SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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 ragged 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 a smooth and detailed surface. More results can be found in the Appendix C.
4.2 Reconstruction from noisy data
In this section, we analyze whether the proposed PINC (13) produces robust results to the presence of noise in the input point data. In many situations, the samples obtained by the scanning process contain a lot of noise and inaccurate surface normals are estimated from these noisy samples. Therefore, it 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.
This section is devoted to ablation analyses which show that each part of the proposed loss function  $\mathcal{L}_{\mathrm{total}}$  in conjunction with the divergence-free splitting architecture plays an important role in high-quality reconstruction.
Effect of curl-free constraint We first study the effect of 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 function  $\mathcal{L}_{p\text{-Poisson}}$  (8). The results on the SRB dataset are reported in Table 3 and Figure 5. Figure 5
Figure 4: Reconstruction results from noisy observations. Two levels of additive Gaussian noise with standard deviations  $\sigma = 0.005$  (low) and 0.01 (high) are considered.
shows that the variable splitting method, which satisfies the  $p$ -Poisson equation as a hard constraint (without the curl-free condition), recovers a fairly decent surface, but it generates oversmoothed surfaces and details are lost. However, as we can see from the qualitative result reconstructed with 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 3 also provide clear evidence of the need for the curl-free term. To further examine the necessity of another auxiliary variable  $\tilde{G}$ , we conduct an additional experiment by applying the curl-free loss term directly on  $G$  without the use of  $\tilde{G}$ . The results are presented in the second row of the Table 3. The results indicate that taking curl on  $G$ , which is constructed by taking curl on  $\Psi$  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  $\tilde{G}$ .
Table 3: Quantitative results on the ablation study of the curl-free term.
Model	GT	Scans
dC	dH	d→C	d→H
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 G	0.16	4.78	0.05	0.80
Figure 5: Comparison of surface reconstruction without (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 is considered. Figure 6 clearly shows this effect. Some parts in the daratech of SRB have a 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 a 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 are point clouds sampled from a closed surface and also are not related to hole filling. Indeed, we
empirically observe that the results are quite similar with and without the minimal area term for all data other than daratech.
(a) wo/ area loss
(b) w/ area loss
Figure 6: Comparison of surface recovery without (a) and with (b) minimum area criterion.
Effect of large  $p$  The  $p$ -Poisson equation (1) draws the SDF as  $p$  becomes infinitely large. Therefore, it is natural to think that it would be good to use a large  $p$ . Here, we conducted experiments on the effect of  $p$ . We define  $G$  with various  $p = 2, 10,$  and 100 and learn the SDF with it. Figure 7 shows surfaces that were recovered from the Gargoyle data in the SRB with different  $p$  values. When  $p$  is as small as 2, it is obvious that it is difficult to reconstruct a compact surface from points. When  $p$  is 10, a much better surface is constructed than that of  $p = 2$ , but the by-products still remain on the small holes. Furthermore, a large value of  $p = 100$  provides a quite proper reconstruction. This experimental result demonstrates that a more accurate approximation can be obtained by the use of a large  $p$ , which is consistent with the theory. This once again highlights the advantage of the variable splitting method we have proposed, which allows an arbitrarily large  $p$  to be used. This highlights the advantage of the variable splitting method (7) we have proposed in Section 3.1, which allows an arbitrarily large  $p$  to be used. Note that the previous approaches have not been able to use 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 to large  $p$  and hence the reconstruction becomes closer to the point cloud.
We presented a  $p$ -Poisson equation-based shape representation learning, termed PINC, that reconstructs high-fidelity surfaces using only the locations of 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 a hard constraint. The curl-free constraint was also used to provide a more accurate representation. Furthermore, the minimal surface area regularization was considered to provide a compact surface and overcome the ill-posedness of the surface reconstruction problem caused by unobserved points. The proposed PINC successively achieved a faithful surface with intricate details and was robust to noisy observations.
The minimization of the surface area is used to reconstruct missing parts of points under the assumption that a point cloud is measured by a 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 to closed surfaces and is inadequate to reconstruct open surfaces. We leave the development to open surface reconstruction as future work. Establishing a 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 or computational cost of utilizing auxiliary variables versus not utilizing them is a worthwhile direction for future research.
6 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 are aware that accurate 3D surface reconstruction
(a)  $p = 2$
(b)  $p = 10$
(c)  $p = 100$
Figure 7: Surface reconstruction of anchor data with various  $p$ . The results show the importance of using a sufficiently large  $p$  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 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.
This work was supported by the NRF grant [2012R1A2C3010887] and the MSIT/IITP ([1711117093], [2021-0-00077], [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 Skłodowska-Curie grant agreement No. 945478.
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A More discussion on curl-free term
This section is devoted to both theoretically and empirically validate the necessity of the curl-free loss term. One might think that the curl-free term is unnecessary, since a curl-free  $G$  can be obtained by reducing the  $L^2$  penalty term for the variable splitting constraint  $\nabla u = G$ . However, this penalty term is not sufficient to train  $G$  as a 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  $L^2$  energy of  $\| \nabla u - G\|$  in (8) without the curl-free term is not sufficient to obtain a conservative vector field  $G$ .
Theorem A.1. There is a sequence  $\{u_n, G_n\}_{n \in \mathbb{N}}$  such that  $\int_{\Omega} \| \nabla u_n - G_n \|^2 d\mathbf{x} \to 0$  as  $n \to \infty$ , but  $G_n$  does not converge to a curl-free vector field.
Proof. For every  $\{u_n\}_{n\in \mathbb{N}}$  defined on  $\Omega = [0,1]^3$ , set
$$
G _ {n} (x, y, z) = \nabla u _ {n} (x, y, z) + \left(0, \frac {1}{n} \sin (2 \pi n x), 0\right) \in \mathbb {R} ^ {3}.
$$
$$
\begin{array}{l} \int_ {\Omega} \left\| \nabla u _ {n} - G _ {n} \right\| ^ {2} d \mathbf {x} = \int_ {\Omega} \left\| \left(0, \frac {1}{n} \sin (2 \pi n x), 0\right) \right\| ^ {2} d \mathbf {x} (14) \\ = \frac {1}{n ^ {2}} \int_ {\Omega} \sin^ {2} (2 \pi n x) d \mathbf {x} (15) \\ \rightarrow 0, (16) \\ \end{array}
$$
as  $n\to \infty$  . However,
$$
\nabla \times G _ {n} (x, y, z) = (0, 0, \cos (2 \pi n x))
$$
does not converge to zero.
Remark A.2. Note that for a  $G_{n}$  set in the proof of the above theorem A.1,  $\int_{\Omega} \| \nabla \times G_{n} \|^{2} d\mathbf{x} = \frac{1}{2}$  is a positive constant independent of  $n$ . This implies that we 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 a simple example of a sphere of radius 0.5 centered at the origin. Figure 8 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 SDF. 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 8 and 9 are of the opposite sign of the trained  $u$ . As shown in the Figure 8, the model trained without the curl-free term learns a vector field  $G$  that is not curl-free, resulting in  $G$  being distinct from the true gradient field. This ultimately impedes  $u$  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.
In this section, we provide more details about the implementation for reproducibility. Note that our code is built on top of IGR  $^{2}$  (MIT License).
2https://github.com/amosgropp/IGR
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  $u$  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  $\mathcal{L}_{\mathrm{total}}$  (13) is a weighted sum of five loss terms with four regularization parameters  $\lambda_1, \lambda_2, \lambda_3$ , and  $\lambda_4$ . In all surface reconstruction experiments, we use  $\lambda_1 = 0.1$ ,  $\lambda_2 = 0.0001$ ,  $\lambda_3 = 0.0005$ , and  $\lambda_4 = 0.1$ . In the proposed model,  $p$  is also a hyperparameter to be chosen. Considering the theoretical fact that  $p$  should be infinitely large and numerical simplicity, we set  $p = \infty$ . We empirically confirm no significant difference between when  $p = 100$  and when  $p = \infty$ . Moreover, we set the smoothing parameter  $\epsilon = 1$  for approximating Dirac delta in (13).
Network Architecture As in previous studies [44, 23, 37], we represent the primary and auxiliary outputs by a single 8-layered multi-layer perceptron (MLP)  $\mathbb{R}^3\to \mathbb{R}^7$  with 512 neurons and a skip connection to the fourth layer, but only the output dimension of the last layer is increased by six due to the two auxiliary variables; see Figure 2. We use softmax activation function  $\alpha (x) = \frac{1}{\beta}\ln \left(1 + e^{\beta x}\right)$  with  $\beta = 100$ . Network weights are initialized by the geometric initialization proposed in [3].
Training details The gradient and the curl of networks are computed with auto-differentiation library (autograd) [46]. In all experiments, we use the Adam optimizer [32] with learning rate  $10^{-3}$  decayed by 0.99 every 2,000 iterations. At each iteration, we uniform randomly sample 16,384 points  $\mathbf{x} \in \mathcal{X}$  from the point cloud  $\mathcal{X}$ . We sample the collocation points of  $\Omega$  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 a 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  $U(-\eta, \eta)$  with  $\eta = 1.1$ .  $F = \frac{1}{3}\mathbf{x}$  is utilized in all experiments.
Baseline models For baseline models on the Thingi10K dataset, we use the official codes of IGR  $^{2}$  (MIT License), SIREN  $^{3}$  (MIT License), and DiGS  $^{4}$  (MIT License). We faithfully follow the official implementation to train each model without normal prior. For the variable splitting representation of the eikonal equation (9), there is a single auxiliary output. Consequently, we use the same 8 layer MLP with 512 nodes, but a network with an output dimension of 4. We normalize the auxiliary output to make it a unit norm, and use the normalized one to represent  $H$ .
Metrics We measure the distance between two point clouds  $\mathcal{X}$  and  $\mathcal{Y}$  by using the standard one-sided and double-sided  $\ell_1$  Chamfer distances  $d_{\vec{C}}$ ,  $d_C$  and Hausdorff distances  $d_{\vec{H}}$ ,  $d_H$ . Each are defined as follows:
$$
d _ {\vec {C}} (\mathcal {X}, \mathcal {Y}) = \frac {1}{| \mathcal {X} |} \sum_ {\mathbf {x} \in \mathcal {X}} \min  _ {\mathbf {y} \in \mathcal {Y}} \| \mathbf {x} - \mathbf {y} \| _ {2},
$$
$$
d _ {C} \left(\mathcal {X}, \mathcal {Y}\right) = \frac {1}{2} \left(d _ {\vec {C}} \left(\mathcal {X}, \mathcal {Y}\right) + d _ {\vec {C}} \left(\mathcal {Y}, \mathcal {X}\right)\right),
$$
$$
d_{\vec{H}}\left(\mathcal{X},\mathcal{Y}\right) = \max_{\mathbf{x}\in \mathcal{X}}\min_{\mathbf{y}\in \mathcal{Y}}\left\| \mathbf{x} - \mathbf{y}\right\|_{2},
$$
$$
d _ {H} \left(\mathcal {X}, \mathcal {Y}\right) = \max  \left\{d _ {\vec {H}} \left(\mathcal {X}, \mathcal {Y}\right) + d _ {\vec {H}} \left(\mathcal {Y}, \mathcal {X}\right) \right\}.
$$
When we estimate the distance from a 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) [41] between the learned  $G$  and the surface normal as follows: from given an oriented point cloud  $\mathcal{X} = \{\mathbf{x}_i,\mathbf{n}_i\}_{i = 1}^N$  comprising of sampled points  $\mathbf{x}_i$  and the corresponding outward normal vectors  $\mathbf{n}_i$ , NC is defined by
$$
N C (G, \mathbf {n}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left| G \left(\mathbf {x} _ {i}\right) ^ {\mathrm {T}} \mathbf {n} _ {i} \right|, \tag {17}
$$
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 a trained neural network  $u$  by using the classical marching cubes meshing algorithm [38] on a  $512 \times 512 \times 512$  uniform grid.
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 a cube centerd at the origin with an edge of the length 1. We consider IGR [23] as an eikonal-rooted baseline, and we 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 two different random seeds. The results are 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 a comparison with models that do not use the surface normal  $\mathbf{n}$  as a supervision. Here, we additionally consider a comparison with models that leverage normal supervision. We
<sup>3</sup>https://github.com/vsitzmann/siren
4https://github.com/Chumbyte/DiGS
Figure 9: Experimental results show whether a method can find the SDF from different network initializations. IGR and PINC are trained on the synthetic cube data with three network initializations: geometric initialization (initialization 1) and Kaiming initialization with two different random seeds (initializations 2 and 3). Each depicts the trained level-set contours of a cross-section cut in the plane  $x = 0$ . Red contours depict the trained zero level set of numerical solutions.
Table 4: Comparison with models that use the surface normal supervision  $\mathbf{n}$  on SRB. The proposed model PINC did not utilize the surface normal.
Model	Anchor	Daratech	DC	Gargoyle	Loard Quas
GT	Scans	GT	Scans	GT	Scans	GT	Scans	GT	Scans
dC	dH	dC	dH	dC	dH	dC	dH	dC	dH	dC	dH	dC	dH	dC	dH	dC	dH	dC	dH
w/n	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
IGR	0.22	4.71	0.12	1.32	0.25	4.01	0.08	1.59	0.17	2.22	0.09	2.61	0.16	3.52	0.06	0.81	0.12	1.17	0.07	0.98
SAP	0.34	8.83	0.09	2.93	0.22	3.09	0.08	1.66	0.17	3.30	0.04	2.23	0.18	5.54	0.05	1.73	0.13	3.49	0.04	1.17
wo/n	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
consider three baseline models as 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], a model that revisits the classical Poisson Surface Reconstruction (PSR) [31] using deep learning. The results are reported in Table 4. We can see that the proposed model performs on par with baselines, despite not utilizing the surface normal. Considering that all of the baselines are PDE-based INR models, the results exhibit the effectiveness of the proposed model (11) in reconstructing a surface from the sole use of raw point clouds.