$p$-Poisson surface reconstruction in curl-free flow from point clouds
Authors: Yesom Park
Taekyung Lee
Jooyoung Hahn
Myungjoo Kang
Venue: NeurIPS 2023 poster (2023)
Keywords: Surface reconstruction
Signed distance function
Implicit neural representations
Point cloud
TL;DR: $p$-Poisson eqquation-based implicit representation of surfaces that not only provides smooth reconstruction but also recover high-frequency features only from raw point clouds.

Abstract: The aim of this paper is the reconstruction of a smooth surface from an unorganized point cloud sampled by a closed surface, with the preservation of geometric shapes, without any further information other than the point cloud. Implicit neural representations (INRs) have recently emerged as a promising approach to surface reconstruction. However, the reconstruction quality of existing methods relies on ground truth implicit function values or surface normal vectors. In this paper, we show that proper supervision of partial differential equations and fundamental properties of differential vector fields are sufficient to robustly reconstruct high-quality surfaces. We cast the $p$-Poisson equation to learn a signed distance function (SDF) and the reconstructed surface is implicitly represented by the zero-level set of the SDF. For efficient training, we develop a variable splitting structure by introducing a gradient of the SDF as an auxiliary variable and impose the $p$-Poisson equation directly on the auxiliary variable as a hard constraint. Based on the curl-free property of the gradient field, we impose a curl-free constraint on the auxiliary variable, which leads to a more faithful reconstruction. Experiments on standard benchmark datasets show that the proposed INR provides a superior and robust reconstruction. The code is available at https://github.com/Yebbi/PINC.

p- p-Poisson surface reconstruction in curl-free flow from point clouds
Yesom Park
Taekyung Lee Jooyoung Hahn', Myungjoo Kang'
Department of Mathematical Sciences, Seoul National University
2 Interdisciplinary Program in Artificial Intelligence, Scoul National University
Department of Mathematics and Descriptive Geometry, Slovak University of Technology in Bratislava {yeisom, dlxorud1231, mkang) @snu. ac kr jooyoung hahn@stuba.sk
Abstract
The aim of this paper is the reconstruction of smooth surface from unorga- nized point cloud sampled by a closed surface, with the preservation of geometric shapes, without any further information other than the point cloud. Implicit neural representations (INRs) have recently emerged as a promising approach to surface reconstruction. However, the reconstruction quality of existing methods relies ground truth implicit function values or surface normal vectors. In this paper, we show that proper supervision of partial differential equations and fundamental prop- erties of differential vector fields are sufficient to robustly reconstruct high-quality surfaces. We cast the p-Poisson equation to learn signed distance function (SDF) and the reconstructed surface is implicitly represented by the zero-level set of the SDF. For efficient training. we develop a variable splitting structure by introducing a gradient of the SDF as an auxiliary variable and impose the p-Poisson equation directly on the auxiliary variable as hard constraint. Based on the curl-free prop- erty of the gradient field, we impose curl-free constraint on the auxiliary variable, which leads to a more faithful reconstruction. Experiments on standard benchmark datasets show that the proposed INR provides superior and robust reconstruction. The code available at htthttp::// //g::////tttbbtbg www.///be/////iieeoeeoee
1 Introduction
Surface reconstruction from an unorganized point cloud has been extensively studied for more than two decades [10 40
9 62 due to its
many downstream applications in computer vi- sion and computer graphics 8 16 62 58). Clas- sical point cloud or mesh-based representations are efficient but they do not guarantee a water- tight surface and are usually limited fixed geometries. Implicit function-based representa- tions of the surface
2x
14 as level [ Cj of continuous 3
set S = {x E 3 implicit function distance functions (SDFs) or occupancy func- tions, have received considerable attention for providing watertight results and great flexibility
Eiconal
Poisson
P-Possoon curl free
+ IR, such signed
Figure 1: Comparison of reconstruction using an eikonal equation (9), the p-Poisson equation (8) and the proposed p-Poisson equation with the curl-free condition (II).
'Equal contribution authors. Correspondence <mkang@anu. kr.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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in representing different topologies. In recent years, with the rise of deep learning, stream of work called implicit neural representations (INRs) 4 16
53 50 has revisited them by parameterizing the implicit function with neural networks. INRs have shown promising results by offering efficient training and expressive surface reconstruction.
61
19
55
Early INRs [44, 42 16 treat the points-to-surface problem as a supervised regression problem with ground-truth distance values, which are difficult to use in many situations. To overcome this limitation, some research efforts have used partial differential equations (PDEs), typically the eikonal equation, as a means to relax the 3D supervision 23 48 While these efforts have been successful in reconstructing various geometries, they encounter an issue of non-unique solutions in the eikonal equation and rely heavily on the oriented normal vector each point. They often fail capture fine details or reconstruct plausible surfaces without normal vectors. A raw point cloud usually lacks normal vectors or numerically estimated normal vectors [ī] contain approximation errors. Moreover, the prior works are vulnerable to noisy observations and outliers.
The goal of this work is to propose an implicit representation of surfaces that not only provides smooth reconstruction but also recovers high-frequency features only from a raw point cloud. To this end, we provide novel approach that expresses an approximated SDF as the unique solution to the p-Poisson equation. In contrast to previous studies that only describe the SDF as a network, we define the gradient of the SDF as an auxiliary variable, motivated by variable splitting methods 4 60 I2 in the optimization literature. We then parameterize the auxiliary output to automatically satisfy the p- p-Poisson equation by reformulating the equation in a divergence-free form. The divergence-freee splitting representation contributes to efficient training by avoiding deeply nested gradient chains and allows the use of sufficiently large p. which permits an accurate approximation of the SDF In addition, we impose curl-free constraint [25] because the auxiliary variable should be learned as a conservative vector field which has vanishing curl. The curl-free constraint serves to achieve a faithful reconstruction. We carefully evaluate the proposed model on widely used benchmarks and robustness to noise. The results demonstrate the superiority of our model without a priori knowledge of the surface normal at the data points.
2 Background and related works
Implicit neural representations In recent years, implicit neural representations (INRs) 41 16 3
53 [54], which define surface zero level-sets of neural networks, have been extensively studied.
Early work requires the ground-trure truth signed implicit function 44 16 41), which is difficult to obtain
in real-world scenarios. Considerable research 3 is devoted to removing 3D supervision and relaxing it with a ground truth normal vector at each point. In particular, several efforts use PDEs to remove supervision and learn implicit functions only from raw point clouds. Recently, IGR (23) revisits a conventional numerical approach [14] that accesses the SDF by incorporating the cikonal equation into a variational problem by using modern computational tools of deep learning. Without the normal vector, however, IGR misses fine details. To alleviate this problem, FFN (56] and SIREN (55) put the high frequencies directly into the network. Other approaches exploit additional loss terms to regulate the divergence 66] or the Hessian 663) The vanishing viscosity method, which perturbs the eikonal equation with a small diffusion term, is also considered [37 49 to mitigate the drawback that the eikonal loss has unreliable minima. The classical Poisson reconstruction [31] which recovers the implicit function by integration over the normal vector field, has also been revisited to accelerate
the model inference time [88. but supervision of the normal vector field is required. Neural-Pull (39] constructs a new loss function by borrowing the geometrical property that the SDF and its gradient
define the shortest path to the surface.
p- P-Possson equation The SDF is described by a solution of various PDEs. The existing work uses the eikonal equation, whose viscosity solution describes the SDF However, the use of the residual of the eikonal equation as a loss function raises concerns about the convergence to the SDF due to non-unique solutions of the eikonal equation. Recent works (5 o utilize the
12
55 6
notion of vanishing viscosity to circumvent the issue of non-unique solutions. In this paper, we use the p-Poisson equation to approximate the SDF, which is a nonlinear generalization of the Poisson equation (p- = 2):
- --. (IVU)P= Vu) 1 in Ω
(1)
u u= 0
2

---

where 2 2, the computation domain Q C R is bounded, and I is embedded in S2.
The main advantage of using the p-Poisson equation is that the solution (I) is unique in Sobolev space WIP (52) [36]. The unique solution with 2 2 brings a viscosity solution of the eikonal equation in the limit p - oo, which is the SDF, and it eventually prevents finding non-viscosity solutions of the eikonal equation; see a further discussion with an example in Appendix C.I] Moreover, in contrast to the eikonal equation, it is possible to describe solution of [ as a variational problem and compute an accurate approximation [ 20]:
Vull" dx
/ udx.
min
(2)
S2
As p - oC, it has been shown [T 30 that the solution of (I) converges to the SDF whose zero level set is r. As result, increasing / gives a better approximation of the SDF, which is definitely helpful for surface reconstruction. However, it is still difficult to use fairly large p in numerical computations and in this paper we will explain one of the possible solutions to the mentioned problem.
3 Method
In this section, we propose a p- p-PPisson equation based Implicit Neural representation with Curl-free constraint (PINC). From unorganized point cloud X' {x; i = 1, 2, .N3 sampled by closed surface r, a SDF u R - IR whose zero level set is the surface l (x RR u (x) = 0} reconstructed by the proposed INR. There are two key elements in the proposed method: First, using variable-splittngg representation 455 of the network, auxiliary output is used to learn the gradient the SDF that satisfies the p-Poisson equation @ Second, curl-free constraint enforced on an auxiliary variable to ensure that the differentiable vector identity is satisfied.
3.1 p- Poisson equation
loss function in the chysics-informen framework [5T] of the existing INRs for the p-Poisson equation (I) can be directly written:
|u] dx + do V. (IVull vu) + dx,
min
where Ao is regularization constant. To reduce the learning complexity of the second integrand, we propose an augmented network structure that separately parameterizes the gradient of the SDF as auxiliary variable that satisfies the p-Poisson equation
that use neural networks with only one output u for the SDF, we introduce a separate auxiliary network output G for the gradient of the SDF; see that the same principle is used in [45). In the optimization literature, it is called the variable splitting method [47 60 22 [2] and it has the advantage of decomposing complex minimization
Variable-splittnng strategy Unlike existing studies 23 37
into a sequence of relatively simple sub uu--problems. With the auxiliary variable G = Vu and the
penalty method [3), the variational problem 3 converted into an unconstrained problem:
min / |u| dx + Ao / u.G
v (IGIP-2
+ dx + A1 /
|| GII? dx.
(4)
where d is a penalty parameter representing the relative importance of the loss terms.
p- P-Poosson as a hard constraint Looking more closely at the minimization { if G is already gradient to satisfy (T), then the second term in 4 is no longer needed and it brings the simplicity of one less parameter. Now, for function F Ω + R such that V F 1. for example F (x) 1 the p- PPissson equation (I) is reformulated by the divergence-fiee free form:
V (IVull* - Vu + F) 0.
(5)
Then, there exists a vector potential v : R + RR3 satisfying
|G| G p- G
(6)
3

---

where G = Vu. Note that similar idea is used in the neural conservation law [52 to construct
divergence- vector field built on the Helmholtz decomposition B3 [57. From the condition (6)
we have ||G|||=] | = | x V - F'll and G parallel to V x V F, then the auxiliary output G is
explicitly written:
G= V
V x -
This confirms that the minimization problem 44) does not require finding G directly, but rather that can be obtained from the vector potential y. Therefore, the second loss term in 4 can be eliminated by approximating the potential function by neural network and defining the auxiliary output G as
hard constraint To sum up, we use loss function of the form
Sf |u| dx +
Lp-Poisson
|u|
I|Vu - G1|² dx,
(8)
where G is obtained by 77). the first term is responsible for imposing the boundary condition of (T). and the second term enforces the constraint G = Vu between primary and auxiliary outputs. is
worth mentioning that G in (7) is designed exactly satisfy the p-Poisson equation (T.
An advantage of the proposed loss function and the hard constraint 77) is that ( can be solved for sufficiently large p. which is critical to make better approximation of the SDF. It is not straightforward in (3) or because the numeric value (p -pooer with large p easily exceeds the limit of floating precision. On the other hand, in 7 we use (p 2p/(p 1)-power, which allows stable computation even when p becomes arbitrarily large. The surface reconstruction with varying p in Figure 7 shows that using large enough p is crucial to get a good reconstruction. As the p increases, the reconstruction gets closer and closer to the point cloud. Furthermore, is worth noting that the proposed representation expresses the second-order PDE ①T) with first-order derivatives only.
By reducing the order of the derivatives, the computational graph is simplified than (3) or (4).
Note that one think of an approach to directly solve the eikonal equation auxiliary variable H = Vu as an output of the neural network:
Vull 1 with an
min x. #|||
[n|| dx + 7)
Vvu H1
where 7) > O. The above loss function may produce non-unique solution of the eikonal equation, which causes numerical instability and undesirable estimation the surface reconstruction; see Figure T To alleviate such an issue, the vanishing viscosity method is used in [37 49] to approximate
the SDF by Uh as a - 0, a solution of -a + sign (u,) (u, - 1) = O. However, the results
dependent the hyper- peeer--rameterr > 0 related to the resolution of the discretized computational
domain and the order of the numerical scheme [7
3.2 Curl-free constraint
In the penalty method, we have to compute more strictly to ensure that G = Vu by using progressively larger values of A in (8) but practice we cannot make the value of X₁ infinitely large. Now, we can think of yet another condition for enforcing the constraint G = Vu from differential vector identity which says a conservative vector field curl-free:
V
G
(10)
for some scalar potential function While it may seem straightforward, adding a penalty term Sa |/V x G dx at the top of (8) is fraught with problems. Since G is calculated by using curl operation (7). the mentioned penalty term makes a long and complex computational graph. In addition, it has been reported that such loss functions, which include high-order derivatives computed by automatic differentiation, induce a loss landscape that is difficult to optimize [34] 59]. J. In order to relax the mentioned issue, we augment another auxiliary variable G, where G G and V x G constrained.
By incorporating the new auxiliary variable & and its curl-free constraint, we have the following loss function:
LPINC = L.p + A/ / /[lc G|( dx + As / V v
(11)

---

x
ML.P
Primary output
(x))
(xx
D P
Input
P-2
IV X P-1
Auxillary output
(x)
(x)
P(P)
Auxiliary ousput
Figure The visualization of the augmented network structure with two auxiliary variables.
Note that the optimal G should have unit norm according to the eikonal equation. To facilitate training, we relax this nonconvex equality condition into convex constraint G || 1. To this end, parameterize the second network auxiliary output L and define G by
T
G = V
(12)
1,
max
where the projection operator to the dineensinniii unit ball. Appendix A provides further discussion on the importance of the curl-free term to learn conservative vector field.
Figure iillustrates the proposed network architecture. The primary and the auxiliary variables trained a single network, instead of being trained separately in individual networks. The number of network parameters remains almost the same since only the output dimension of the last layer is increased by six, while all hidden layers are shared.
3.3 Proposed loss function
In the case of real point cloud to estimate closed surface by range scanners, is inevitable have occluded parts the surface where the surface has a concave part depending on possible angles of the measurement (35) It ends up having relatively large holes in the measured point cloud. Since there are no points in the middle of the hole, it is necessary to have a certain criterion for how to fill in the hole. In order to focus check the quality of LPINC III in this paper, choose simple rule to minimize the area of zero level set of u:
Liotal LPINC + d4 / 8, (u) Vull dx,
(13)
where A4 > 0 and 5, (x) = 1 tanh" [ is smeared Dirac delta function with > 0. The minimization of the area is used in [21 the point cloud to provide better performance of the reconstruction.
49 and the advanced models [5] 27 63
missing parts of
Experimental results
In this section, we evaluate the performance of the proposed model to reconstruct 3D surfaces from point clouds. We study the following questions: (i) How does the proposed model perform compared existing INRs? (ii) it stable from noise? (iii) What is the role of the parts that make up the model and the loss? Each is elaborated in order in the following sections.
37]. we use an 8-layer network with 512 neurons and skip connection to the middle layer, but only the output dimension of the last layer is increased by six due to the auxiliary variables. For 133. we empirically set the loss coefficients to Å = 0.1, X2 0.0001. A3 0.0005, and Å 0.1 and use p
Implementation
in previous studies
in (7) for numerical simplicity. We implement all numerical experiments on single NVIDIA RTX 3090 GPU. In all experiments, we use the Adam optimizer [32 with learning rate 1 decayed by 0.99 every 2000 iterations.
5

---

Point Cloud
IGR
SIREN
DIGS
PINC(ours)
Anchor
Gargoyle
Buser head
Screwstar
Squirrel
Figure 3: 3D Reconstruction results for SRB and Thingi datasets.
Datasets We leverage two widely used benchmark datasets to evaluate the proposed model for surface reconstruction: Surface Reconstruction Benchmark (SRB) [7] and Thingi10K [65]. The geometries in the mentioned datasets are challenging because of their complex topologies and incomplete observations. Following the prior works, adopt five objects per dataset. We normalize the input data to center at zero and have a maximum norm of one.
Baselines We compare the proposed model with the following baselines: IGR 23 SIREN (55),
SAL [3) PHASE [37] and DiGS (6) All models are evaluated from only raw point cloud data without surface normal vectors. A comparison with models that leverage surface normals as supervision is
included in Appendix C
Metrics To estimate the quantitative accuracy of the reconstructed surface, we measure Chamfer (dc) and Hausdorff (dw) distances between the ground-truth point clouds and the reconstructed surfaces. Moreover, we report one-sided distances dz and dz between the noisy data and the reconstructed surfaces. Please see Appendix B.2 for precise definitions.
4.1 Surface reconstruction
We validate the performance of the proposed PINC (13 in surface reconstruction in comparison to other INR baselines. For a fair comparison, we consider baseline models that were trained without normal prior. summarzzss the numerical comparison on SRB in terms of metrics. We report
the results of baselines from [37 49 6 The results show that the reconstruction quality obtained par with the leading INRs, and achieved state-or-the-arr performance for Chamfer distances.

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Table 1: Results on surface reconstruction of SRB.
Anchor
Daratech
DCC
Gargoyle
Loard Quas
GT
Scuns
GT
Scams
GT
Scans
GT
Scans
GT
Scans
Model
de
du
2p
"B
de
djs
d2
d
de
du
de
dy
2p
d
de
du
2z
IGR
0.45
7,55
0.17
4,55
4.9
42.15
0.7
3.68
0.63
10.35
0.14
344
0.77
17.46
0.18
2.04
0.16
4.22
0.08
1.14
SIREN
0.72
10.98
1.27
0.21
4.37
1.78
0.34
6.27
0.06
2.71
0.46
7.76
0.08
0.68
0.35
8.96
0.06
0.65
SAL
0,42
7.21
0.17
4.67
0.62
13:21
0.11
2.15
0.18
3,06
0.08
2.82
0.45
9.74
0.21
3.84
0.13
414
0.07
4.04
PHASE
0.29
7,43
0.09
1.49
0.35
7.24
0.08
1.21
0.19
4.65
0.05
2.78
0.17
4.79
0.07
1.58
0.11
0.71
0.05
0.74
DIGS
0.29
7.19
0.20
3.72
1.80
0.15
1.70
0.07
2.75
4.10
0.09
0.92
0.91
0.06
0.70
PINC
0.29
7.54
0.09
1.20
0.37
7.24
0.11
1.88
0.14
2,56
0.04
2.73
0.16
4.78
0.05
0.80
0.10
0.92
0.04
0.67
Table 2: Results on surface reconstruction of Thingi 10K.
Squirrel
Buser head
Screwstar
Frogrock
Pumpkin
Model
de
dy
de
dæ
de
dy
de
dæ
de
dæ
IGR
0.36
11.97
0.38
5.95
0.18
3.02
0.48
12.05
0.11
1.13
SIREN
0.47
5.66
0.43
4.81
0.27
4.98
0.78
14.75
0.46
5.03
DIGS
0.50
12.45
0.39
10.64
0.26
6.33
0.45
10.50
0,32
8.03
PINC
0.35
11.55
0.37
6.19
0.17
3.00
0.43
11.06
0.10
1.90
We further verify the accuracy of the reconstructed surface for the Thingi10K dataset by measuring the metrics. For Thingil0K, we reproduce the results of IGR, SIREN, and DiGS without normal vectors using the official codes. Results on Thingil0K presented in Table 2 show the proposed method achieves superior performance compared to existing approaches. PINC achieves similar or better metric values on all objects.
The qualitative results are presented in Figure 3 SIREN, which imposes high-frequency features to the model by using a sine periodic function as activation, restores a somewhat torn surface. Similarly, DiGS restores rough and rogged surfaces, for example, the human face and squirrel body are not smooth and are rendered unevenly. On the other hand, IGR provides smooth surfaces but tends to over-smooth details such as the gargoyle's wings and detail on the star-shaped bolt head of screwstar. The results confirm that the proposed PINC (13) adopts both of these advantages: PINC represents smooth and detailed surface. More results can be found in the Appendix
4.2 Reconstruction from noisy data
In this section, analyze whether the proposed PINC produces robust results the presence of noise in the input point data. In many situations, the samples obtained by the scanning process contain lot of noise and inaccurate surface normals are estimated from these noisy samples. Therefore, is an important task to perform accurate reconstruction using only noisy data without normal vectors. To investigate the robustness to noise, we perturb the data with additive Gaussian noise with mean zero and two standard deviations 0.005 and 0.01.
We quantify the ability of the proposed model to handle noise in the input points. The qualitative results are shown in Figure 4 Compared to existing methods, the results demonstrate superior resilience of the proposed model with respect to noise corruption in the input samples. We can observe that SIREN and DiGS restore broken surfaces that appear to be small grains as the noise level increases. On the other hand, the proposed model produces a relatively smooth reconstruction. Results show that PINC is less sensitive to noise than others.
4.3 Ablation studies
This section is devoted to ablation analyses which show that each part of the proposed loss function Eisai in conjunction with the divergence-free splitting architecture plays important role in high- quality reconstruction.
Effect of curl-free constraint We first study the effect the curl-free constraint on reconstructing high fidelity surfaces. To investigate the effectiveness of the proposed curl-free constraint, we compare the performance of PINC without the curl-free loss term, i.e., the model trained with the loss 8). The results the SRB dataset are reported in Table and Figure 5 Figure
function Poisoonn
7

---

Point Cloud
IGR
SIREN
DIGS
PINC (ours)
Low noise
High moise
Figure 4: Reconstruction results from noisy observations. Two levels of additive Gaussian noise with standard deviations a 0.005 (low) and 0.01 (high) are considered.
shows that the variable splitting method, which satisfies the p-Poisson equation hard constraint (without the curl-free condition), recovers a fairly decent surface, but generates oversmoothed surfaces and details are lost. However, the curl-free constraint, this constraint allows us to capture the details that PINC without the curl-free condition cannot recover. The metric values presented in Table also provide clear evidence of the need for the curl-free term. To further examine the necessity of another auxiliary variable G conduct an additional experiment by applying the curl-free loss term directly on without the use of G. The results are presented in the second row the Table The results indicate that taking curl G. which is constructed by taking curl in 7 leads to a suboptimal reconstruction. This is likely due to a challenging optimization landscape that is difficult to optimize as a result of consecutive automatic differentiation [59] The results provide numerical evidences of the necessity of introducing G.
from the qualitative result reconstructed with
Table 3: Quantitative results on the ablation study of the curl-free term.
GT
Scans
Model
dc
d
d
wo/ curl free
0.20
4.96
0.12
2.98
w/ curl free on G 4.17
52.26
0.48
6.03
w/ curl free on & 0.16
4.78
0.05 0.80
curl-fice
w/ curl-free
Figure 5: Comparison of surface reconstruction with-
out (left) and with (right) curl-free constraint.
Effect of minimal area criterion We study the effect of the minimal area criterion suggested in Section 3.3 In real scenarios, there are defected regions where the surface has not been measured. To fill this part of the hole, the minimum surface area considered. Tigure/Jcleard Some parts in the daratech of SRB have hole in the back. Probably because of this hole, parts that are not manifolds are spread out as manifolds as shown in the left figure without considering the minimal area. However, we can see that adding minimal area loss term alleviates this problem. We would like to note that, except for daratech, we did not encounter this problem because other data point clouds sampled from a closed surface and also not related to hole filling. Indeed,
shows this effect.

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empirically observe that the results are quite similar with and without the minimal area term for all data other than daratech.
(a) wa' area loss
(b) w/ area loss
Figure 6: Comparison of surface recovery without (a) and with (b) minimum area criterion.
Effect large? p The p-Poisson equation (T) draws the SDF as p becomes infinitely large. Therefore, it is natural to think that it would be good to use large Here, we conducted experiments on the effect of p. We define with various p = 2, 10, and 100 and learn the SDF with Figure shows surfaces that were recovered from the Gargoyle data in the SRB with different values. When is as small as
obvious that it is difficult to reconstruct compact surface from points. When is 10, much better surface is constructed than that of p 2, but the by-products still remain on small holes. Furthermore, a large value of p = 100 provides quite proper reconstruction. This experimental result demonstrates that more accurate approximation can be obtained by the of large p. which consistent with the theory. This once again highlights the advantage the variable splitting method we have proposed, which allows arbitrarily large p to be used. This highlights the advantage of the variable splitting method (7) we have proposed Section 3.1 which allows an arbitrarily large p be used. Note that the previous approaches have not been able to large p because the numeric value of p-power easily exceeds the limit of floating precision. On the other hand, the proposed method is amenable large and hence the reconstruction becomes closer the point cloud.
5 Conclusion and limitations
We presented a p-Poisson equation-based shape representation learning, termed PINC, that recon- structs high-fidelity surfaces using only the locations given points. We introduced the gradient of the SDF as an auxiliary network output and incorporated the p-Poisson equation into the auxiliary variable as hard constraint. The curl-free constraint was also used to provide more accurate representation. Furthermore, the minimal surface area regularization was considered to provide compact surface and overcome the ill illeposedsess of the surface reconstruction problem caused by unobserved points. The proposed PINC successively achieved a faithful surface with intricate details and was robust noisy observations.
The minimization of the surface area is used to reconstruct missing parts of points under the assump- tion that a point cloud is measured by closed surface. Regarding the hole-filling strategy, it still needs further discussion and investigation of various constraints such as mean curvature or total variation of the gradient. At present, the proposed PDE-based framework is limited closed surfaces and is inadequate to reconstruct open surfaces. We leave the development to open surface reconstruction as future work. Establishing neural network initialization that favors the auxiliary gradient of the SDF would be an interesting venue. Furthermore, the computational cost of convergence would differ when using and not using auxiliary variables. Analyzing the convergence speed computational cost of utilizing auxiliary variables versus not utilizing them is worthwhile direction for future research.
Societal Impacts
The proposed PINC allows high-quality representation of 3D shapes only from raw unoriented 3D point cloud. It has many potential downstream applications, including product design, security, medical imaging, robotics, and the film industry. We aware that accurate 3D surface reconstruction

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p=
(b) p
p 100
Figure 7: Surface reconstruction of anchor data with various p. The results show the importance of using a sufficiently large for an accurate approximation.
can be used in malicious environments such as unauthorized reproduction of machines without consent and digital impersonation. However, it is not work to develop technique go to abuse, and we hope and encourage users of the proposed model to concenter on the positive impact of this work.
7 Acknowledgements
This work was supported by the NRF grant [2012R A2C3011888) and the MSIT/IIT ([1711117093,, 22021-0000777, [No. 2021-0-01343, Artificial Intelligence Graduate School Program(SNU)), Also, this project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Cure grant agreement No. 945478.
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A More discussion on curl-free term
This section is devoted both theoretically and empirically validate the necessity of the curl-free loss term. One might think that the curl-free term is unnecessary, since curl-free G can be obtained by reducing the L penalty term for the variable splitting constraint Vu = G. However, this penalty term is not sufficient to train G conservative vector field. In subsequent sections, we prove this theoretically and verify experimentally that this is indeed the case in practice.
A.1 Theoretical justification
In the following theorem, we opine that minimizing the L energy of VVu G| in 8 without the curl-free term is not sufficient to obtain conservative vector field G.
Theorem A.1. There is a sequence {un, Gn I neN such that Jq V Gn dx - as n + co, but Gn, does not converge curl-free vector field.
Proof. For every {unInes defined on 52 [0, 1] set
Gn (x, = Vun (z. 2) + (o. sin (2Th.).0) .)) E a...
Then,
S Vun -Gull dxl
/
1
0, n
sin (2mna), dx
(14)
1
sin (2Tnz) dx
(15)
n²
a
+ 0,
(16)
as nl + 00. However,
V x Gn (z, y, = (0, 0, cos (2Tnx))
does not converges to zero.
Remark A.2. Note that for a G, set in the proof of the above theorem A.1 Jg V G.ll dx= is positive constant independent of n. This implies that can prevent the pathological example above by adding the curl-free loss term. Therefore, the curl-free term is necessary to accurately learn the gradient field G.
=
A.2 Empirical Validation
To examine the practical effect of the curl-free term on learning a conservative vector field, we include experimental results on simple example of sphere of radius 0.5 centered at the origin. Figure
depicts the level set contours of the trained u of a cross section cut at the planes x = 0.2 and 0.4, along with the vector field of the trained G projected onto this plane together with the gradient field of the true SDE. We note that the p-Poisson equation (1) gives an SDF that is positive on the interior of the surface and negative on the outside, however, the contours depicted in Figures g a Iand/rre are of the opposite sign of the trained u. As shown in the Figure 8 the model trained without the curl-free term learns vector field that not curl-free, resulting in G being distinct from the true gradient field. This ultimately impedes from correctly learning the SDF On the other hand, it is evident that the model trained with the curl-free term converges fairly close to the true gradient field. This ultimately helps u to accurately learn the SDF.
B Implementation Details
In this section, we provide more details about the implementation for reproducibility. Note that our code is built on top of IGR (MIT License).
15

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x = 0.2
X 0.4
1.40
1.40
0.5
0.95
w/ curl-free
0.05
0.05
-0.40
-0.40
1.40
wo/ curl-free
0.50
0.50
0.05
-0400
Figure 8: The trained results of a cross section cut in planes x 0.2 (left) and x 0.4 (right). The level-sets show the signed distance fields w learned by the proposed model with (top) and without (bottom) the curl-free term. Dashed contours depict the learned zero level set. Quivers represent the vector field of the trained auxiliary variable G' and the true gradient fields are plotted in red arrows.
B.1 Experimental Setup
Parameter Tuning The proposed training loss Liaal (13) is weighted sum of five loss terms with four regularization parameters Å1, A2, Az. and A4. In all surface reconstruction experiments, use À 0.1, X2 0.0001, A3 0.0005, and X 0.1. In the proposed model, p is also hyperparameter to be chosen. Considering the theoretical fact that should be infinitely large and numerical simplicity,
set p We empirically confirm significant difference between when 100 and when p cc. Moreover, we set the smoothing parameter for approximating Dirac delta in (13).
Network Architecture As in previous studies 44 23
37), we represent the primary and auxiliary
outputs by single 8-layered multi-layer perceptron (MLP) R$ IR? with 512 neurons and skip connection to the fourth layer, but only the output dimension of the last layer is increased by six due to the auxiliary variables; Figure 2 We use softplus activation function a (1) = In (1((1 82)
with 3 100. Network weights are initialized by the geometric initialization proposed in
Training details The gradient and the curl of networks are computed with auto-differentiation library (autograd) [46]. In all experiments,
use the Adam optimizer [32) with learning rate decayed by 0.99 every 2,000 iterations. At each iteration, we uniform randomly sample 16,384 points x É X' from the point cloud X. We sample the collocation points of I as provided in (23] The collocation points consist of global points and local points. The local collocation points are sampled by perturbing each of the 16,384 points drawn from the point cloud with a zero mean Gaussian distribution with standard deviation equal to the distance to the 50th nearest neighbor. The global collocation points are made up of approximately 2,000 points from the uniform distribution U (-1. with 7 1.1. F f utilized in all experiments.
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Baseline models For baseline models on the Thingi 10K dataset, we use the official codes of IGR (MIT License). SIREN License), and DiGS (MIT License). We faithfully follow the official implementation to train each model without normal prior. For the variable splitting representation of the eikonal equation ( there is single auxiliary output. Consequently, we use the same 8 layer MLP with 512 nodes, but network with an output dimension of 4. We normalize the auxiliary output to make a unit norm, and use the normalized one to represent H.
B.2 Evaluation
Metrics We measure the distance between two point clouds X' and y by using the standard one- sided and double-sided & Chamfer distances dzz de and Hausdorff distances dz- dg- Each defined as follows:
(x,))
Σ
min |X yl12. yEy
X
XEX
do (X23) 1
2 (dg (K2 + 2p (2)
dz (X.)) = max min ||x - yl12. XE yE)
dæ ()) max {d (X,J) + dz (2,
When we estimate the distance from surface, we sample 10M uniformly random points from the surface and then measure the distance from the sampled point clouds by the metrics defined above.
Furthermore, in order to measure the accuracy of the trained gradient field, we evaluate Normal Consistency (NC) 4I between the learned G and the surface normal follows: from given oriented point cloud X' {x,, n.);=1 comprising of sampled points x, and the corresponding outward normal vectors n,, NC is defined by
N NC n) = Σ i=1
G(x)"
(17)
the average of the absolute dot product of the trained G and the surface normals.
Level set extraction We extract the zero level set of trained neural network by using the classical marching cubes meshing algorithm (38 on a 512 x 512 x 512 uniform grid.
C
Additional Results
C.1 Uniqueness of the solution of p-Poisson equation
In this section, we provide a numerical example supporting the strength of the proposed model regarding the uniqueness of the solution to the p-Poisson equation. The given points are located on cube centerd at the origin with an edge of the length 1. We consider IGR [23] as an eikonal- rooted baseline, and train IGR and the proposed PINC with the following three different network initializations: The geometric initialization [23] that IGR originally used and the Kaiming uniform initialization [26] with different random seeds. The results summarized in Figure 9 The results show that IGR converges to different solutions depending on the model initializations. In particular, IGR fails to learn the SDF of the cube except for the geometric initialization. On the other hand, the results of PINC with the same initializations show that the proposed model converges to the SDF in all three cases. The numerical results of the chosen example show that the proposed method can pursue the unique solution of the PDE.
C.2 Additional comparison with models utilizing surface normals
In Section 4 we made comparison with models that do not use the surface normal supervision. Here, we additionally consider a comparison with models that leverage normal supervision. We
http:://ggecabbb.cc////Cuuyyyy/p
17

---

IGR (Initialization 1)
IGR (Initialization 2)
IGR (Initialization 3)
0.72
PINC (Initialization 1)
PINC (Initialization 2)
PINC (Initialization 3)
Figure 9: Experimental results show whether a method can find the SDF from different network initializations. IGR and PINC are trained the synthetic cube data with three network initializations: geometric initialization (initialization I) and Kaiming initialization with two different random seeds (initializations 2 and 3). Each depicts the trained level-set contours a cross-section cut in the plane x O. Red contours depict the trained zero level set of numerical solutions.
Table Comparison with models that use the surface normal supervision n SRB. The proposed model PINC did not utilize the surface normal.
Anchor
Daratech
DC
Gargoyle
Loard Quas
Scans
GT
Scans
Scuns
GT
Scans
Scans
Model
dg
dy
dg
de
dz
dg
de
du
dg
"
de
dy
d
dc
dg
dg
d
VisCO
0.21 3.00 0.15
1.07
0.26
4.06
0.14
1.76
0.15
2.22
0.09
2.76
0.17
4,40
0.11
0.96
0.12
1.06
0.7
0.64
w/n
IGR
0,22
4.71
0.12
1.32
0.25
4.01
0.08
1.59
0.17
2.22
0.09
2661
0.16
3.5
0.06
0.81
0.12
1.17
0.07
0.98
8.83
293
0.22
1.66
0.04
0.18
5.54
1.73
0.13
0.04
1.17
wo/n
PINC
0.29 7.54 0.09
0.37
7.24
0.11
1.88
0.14
2.56
0.04
2.73
0.16
4.78
0.80
0.10
0.04
0.67
consider three baseline models follows: (i) IGR that evaluates using the surface normal, (ii) VisCo [49]. a grid-based method based on the viscosity regularized eikonal equation, and (iii) Shape As Points (SAP) [48). model that revisits the classical Poisson Surface Reconstruction (PSR) [3] using deep learning. The results are reported in Table 4 We can see that the proposed model performs on par with baselines, despite utilizing the surface normal. Considering that all of the baselines are PDE-haaed INR models, the results exhibit the effectiveness of the proposed model (TT) in reconstructing surface from the sole use of raw point clouds.
is worth note that the proposed model may be interpreted as PSR because of More precisely, the Euler-Lagrange equation of (8) says that the variational problem (8) for finding a scalar function u whose gradient best approximates a given vector field transforms into the following Poisson problem:
Au V u u=
G in Ω
(18)
o.
In the conventional PSR, the gradient field is set to surface normals. Thus, the auxiliary variable G' can be regarded playing role of surface normals for PSR. However, the vector field G in the proposed model is not obtained from the oriented point cloud, trained with u at the same time. Moreover, since we bake the p-Poisson equation into G as a hard constraint (7), we obtain continuous SDF rather than an indicator function like PSR and SAP. The results confirm that simultancous training of the gradient field and the SDF, that is, the variable
learnable function that
18

---

Table 5: Normal consistency of reconstructed surfaces on SRB.
Model
Anchor
Daratech
DC
Gargoyle
Lord Quas
IGR 0.9706
0.8526
0.9800
0.9765
0.9901
SIREN
0.9438
0.9682
0,9735
0,9392
0.9762
DiGS
0.9767
0.9680
0.9826
0.9788
0.9907
SAP 0.9750
0.9414
0,9636
0.9731
0.9838
PINC
0.9754
0.9311
0.9828
0.9803
0.9915
Table 6: Normal consistency of reconstructed surfaces on Thingi 10K.
Model
Squirrel Pumpkin
Frogrock
Screstar Buser head
IGR 0.9820
0,9565
0,9509
0,9709
0.9249
SIREN 0.9529
0.8996
0.9035
0,9142
0.8860
DIGS 0.9557
0,9353
0.9468
0,9386
0.9171
SAP 0.9791
0.9520
0.9319
0.9767
0.9004
PINC
0.9816
0.9583
0.9545
0.9805
0.9376
splitting method, achieves similar even better surface restoration than SAP, without using the given surface normal n.
C.3 Additional quantitative results
We reported Chamfer distances and Hausdorff distances in Tables and 2 but these metrics do not reflect the complete quality of the restored surface. Here, we evaluate Normal Consistency (NC) (17) which measures how well the model can capture higher order information of the surface. The results on both SRB and Thingil0K datasets are summarized in Tables 5 and 0 respectively. Overall, the proposed model achieves a better NC score than baseline models. In particular, the results show that the proposed model achieves superior NC for the tested examples than SAP, even though it does not employ surface normal supervision.
Moreover, we measure Chamfer distance and Hausdorff distance for ablation studies reported in Figures and of Section 4.3 and summarized them in the Tables 7 and respectively.
More results effeec of Theoretically, accurate SDF can be obtained grows infinitely. That the same story continues in practice confirmed by the results shown in Figure I0 We can see that the larger p induces a better reconstruction. This phenomenon is also observed in Figure 7 Moreover, it can be seen that p = oo, which we used the implementation, gives a similar qualitative result
100. These experimental results once again remind us how important it is to be able to use a large p.
Furthermore, provide numerical verification for the use of oo in Figure 11 convenience, we use the subscript Up to denote the dependence the solution on the parameter p- Figure TT depicts graphs of the mean squared error (MSE) of Up and t over different MSEs are computed by discretizing the computational domain Q into a 100 x 100 x 100 uniform grid. The results show that the MSE decreases as p increases. In other words, it confirms that Up is getting closer to Uho as p grows, which supports the justification for using p = co.
For notational
Table 7: Quantitative results on the effect of area loss on daratech.
GT
Scans
Model
de
dæ
dg
z
PINC wo/ area loss
4.26
53.34
0.20
2.81
PINC w/ area loss
0.37
7.24
0.11
1.88
19

---

Table 8: Quantitative results on the ablation study of p.
Gargoyle
Anchor
GT
Scans
GT
Scans
Model
de
dæ
c
77
de
7
p
3,96
43.13
0.51
6.31
4.32
46.66
14.58
p
0.22
8.14
0.10
0.50
7.24
0.12
3.02
p 100
0.17
4.90
0.10
0.82
0.31
7.20
1.80
p oc
0.16
4.78
0.80
0.29
7.19
1.17
p=2
p= 10
100
Figure 10: Quality of surface reconstruction with varying p from 2 to p
-1.0
-1.5
2.0
log MSE(U 8
-2.5
Anchor Gargoyle
-3.0
-3.5
-4.0
-4.5
0.5
1.0
1.5
2.0
2.5
3.0
log p
Figure 11: MSEs of and uho over different p.
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---

Point Cloud
IGR
SIREN
DIGS
PINC(ours)
Lord quas
De
Daratech
Frogrock
Pumpkin
Figure 12: Additional qualitative results of the surface reconstruction on SRB and Thingi10K datasets.
C.4 Additional qualitative results
Figure 12 provides additional qualitative results of surface reconstruction on SRB and Thingi 10K discussed in Section
Reconstruction of large point clouds We further provide qualitative results for surface reconstruc- tion from large models taken from Thingi10K. The adopted point clouds consist of from 35K to 980K vertices. Figure 13] depicts the qualitative reconstruction results of PINC these large point clouds. The model is trained with the same configuration used in Section 4.1
C.5 Training/Inference time
To investigate the computational time of the proposed model, We carefully measured average execution time compared to baselines. In the Table we report the average training time per iteration and inference time resolution of 323 voxels. The proposed model requires more computational cost than baseline models because the computation on curl using automatic differentiation.
21

---

Figure 13: Reconstructed surfaces of large point clouds from Thingi 10K.
Table 9: Training and inference times for surface reconstruction on SRB,
Model
IGR
SIREN
DIGS
PINC
Training time (ms/iteration)
48.34
13.11
52.34
295.01
Inference time (ms)
6.86
3.51
4.39
6.93
22


Reviews:
[{"review_id": "YM56ITqHGJ", "reviewer": "Reviewer_rqMu", "summary": "The paper considers the task of surface reconstruction from point clouds without normals with INRs. They consider solving for the SDF by solving the p-Poisson equation (with manifold constraint) as p\\to\\infty. However as the obvious loss function form of this is difficult to optimise, they consider a variable splitting strategy. They also do this w.r.t. ensuring that their gradient solution is conservative. Finally they add a minimal surface area regularisation term.", "questions": "- What is $F$? Is it kept to the example given, $1/3x$?\n- I am confused why the curl-free constraint needed. By equation 10, the new curl-free constraint ensures that $G=\\nabla v$ for some $v$, but why allow it to be some $v$ that is not the $u$ being outputed as the INR value, and/or instead constrain it to be similar to the current INR value $u(x)$?\n- I would like more intuition on the role of the curl-free constraint. Figure 5 seems to show that it forces the model to pay attention to detail more, however I don't see why this is the case theoretically. Is it because the eikonal term is loosely being enforced within the curl-free objective by construction?\n- The good results of IGR on Thingi10K seems unlikely given its bad performance on SRB, it is almost as good as your model. Is there a property of Thingi10K that causes this? Is there a reason your model and IGR would be so similar on Thingi10K?\n- The performance of your model on Daratech in SRB seems a bit confusing, in the results it does really badly on $d_C$ however it looks fairly good in Figure 6?", "limitations": "Some discussion given.\nNo clear potential negative societal impact or broader societal impacts to discuss", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Variable splitting is a nice approach to the issues with neural networks and automatic differentiation\n- Model is backed strongly by theoretical intuition\n- Auxilary variables sharing the network structure is nice\n- Decent results", "weaknesses": "- It would be nice to have some intuition as to why you propose to use the p-Poisson equation rather than other PDEs like the Eikonal equation. At first glance it seems that the reason is because it is possible to describe it as a variational problem as shown in equation 2, however that doesn't get used by your method. Another argument you posit is that without the vanishing viscosity method a normal eikonal PDE based solution may produce a non-unique weak solution, why is it clear that your method does not produce a non-unique weak solution? Is it because of the curl-free constraint being enforced?\n- How important is the enforcement of the loose eikonal constraint within the curl-free constraint? It somewhat diminishes the story of trying to solve the SDF problem using a different PDE to the eikonal equation.\n- It is not clear why the curl-free constraint is needed. $\\nabla u$ is curl-free by design, so isn't minimising $||\\nabla u - G||$ in (8) sufficient? Why is a separate auxiliary variable necessary, apart from enforcing a loose Eikonal constraint by construction? Doesn't the argument about needed to set $\\lambda_1$ infinitely large to enforce the constraint apply to $\\lambda_3$ as well?\n- The qualitative diagrams (Figure 5-7) for the ablation study are great for intuition and understanding, however you should have quantitative results on what happens when you remove each of those components (especially for curl-free)\n\nI like this type of approach, willing to increase the score if clarity on the necessity of both the curl-free constraint and the eikonal construction constraint as well as quantitative ablations are given.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful feedback. We hereby carefully address your questions as follows:\n\n**Q1. It would be nice to have some intuition as to why..**\n\n**Reply**:  Please check the answer to the common question above.\n\n**Q2. How important is the enforcement of the loose eikonal constraint..**\n\n**Reply**: Imposing the curl-free condition helps to learn $G$ and $u$ accurately. The necessity of the curl-free constraint is explained in detail in the answer to Q3.\n\n**Q3. It is not clear why the curl-free constraint is needed..** \n\n**Reply**: Minimizing $L_2=\\int_\\Omega \\parallel \\nabla u_n - G_n\\parallel^2 dx$ in (8) is not a pointwise-manner. There is a sequence {$u_n,G_n$} such that $L_2 \\rightarrow 0$ but $G_n$ does not converge to a curl-free field. For every $u_n$ defined on $\\Omega=[0,1]^3$, set $G_n(x,y,z)=\\nabla u_n(x,y,z) + (0,1/n\\sin(2\\pi nx),0)$. Then, $L_2\\rightarrow 0$ but $\\nabla\\times G_n=(0,0,\\cos(2\\pi nx))\\nrightarrow 0$. Note that $\\int_\\Omega\\parallel\\nabla\\times G_n\\parallel^2=1/2$ is constant. This implies that we can prevent the pathological example above by adding the curl-free loss term.\nTherefore, the curl-free term is necessary to accurately learn $G$.\n\nIf we impose the curl-free loss directly on $G$ without using $\\tilde{G}$, we have to take curl on $G$, which is constructed by computing curl on $\\Psi$ in (7). However, applying automatic differentiation (AD) consecutively leads to excessive memory consumption and computational inefficiency. In addition, the objective with a high-order derivative using AD has a challenging loss landscape that is difficult to optimize (Wang, 2021).\nWe introduced the additional auxiliary variable $\\tilde{G}$ to avoid these problems. We conducted an additional experiment by imposing the curl-free loss term directly on $G$ without using $\\tilde{G}$. The results are reported in the Table A1 of the attachment. The results shows of the necessity of introducing $\\tilde{G}$.\n\nThe requirement of $\\lambda_1 \\rightarrow \\infty$ is theoretically important to penalize the constraint in the penalty method. We may gradually increase $\\lambda_1$ during training. However, as $\\lambda_1$ becomes larger, the balance between the loss terms becomes imbalanced and other terms could be ignored. It means that the condition $u=0$ on $\\Gamma$ is not properly enforced and it is also difficult to obtain the curl-free constraint.\n\n**Q4. The qualitative diagrams..**\n\n**Reply**: We summarized quantitative metrics (Chamfer and Hausdorff distances) in the Tables A1, A2, and A3 in the attachment.\nTo further investigate the effect of the curl-free term on the learning of $G$, we measure the difference between the given surface normal $n$ and the learned $G$. Given point cloud with normals {$x_i,n_i$},\nwe estimate the cosine similarity (CS) $G^Tn:=\\frac{1}{N}\\sum_{i=1}^N |G(x_i)^T n_i |$, and report it in the Table A4 in the attachment. The results show the angle difference of $G$ and $n$ differing by an average of almost 1.50 (Anchor) and 3.57 (Gargoyle) degrees. Also, $\\nabla u$ also has similar differences with $n$. It numerically validates that the curl-free term brings more accurate results to learn $G$ and $u$ for the given test cases. We would like to emphasize that CS does not reflect errors that occur when the 0-level set of $u$ is far from the given point cloud, as it is computed only at points where the $n$ is defined. So, it does not fully describe the quality of the trained surface and gradient fields, but we evaluate it to show the effect of curl-free term on learning accurate gradient fields where the $n$ is defined.\n\n**Q5. What is $F$?..**\n\n**Reply**: Yes, $F=1/3x$ was used in all experiments. In the revised manuscript, we'll make it clear.\n\n**Q6. I am confused why the curl-free constraint needed..**\n\n**Reply**: The necessity of the curl-free constraint is explained in the answer of Q3. In other aspects, for a given $G$ in (10), we agree with the reviewer that $u$ satisfying $G=\\nabla u$ is not unique. However, the first term in (8) brings the uniqueness of $u$ such that $G=\\nabla u$.\n\n**Q8. The good results of IGR on Thingi10K..**\n\n**Reply**: The similar tendency between IGR and the proposed model seems to be due to the initialization and activation function of the network. Both IGR and our model use the softplus activation and initialize the network to be approximately the SDF of a sphere. On the other hand, SIREN and DiGS use a sine activation and initialize the network in different ways. These are almost the only difference between IGR and SIREN, but the results show that IGR restores smoother surfaces. Therefore, IGR and our model seem to have a tendency to restore smooth surfaces due to these differences.\n\n**Q9. The performance of your model on Daratech..**\n\n**Reply**: A given point cloud of Daratech has an empty part at the back. The proposed model restores the surface by filling in this part, which is why the metric could be high. After the submission of the paper, we have found that $\\beta=0.1$ restores this part better. We have reported the additional metric values in Table A2 in the attachment. \n\n**Q10. No clear potential negative societal impact..**\n\n**Reply**: We will specify societal impacts in the revised manuscript as follows: \"The proposed PINC allows high-quality representation of 3D shapes only from raw unoriented 3D point cloud. It has many potential downstream applications, including product design, medical imaging, and the film industry. We are aware that accurate 3D surface reconstruction can be used in malicious environments such as  unauthorized reproduction of machines without consent and digital impersonation. However, it is not a work to develop a technique to go to abuse, and we hope and encourage users of the proposed model to concenter on the positive impact of this work.\"\n\n**Reference** \n\nS. Wang et al. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM, 2021."}, {"review_id": "9mzeBkkYb4", "reviewer": "Reviewer_ymPA", "summary": "The paper presents a surface reconstruction method that uses only raw point clouds. It enforces the Poisson surface equation implicitly over the SDF representation of the surface. As a consequence, it obtains smooth surfaces with preserved details without any 3D supervision or apriori knowledge of normals. The experiments show that the performance is comparable to sota methods which require data beyond raw point clouds.\n", "questions": "see weakness section", "limitations": "to some extent, the limitations are discussed.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The use of p-poisson equation to describe SDF is well-motivated.\nThe use of auxiliary variables relating to the gradient and curl of SDF is interesting and convincingly reduces the computational complexity.\nAblation study is a plus.\nThe performance is comparable to the methods that use either 3D supervision or oriented normals.\nThe performance on noisy data is generally better than sota. I think it is due to the fact the method uses only raw point clouds that serves as a benefit here. Normal computation on noisy point clouds can be disproportionately erroneous.", "weaknesses": "There is no theoretical motivation/argument provided to choose p-poisson over eikonal equation to describe SDFs.\n\nIGR[23] can perform surface reconstruction without normals as well. It is not clear whether the authors used the Normals while evaluating IGR. A comparison with both: IGR with and without normals should have been considered.\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable feedback. We carefully address your questions as follows:\n\n**Q.1 There is no theoretical motivation/argument provided to choose p-poisson over eikonal equation to describe SDFs.**\n\n**Reply**: For a detailed response to the reviewer's question about the theoretical motivation for adopting the $p$-Poisson equation instead of the eikonal equation to describe SDFs, please check the response to the common question above.\n\n**Q2.IGR[23] can perform surface reconstruction without normals as well..**\n\n**Reply**: We compared the performance with IGR without normal vectors $n$. As the reviewer recommended, we present the comparison of results with IGR with and without $n$, and the proposed model in the Table A5 of the attachment."}, {"review_id": "J2obBm3aQL", "reviewer": "Reviewer_FaEi", "summary": "This paper considers the problem of reconstruction of a smooth surface from an unorganised point cloud. The proposed approach is based on neural implicit function while without normal information. The main contribution of this work is that they demonstrate that proper supervision of partial differentiable equation and fundamental properties of differential vector fields are enough to reconstruct high-quality surfaces. A novel part is to develop a variable splitting structure by introducing a gradient of the SDF as an auxiliary variable and a curl-free constraint on the auxiliary variable.\n\nThe experimental results somehow demonstrate the effectiveness on some aspects.", "questions": "With additional auxiliary variable, does the optimization take long time to converge? If yes, please list the computation time with more details.", "limitations": "As listed in the above.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I like the idea by introducing auxiliary variable to solve the optimization problem under the framework of neural implicit function. Actually, they all both solve optimization problem. Therefore, the optimization strategy in numerical algorithm can be adopted for neural implicit function based representation. This paper shows a good example in this aspect, and might inspire some interesting along this direction.", "weaknesses": "The results shown in Tab. 1 and Tab. 2 are not good enough. Can you explain why the performance is not good enough for different dataset and metrics?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the valuable comments. Below, we carefully address the reviewer's comments:\n\n**Q1. The results shown in Tab. 1 and Tab. 2 are not good enough..**\n\n**Reply**: Each shape of data has a different level of difficulty due to its own challenging characteristics, such as complex topology, missing data, sharp corners, a high level of detail, and so on.\nTherefore, it is natural for the model to perform differently on these different data. The proposed model has the advantage of learning an SDF based on the $p$-Poisson equation, which implicitly represents the surface, allowing for accurate and smooth reconstruction of closed surfaces. Therefore, it is difficult to say that surface reconstruction of the proposed model performs better than other models on data that cannot highlight these advantages. We only confirmed that the experimental results of the proposed model achieved comparable results to leading INR models across the various data. Moreover, different metrics quantify different features.\nThe proposed model seems to show better results with Chamfer distances than with Hausdorff distances.\nHowever, these two metrics do not reflect the complete quality of the restored surface. We evaluate normal consistency (L. Mescheder, 2019) as recommended by the reviewer WkvV (Please see Tables 2 and 3 below the answer to Q4). We would like to note that the proposed model achieves better normal consistency for the tested examples. \n\n**Q2. With additional auxiliary variable, does the optimization take long time to converge? If yes, ..**\n\n**Reply**: We agree with the reviewer that, in general, the computational cost of using and not using auxiliary variables is undoubtedly not the same. However, we have technical difficulties to make a fair comparison on this issue. In the case of auxiliary variable $\\tilde{G}$, the results show a significant difference in performance with and without using it; see more details in Figure 5. It means that the convergent point may not be the same with and without using $\\tilde{G}$, so it is difficult to compare convergence speeds under equivalent conditions when the exact solution is not specifically known. In the case of auxiliary variable $G$ cannot be excluded due to the construction of the proposed model. Nevertheless, analyzing the convergence speed or computational cost of using and not using auxiliary variables is indeed a crucial topic and it should be studied rigorously and mathematically with a very simple and meaningful loss function. We thank the reviewer for pointing out such a worthwhile future research topic. We will add this as future work in Section 5."}, {"review_id": "Mj3eI5IA86", "reviewer": "Reviewer_WkvV", "summary": "The paper introduces an intriguing approach to surface reconstruction, but further enhancements in terms of completeness and evaluations would strengthen its contribution to the field. Also, the gradient of the SDF acts as an auxiliary network output and incorporated the Poisson equation as a hard constraint. The proposed method was also used to propose a more accurate representation. They perform some experiments on standard benchmark datasets to demonstrate superior and robust reconstruction. In my view, the proposed method cannot achieve the best one numerically on average (table 1).", "questions": "1. Your method appears to be computationally intensive. Can you provide information on the training and inference times compared to other methods?\n\n2. While your results look promising, including quantitative results on surface quality would be valuable.\n\n3. It would be unfair to directly compare the proposed method with approaches that utilize normal information. However, conducting experiments and comparisons with methods such as \"shape as points\" from the URL [https://pengsongyou.github.io/sap] would provide valuable insights and contribute to a more comprehensive evaluation of the proposed method.\n\n4. What are lines 457-460?", "limitations": "The limitation has been roughly discussed in the paper.\nPoisson-based methods, including the proposed approach, are unable to handle open surfaces.", "rating": 5, "confidence": 5, "soundness": 1, "presentation": 1, "contribution": 3, "strengths": "This paper proposes a novel surface reconstruction method using the p-Poisson equation and a curl-free constraint, which is highly interesting. It demonstrates superior performance compared to previous works.", "weaknesses": "The authors have submitted a revision of the full paper in the supplementary material, it may be considered a violation of the rules. It may be appropriate to consider resubmitting the paper to another venue due to the violation.\n\nThere are several incomplete pieces of information.\n\nIt would be beneficial to include evaluations to measure surface quality, such as normal consistency, as well as provide details on training and inference times.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the thoughtful comments. Below, we carefully address the reviewer's comments:\n\n**Q1. The authors have submitted a revision of the full paper in the supplementary material..**\n\n**Reply**: The reviewer's point is indeed correct if the main argument, main idea, or computational results of the proposed algorithm had been changed between the original paper and the paper with the supplementary material. The paper with the supplementary material has precisely the same arguments and ideas as the original paper. The only changed parts are i) the results of SIREN, ii) a typographical error in Figure 7, and iii) some of the incomplete information in the reference. The superiority of the results remains almost the same, so none of the main arguments in the paper are affected. We would like to note that we revised the manuscript to provide accurate values for the reviewer's better judgment, not because we wanted to change or strengthen the main argument.\n\n**Q2. It would be beneficial to include evaluations to measure surface quality,..**\n\n**Reply**: Thanks for the comments on further evaluations. We provide the related answers in Q4 for a measure of surface quality regarding surface quality; see Tables 2 and 3. Training/inference times are reported in the answer to Q3; see Table 1.\n\n**Q3. Your method appears to be computationally intensive..**\n\n**Reply**: We investigate the training/inference times of the proposed model compared to other models. In the Table 1 below, we report the average training time per iteration on SRB dataset and inference time at a resolution of $32^3$ voxels. \nAs the reviewer mentioned, the proposed model requires more computational cost than baseline models because of the computation on curl using automatic differentiation.\n\n[Table 1] Training/Inference times\n|Time|IGR|SIREN|DiGS|PINC (ours)|\n|:---|:---:|:---:|:---:|:---:|\n|Training time (ms/iteration)|48.34| 13.11 | 52.34 | 295.0|\n|Inference time (ms)| 6.86 | 3.51 | 4.39 | 6.93|\n\n**Q4. While your results look promising, including quantitative results on surface quality would be valuable.**\n\n**Reply**: Thank you for the suggestion. As normal consistency (L. Mescheder, 2019) is recommended in Q2, it is tested and the results are reported in the Tables 2 and 3 below. Overall, compared to other models, the proposed model achieves a better normal consistency.\n\n[Table 2] Normal Consistency on SRB\n|Model|Anchor|Daratech|DC|Gargoyle|Lord Quas|\n|:---:|:---:|:---:|:---:|:---:|:---:|\n|IGR|0.9706|0.8526|0.9800|0.9765|0.9901|\n|SIREN|0.9438|**0.9682**|0.9735|0.9392|0.9762|\n|DiGS|**0.9767**|0.9680|0.9826|0.9788|0.9907|\n|SAP|0.9750|0.9414|0.9636|0.9731|0.9838|\n|PINC (ours)|0.9754|0.9311|**0.9828**|**0.9803**|**0.9915**|\n\n\n[Table 3] Normal Consistency on Thingi10K\n|Model|Squirrel|Pumpkiin|Frogrock|Scrwstar|Buser head|\n|:---:|:---:|:---:|:---:|:---:|:---:|\n|IGR|**0.9820**|0.9565|0.9509|0.9709|0.9249|\n|SIREN|0.9529|0.8996|0.9035|0.9142|0.8860|\n|DiGS|0.9557|0.9353|0.9468|0.9386|0.9171|\n|SAP|0.9791|0.9520|0.9319|0.9767|0.9004|\n|PINC (ours)|0.9816|**0.9583**|**0.9545**|**0.9805**|**0.9376**|\n\n**Q5. It would be unfair to directly compare the proposed method with approaches that utilize normal information..**\n\n**Reply**:  Thank you for the comment. Following the reviewer's suggestion, we include the comparison with Shape As Points (SAP) on both SRB and Thingi10K dataset using three metrics for quantitative evaluation: Chamfer distance (CD) and Hausdorff distance (HD) are summarized in Tables A5 and A6 in the attachment, and evaluation of normal consistency (NC) is reported in Tables 2 and 3 in the response to Q4. The results show that the CD and HD of the proposed model are similar to SAP, despite not utilizing the given surface normal. Furthermore, the proposed model, which learns the gradient field of the $p$-Poisson equation instead of using the given surface normal, achieves a better overall NC.\n\nSAP is based on Poisson Surface Reconstruction (PSR; Kazhdan, 2006). The proposed model may be interpreted as PSR because of (8). However, the vector field $G$ in the proposed model is not obtained from the oriented point cloud, but the learnable function that is trained with $u$ at the same time. Moreover, since we bake the $p$-Poisson equation into $G$ as a hard constraint in (7), we obtain a continuous SDF rather than an indicator function like PSR and SAP. The results confirm that simultaneous training of the gradient field and the SDF, that is, the variable splitting method, achieves similar or even better surface restoration than SAP, even without using the given surface normal.\n\n**Q6. What are lines 457-460?**\n\n**Reply**: Thank you for pointing out the mistake. We will make sure that it will be deleted in the revised version. \n\n**References**\n\nL. Mescheder et al. Occupancy networks: Learning 3d reconstruction in function space. IEEE/CVF, 2019.\n\nM. Kazhdan et al. Poisson surface reconstruction. In Proceedings of the fourth Eurographics symposium on Geometry processing, 2006."}, {"review_id": "fJmxDymQIL", "reviewer": "Reviewer_2pnE", "summary": "This content seems interesting. I like that the authors gave considerable thought to improving surface reconstruction from the perspective of vector field processing. The paper targets two challenging problems in surface reconstruction: (1) removing the requirement for surface normal, and (2) improving the over-smoothness. However, there are technical details that need to be addressed before I can accept it.", "questions": "major:\n\n(1) Add a comparison for the learned vector field. From equation (8), your G plays the role of the vector field derived from the oriented point cloud in PoissonRecon. It might be beneficial to compare G with the vector field formulated by surface normal. \n\n(2) In section 3.2, you mentioned the goal is to progressively enlarge lambda to infinity. If so, have you tried getting rid of the first term of equation (8)? I don’t think the method will fall apart. You would only be downgrading from ScreenedPoissonRecon to PoissonRecon.\n\n(3) The result in Figure 3 is a bit confusing. What’s your sampling strategy? Are you sampling all the vertices from the mesh? Because from the Screwstar, it seems you are not sampling uniformly. Could that partially be the reason the SIREN has a broken surface? Can you try to uniformly sample the star and see if there’s any improvement in the completeness?\n\n(4) The result in Figure 7 (a) had some clear artifacts. Why did that happen? When p = 2, your energy looks almost the same as ScreenedPoissonRecon. However, when I ran ScreenedPoissonRecon on this exact model, I did not see this artifact.\n\nminor:\n\n(1) The statement “irrotational flow” only appeared once in the title. If you want to use the statement, you need to be consistent in the writing. At least mention and explain it in the introduction.\n\n(2) In Chapter 4.3, you used the subtitle “Effect of curl-free constraint” twice. I believe you meant to say “Effect of minimal surface area constraint” for the second one.", "limitations": "yes", "rating": 7, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "(1) The problems are challenging.\n(2) Methodology is novel.\n(3) Improvements are observed.\n", "weaknesses": "In a p-Poisson equation, you have the p-Laplacian instead of the Laplacian. This is equivalent to giving weights to the gradient of an implicit function. And this weight is based on the magnitude of the gradient itself. The geometric intuition behind this weight is not clearly stated. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the constructive comments. Below, we carefully address the reviewer's comments:\n\n**Q1. In a p-Poisson equation, you have the p-Laplacian instead of the Laplacian..**\n\n**Reply**: In the $p$-Poisson equation $-\\triangle_p u = 1$, by letting $p$ in the weight $\\parallel\\nabla u \\parallel^{p-2}$ grows infinitely, we can obtain the SDF without normal supervision. More precisely, the $p$-Laplacian $\\triangle_p$ can be decomposed as follows (Kawohl, 2016):\n$$\\frac{1}{p}\\triangle_p u= \\frac{1}{p}\\parallel\\nabla u \\parallel^{p-1}\\triangle_1 u+ \\frac{p-1}{p}\\parallel\\nabla u \\parallel^{p-2}\\triangle_\\infty u,$$\nwhere $\\triangle_1 u = \\nabla \\cdot \\left( \\frac{\\nabla u}{\\parallel\\nabla u\\parallel} \\right)$ geometrically represents the mean curvature of the isosurfaces of $u$ and $\\triangle_\\infty u = \\nabla u^T H\\nabla u$ stands for the second derivative in the steepest ascending direction. Here, $H$ denotes the Hessian matrix of $u$. The parameter $p$ directly represents the weights between these two terms. As $p$ becomes larger, the weight of the second part is larger and eventually converges to the infinite Laplacian $\\triangle_\\infty u$. On the other hand, if we set $p=2$ (the Laplacian), we couldn't get the SDF.\n\n**Q2. Add a comparison for the learned vector field..**\n\n**Reply**: We agree that $G$ plays the role of normal field in PoissonRecon because we find a scalar function $u$ whose gradient best fit the vector field $G$ by minimizing $\\int_\\Omega \\left\\Vert \\nabla u - G\\right\\Vert^2$. Following the reviewer's recommendation, we measure the difference between the given surface normal $n$ and the learned $G$ on SRB dataset. Given an oriented point cloud together with surface normals \n$\\lbrace x_i, n_i \\rbrace$, $i=1,\\cdots,N$, we measure the cosine similarity $G^Tn\\coloneqq\\frac{1}{N}\\sum_{i=1}^N | G\\left(x_i\\right)^T n_i |$. In addition, we also report the cosine similarity of the gradient field of learned SDF $u$ and the surface normal. Results are reported in the Table below. The results confirm that $G$ is accurately trained. \n\n|Model|Anchor|Daratech|DC|Gargoyle|Lord Quas|\n|:---:|:---:|:---:|:---:|:---:|:---:|\n|$G^Tn$|0.9868|0.9544|0.9941|0.9924|0.9965|\n|$\\nabla u^Tn$|0.9870|0.9579|0.9946|0.9929|0.9966|\n\n\n**Q3. In section 3.2, you mentioned the goal is to progressively enlarge lambda to infinity..**\n\n**Reply**: In the standard penalty method, using a progressively larger value of the penalty parameter is theoretically necessary to obtain a solution of the constrained optimization. In the numerical simulation, we could make a procedural sequential approach to increase $\\lambda_1$ in (8). However, in practice, we cannot make the value of $\\lambda_1$ infinitely large (the line number 147 of the paper). As $\\lambda_1 \\rightarrow \\infty$, the balancing between $\\parallel \\nabla u - G\\parallel$ and $\\parallel \\nabla \\times \\tilde{G}\\parallel$ is going to be broken and then it is difficult to obtain the curl-free constraint which is one of crucial parts of the proposed method; please check Q3 of the author response to the reviewer rqMu for the necessity of the curl-free term. Moreover, without the first term, the solution is up to constant and then there should be an extra step to find a unique solution. Since the first term is the fidelity working like Dirichlet boundary condition, it enforces the implicit function should be zero at the point cloud. \n\n**Q4. The result in Figure 3 is a bit confusing..**\n\n**Reply**: When we trained SIREN, points on $\\Gamma$ were uniform randomly sampled from the original point cloud and we set collocation points of $\\Omega$ as the uniform grid. The reason for SIREN to restore the broken surface seems to be the choice of activation function and network initialization. We can see from Figure 3 that IGR has smooth surfaces, but the difference between SIREN and IGR is the choice of activation function and network initialization. IGR uses the softplus activation function and initializes the network to be the SDF for the sphere. SIREN, on the other hand, adopts the sine function as the activation and an initialization that preserves the distribution of activations through its layers.\n\n**Q5. The result in Figure 7 (a) had some clear artifacts..**\n\n**Reply**: In the proposed model, there are a few differences with ScreenedPoissonRecon (SPR). Firstly, the corresponding Euler-Lagrange (EL) equation of the proposed objective is not reduced to the screened Poisson equation because the first loss term considers $\\left\\vert u\\right\\vert$ rather than $\\left\\vert u\\right\\vert^2$ and the integration is computed over $\\Gamma$ rather than the whole computational domain $\\Omega$. The first term in (8) corresponds to the boundary condition. Secondly, $G$ is not the vector field obtained from the oriented point cloud, but the learnable function that is simultaneously trained with $u$. Moreover, since we impose the $p$-Poisson equation on $G$ as a hard constraint (7), we obtain an SDF rather than an indicator function like SPR. When we use small $p$ in the view of finding the SDF, the construction of $G$ leads to degraded results; see Figure 7. When $p=2$, the optimal solution $u$ is the following PDE:\n$$\\triangle u = \\nabla\\cdot G= -1$$\n\nwith Dirichlet boundary condition $u=0$ on $\\Gamma$, which is far away from the SDF.\n\n**Q6. The statement “irrotational flow” only appeared once in the title..**\n\n**Reply**: We completely agree with the reviewer. We will revise the manuscript as recommended.\n\n**Q7. In Chapter 4.3, you used the subtitle “Effect of curl-free constraint” twice..**\n\n**Reply**: Thank you for pointing this out. We will revise the manuscript as recommended.\n\n**Reference**\n\nB. Kawohl et al. On the geometry of the $ p $-Laplacian operator. arXiv preprint arXiv:1604.07675, 2016."}]

Author rebuttal:
**General Response to All Reviewers**

We sincerely thank all the reviewers for their valuable comments, recommendations, and suggestions. The opinions of the reviewers are carefully considered and answering their questions has improved the paper. We first address a common question raised by the reviewers rqMu and ymPA. Then, we address the individual response to each reviewer below. We also attach a supplementary file for Tables. Hopefully, the replies could address all the questions.

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**Common Question:** Theoretical motivation/intuition for choosing  $p$-poisson instead of eikonal equation to describe SDFs.

**Reply**: The main advantage of using the $p$-Poisson equation is that $u_p$ (solution of (1)) is unique in $W^{1,p}$ (Lindqvist, 2017), which prevents from non-unique weak solution in the eikonal equation $\left\Vert\nabla u\right\Vert=1$. A numerical challenge is to deal with $p \rightarrow \infty$ in order to get a good approximation of the viscosity solution. When the variational formulation (2) is used, the difficulty of using a large $p$ is still persistent numerically. However, it was resolved by using (7), which is one of the main advantages of the proposed algorithm.

**Reference**

P. Lindqvist. Notes on the $p$-Laplace equation. No. 161. University of Jyväskylä, 2017.

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**Attachment $\downarrow$**

Meta-review: The paper received mixed ratings initially. After rebuttal and reviewer-author discussions, all reviewers became positive. The paper presents a novel PDE-based formulation that can incorporate a variety of constraints. The approach does not require oriented normals as inputs, making it suitable for many application settings. The optimization procedure, which is motivated from variable splitting, is interesting and effective. Therefore, it is recommended to accept the paper due to the technical contributions. Please follow the rebuttal and reviewer's comments to prepare the final version.
Decision: Accept (poster)