in linear+MSP and PointMLP + MSP. As shown in Tab. F, the combination of different feature extraction networks and MSP achieved better results in terms of both CD and NC metrics. We further demonstrate the effectiveness of MSP in Fig. A and Fig. B in the PDF. PointMLP / linear+MSP can generate finer local details compared to PointMLP / linear + Pull. \n\n##### **(5) Typo correction.**\n\n> S is used for both raw point cloud in L91 and SDF surface on L99. D is used for both feature dimensions and distance. Minor: The number reported on the last row of Table 7 appears to be wrong\n\n**RESPONSE**: Thank you for the typo errors raised by the reviewer. We will correct these errors in future versions. We apologize for any inconvenience caused during your reading.\n\n[1] Ma X, Qin C, You H, et al. Rethinking network design and local geometry \nin point cloud: A simple residual MLP framework[J]. arXiv preprint \narXiv:2202.07123, 2022.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "H6CB4G0MCj", "reviewer": "Reviewer_VUUa", "summary": "The authors propose a method to reconstruct neural multi-scale SDFs from point clouds using an optimization-based approach. The multi-scale SDF is represented with a network architecture based on previous work [11] and optimized using a new iterative pulling approach, where the SDF for each scale is optimized to pull a query point to the 0-level-set (using the gradient and value of the SDF at the query), starting with a relatively distant query point for the coarsest SDF, and using the updated query position from the result of the previous scale in the next-finer SDF. Losses based on GridPull [20], but adapted to the multi-scale setting are used as objective for the optimization. The authors show that this approach performs better than several exisiting surface reconstruction methods on several synthetic and real-world datasets.", "questions": "A clarification of the design choices could help, as well as a discussion of the differences and advantages/disadvantages compared to the missing related work.", "limitations": "Limitations in the long computation time are discussed. I don't foresee any particular negative societal impacts.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The iterative multi-scale pulling approach seems novel as far as I can tell, and does make sense to me (apart from some details). It effectively optimizes coarser SDFs to describe regions more distant from the surface but to miss details close to the surface, and more detailed SDFs to describe regions close to the surface with more details.\n- The evaluation is quite comprehensive, including several datasets and a good ablation study (although some relevant recent methods are not included), and shows a good improvement over existing work.", "weaknesses": "- The evaluation is missing some relevant recent methods for optimization-based surface reconstruction. (See details below)\n- Some relevant related work is not cited. (See details below)\n- The exposition is missing motivation for some design choices and is hard to understand in some parts. (See details below)\n\nOverall, even though the authors missed some related work, results look quite promising and I think the idea is novel and interesting enough to make me lean towards acceptance. The exposition can probably be improved somewhat in the final version.\n\nDetails:\n\n- The field of surface reconstruction from point clouds is quite vast, so the authors missed several works, for example (see also related work within these papers to get more missing work):\n  -Training-free methods:\n    - Neural-Singular-Hessian: Implicit Neural Representation of Unoriented Point Clouds by Enforcing Singular Hessian, Wang et al., TOG 2023\n    - Iterative Poisson Surface Reconstruction (iPSR) for Unoriented Points, Hou et al., TOG 2022\n  - Learning-based methods:\n    - 3DShape2Vecset: A 3D Shape Representation for Neural Fields and Generative Diffusion Models, Zhang et al., TOG 2023\n    - Geoudf: Surface reconstruction from 3d point clouds via geometry-guided distance representation, Ren et al., TOG 2023\n  - Given good normals, surface reconstruction becomes much easier - the normal computation could be followed by Poisson reconstruction based on the normals for example. Therefore papers to compute oriented normals are relevant, such as:\n    - Orienting Point Clouds with Dipole Propagation, Metzer et al., TOG 2021\n    - Globally Consistent Normal Orientation for Point Clouds by Regularizing the Winding-Number Field, Xu et al., TOG 2023\n    - SHS-Net: Learning Signed Hyper Surfaces for Oriented Normal Estimation of Point Clouds, Li et al., CVPR 2023\n\n- Several design choices are not motivated well enough:\n  - It is unclear to me why the SDF evaluated at Q_1 uses a feature y_1 as input that was computed from a query position Q_0, rather than a feature computed at query position Q_1. This is an interesting design choice that could use more discussion. An ablation would also be interesting: how would the method perform when re-evaluating h_0, ...,  h_i at Q_i instead of Q_0 when computing y_i for f(Q_i, y_i) (in that case inputting Q_i a second time to the LSNN might also not be necessary)?\n  - In Eq. 6, the design choice of only scaling D_1 and D_2 with the error-based factor, but not D_3 needs to be motivated. Also, as Section 3.2 was written for a general number of layers N_L, it would be good to continue this generality in Section 3.3, if possible. How would Eq. 6 look for a general N_L., for example?\n  - In Eq. 7, the design choice for using the minimum over the three levels.\n  \n- The exposition is hard to understand in some parts and could be improved:\n  - Several passages are currently hard to understand and could use clarification. For example: \"The is because Q_i  locate at different scales in space,, making it difficult to simultaneously account for the optimization step sizes at different scales with the inconsistent gradient directions caused by continuous movement.\" Its possible to get a good guess at what the authors mean here, but it requires re-reading a few times and is not 100% clear even then. Partially this seems to be due to bad grammar, partially due to a somewhat convoluted formulation. Generally the paper text could use more passes to avoid these issues.\n  - In Section 3.1, it should be mentioned early on that the FFT module is based on (or at least very similar to) MFN [11]. Also, the exact differences between the proposed approach in Section 3.1 and the approach of MFN [11] should be discussed and clarified.\n  - In Section 3.3, it would be good to mention that the loss functions are based on GridPull [20]. Also, it would be good to discuss the exact differences to GridPull (I assume the adpation to the multi-scale setting is the main difference).\n  - The architecture of the Linear Sequence Neural Network (LSNN) is never described. If it is taken from existing work, this work should be cited, otherwise it needs to be described, at least in Appendix A.\n  - In the paragraph starting at line 112 and Eq. 4, it is unclear which weights are initialized according to this strategy (the omega_i in Eq. 1? The W_i in Eq. 3? The offset theta_i and b_i?), and which parameters are plotted in Figure 3 (the L_i are not defined clearly).\n  - In Eq. 4, \\eta is not defined.\n  - Near line 138, the query point update should probably use Q_{i-1} everywhere on the right-hand-side (instead of Q_i, which is not known yet). Or alternatively, Q_{i+1} on the left-hand side.\n  - In Eq. 1, I don't follow why the predicted signed distance f(Q_1, y_1) is in the loss. Why should that be optimized to be 0? This should be clarified.\n  - in Eq. 1, \", i \\in [1,I]\" can be removed, since it is redundant with the subscript in the sum\n  - In the section starting at line 160, how is the target surface point q determined? This is described in the appendix, but should be shortly mentioned in the main paper as well.\n  - In Eq. 7, argmin should probably be min instead. Also, the \"1 - ...\" in both lines seems odd, should it only be in one of the lines (otherwise the loss would be *higher* the more similar the gradient directions are, instead of lower).\n  - Near Eq. 8, the authors mention that L_surf encourages the SDF at the *final* query point to be close to 0, but Eq. 8 does not show an index for Q. Is L_surf applied to Q_{N_L} only or to all Q_i? This should be clarified.\n  - Near line 202: Fig. 4.1 should probably be  Fig. 5\n  - In Table 7, Step 5 should probably have CD_L2 of 0.0357 (not 0.357)\n  - SAL [15] is in the bibliography, but not referenced from Section 2.\n  - Citations [45] and [59] are duplicates.\n\n- An inference time comparison between methods should be included.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your acknowledgements in our method and results. In the following, we respond to the comments with respect to the weaknesses and questions in turn. All experiments presented in the rebuttal are based on the FAMOUS dataset.\n\n##### **(1) Further comparison with related works.**\n\n> The field of surface reconstruction from point clouds is quite vast, so the authors missed several works\n\n**RESPONSE**: We follow the reviewer's suggestion to make a further comparison of the related works. Due to the time limit, we just select overfitting based method Neural-Singular-Hessian [1] for visual comparison, and we will add additional comparison results of other work in our version. As shown in Fig. C in the PDF, our method still has competitive performance compared to Neural-Singular, and we perform better in generating local details.\n\n##### **(2) Concern about frequency features.**\n\n> It is unclear to me why the SDF evaluated at $Q_1$ uses a feature $y_1$ as input that was computed from a query position $Q_0$, rather than a feature computed at query position $Q_1$. An ablation would also be interesting: how would the method perform when re-evaluating $h_0$, ..., $h_i$ at $Q_i$ instead of $Q_0$ when computing $y_i$ for f($Q_i$, $y_i$)\n\n**RESPONSE**: All frequency features y in the FFT are obtained from the query point $Q_0$. we transform the query point $Q_0$ into frequency features as conditions to guide the movement of $Q_0$, $Q_1$, and $Q_2$. The deeper the features in the FFT, the more comprehensive the information contained. We hope to use this movement strategy to enable MSP to learn the global information of the input from low-frequency features, and then optimize local details through high-frequency features. In summary, the frequency features of FFT come from the same set of query points. If we re-evaluate the $Q_i$ of each stage into FFT, it will disrupt the consistency of the features. Therefore, FFT cannot simultaneously accept multiple point clouds for frequency feature encoding.\n\n##### **(3) Details of $\\mathcal{L}_{recon}$.**\n\n>  In Eq. 6, the design choice of only scaling $D_1$ and $D_2$ with the error-based factor, but not $D_3$ needs to be motivated. How would Eq. 6 look for a general $N_L$., for example?\n\n##### **(4) Design choice for $\\mathcal{L}_{grad}$.**\n\n> In Eq. 7, the design choice for using the minimum over the three levels.\n\n**RESPONSE**: In order to maintain consistency in the moving direction of query points at different steps during continuous offset, we naturally choose gradient similarity $L_{grad}$ as a constraint term. Furthermore, we force the network to focus on the step where the gradient direction changes the most. We conduct experiments according to reviewer uTM5 to demonstrate the validity in Tab. A.\n\n**RESPONSE**:  We force network to focus more on outliers and points close to the surface to accelerate network convergence in $L_{recon}$. Therefore, we make the error-based factors of $D_3$ higher than  and $D_1$ and $D_2$. We define a set $D$={$D_{1}, D_{2}, D_{3}$} and its corresponding weights $w$={$\\alpha, (1- \\alpha )^{\\gamma}, 1$}, then $L_{recon}$ can be expressed as: $L_{recon}$ = $\\sum^{3}_{i=1}w_i D_i$.\n\n##### **(5) Inference time comparison.**\n\n> An inference time comparison between methods should be included.\n\n**RESPONSE**: In Tab. G, we further evaluate inference time with NP and PCP. The results show that the local based method (PCP) has the longer inference time than the global based method (NP, Multiplaull), with NP having the fastest inference time.\n\n##### **(6) Explaination about details.**\n\n> The exposition is hard to understand in some parts and could be improved.\n\n**RESPONSE**: Thank you for the reviewer's suggestions for writing revisions and reminders for adding details. Here we will present the issues that require further discussion, and we will refine their writing suggestions and revision suggestions in the next version.\n\n##### **Differences of GridPull.**\n\nOur method uses pulling based loss like GP [2]. However, our optimization strategy is different from GP. GP uses grid points to perform trilinear interpolation on point inputs to predict the SDFs, which does not require any neural network for optimization. Instead, We need multi-scale frequency features as conditions to move the query points toward the underlying surface of the point cloud in multiple steps. This process uses LSNN network to learn SDFs and gradients at different steps. As shown in the visual comparison in the main text, optimization based methods can learn smoother surfaces.\n\n##### **Architecture of LSNN.**\n\nLSNN consists of 8 linear layers, with seven layers having a dimension of 512, and the last layer having a dimension of 1. We use LSNN to predict the signed distance value $f (Q_i, y_i)$ of the query point $Q_i$ under the frequency feature $y_i$ after the $i-th$ step of movement. We will add relevant content in the subsequent appendix. We apologize for any inconvenience caused to your reading.\n\n##### **Construction of query points.**\n\nWe construct query points based on the sampling method of GridPull [2]. Due to space limitations, reviewer can refer to reviewer NaoM's question (6) for a detailed introduction.\n\n##### **Scope of use of $\\mathcal{L}_{surf}$.**\n\nWe only use $\\mathcal{L}_{surf}$ at $f(Q_3)$ to make the network more confident in predicting the zero-level-set on the point cloud surface.\n\n[1] Wang Z, Zhang Y, Xu R, et al. Neural-Singular-Hessian: Implicit Neural \nRepresentation of Unoriented Point Clouds by Enforcing Singular \nHessian[J]. ACM Transactions on Graphics (TOG), 2023, 42(6): 1-14.\n\n[2] Chen C, Liu Y S, Han Z. Gridpull: Towards scalability in learning \nimplicit representations from 3d point clouds[C]//Proceedings of the \nieee/cvf international conference on computer vision. 2023: 18322-18334.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all the reviewers for their thoughtful comments and valuable feedback. We sincerely thank reviewers uTM5 and NaoM for their supplementary requirements on the paper details and our contribution experiments. Through these experiments, we further demonstrate the scalability and unique contributions of our method. We also appreciate the additional requirements from ZVf1 and VUUa for the experimental content, as well as the analysis of exploratory experiments. These additional experiments significantly enhance the quality and readability of the manuscript . **We provide detailed results in rebuttal and PDF.**

We also value the reviewers' recognition of our paper's writing, presentation, and metrics. We will respond each of the reviewers' questions individually. The additional results and discussions will be added in the revised draft or supplementary materials.

Meta-review: The paper proposes MultiPull, a multi-scale / multi-step pulling approach to reconstruct a mesh surface from point cloud.  MultiPull trains a neural network to learn a SDF by using multi-scale frequency features. Experiments on several datasets shows that compared to baseline methods, the proposed method has higher reconstruction accuracy and lower errors.

Reviewers are split in their ratings on this work, with three indicating acceptance (ZVf1,VUUa,NaoM) and one voting for reject (uTM5).

The main criticism from reviewer uTM5 is that the proposed method is a combination of pulling-based SDF Neural-Pull [18] and the multiplicative Fourier level of detail MFLOD [14].  In their response, the authors note that it was not trivial to combine the two approaches, as to make use of features from MFLOD and pulling operation from NeuralPull, details of how to initialize and the design of multi-step pulling module was required.  The AC also notes that novel combination of techniques (whether well-known or not) can be valuable (as also noted in the NeurIPS 2024 reviewer guidelines).  As most reviewers find the work to be interesting and there does not appear to be significant flaws with the evaluation, the AC believe this work would be of interest to the NeurIPS community and recommends acceptance.

The authors are encouraged to follow reviewer suggestions to improve the paper:
- Include additional experiments and ablations provided during rebuttal
- Improvements to writing (see detailed comments from VUUa) including adding recent relevant work, discussion of design choices, improve clarity of exposition.
- Please proofread for typos and grammar.  Some (non-exhaustive) examples (in addition to the suggestions from VUUa) are given below:
   - Reword copied sentence (L82-83)   
   - Equations
      - Improve notation (see R-ZVf1 comments)
      - L138: Use "\cdot" not "\times"
      - Improve typesetting (use `\sin', '\cos' for sin/cos, '\text{pull}', '\text{recon}', etc for words in equations)
   - Correct number in last row of Table 7 (ZVf1)
   - Line 2: "Latest" => "the latest" (or just "Recent")
   - Line 4: "distnaces" => "distances"
   - Figure 1: "Setp 2" => "Step 2"
   - Table 1: "ShapNet" => "ShapeNet"
   - Line 211: "compare" => "compared"
Decision: Accept (poster)