SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$p$ -Poisson surface reconstruction in curl-free flow from point clouds
Yesom Park $^{1*}$ , Taekyung Lee $^{2*}$ , Jooyoung Hahn $^{3}$ , Myungjoo Kang $^{1}$
$^{1}$  Department of Mathematical Sciences, Seoul National University
$^{2}$  Interdisciplinary Program in Artificial Intelligence, Seoul National University
$^{3}$  Department of Mathematics and Descriptive Geometry,
Slovak University of Technology in Bratislava
{yeisom，dlxorud1231，mkang}@snu.ac.kr
jooyoung.hahn@stuba.sk
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.
Surface reconstruction from an unorganized point cloud has been extensively studied for more than two decades [10, 29, 40, 9, 62] due to its many downstream applications in computer vision and computer graphics[8, 16, 62, 58]. Classical point cloud or mesh-based representations are efficient but they do not guarantee a watertight surface and are usually limited to fixed geometries. Implicit function-based representations of the surface [28, 64, 43, 14] as a level set  $S = \{\mathbf{x} \in \mathbb{R}^3 \mid u(\mathbf{x}) = c\}$  of a continuous implicit function  $u: \mathbb{R}^3 \to \mathbb{R}$ , such as signed distance functions (SDFs) or occupancy functions, have received considerable attention for providing watertight results and great flexibility
$p$  -Poisson
$p$ -Poisson + curl-free
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 (11).
*Equal contribution authors. Correspondence to: <mkang@snu.ac.kr>.
in representing different topologies. In recent years, with the rise of deep learning, a stream of work called implicit neural representations (INRs) [2, 44, 16, 61, 19, 55, 53, 50] has revisited them by parameterizing the implicit function  $u$  with neural networks. INRs have shown promising results by offering efficient training and expressive surface reconstruction.
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, 37, 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 at each point. They often fail to capture fine details or reconstruct plausible surfaces without normal vectors. A raw point cloud usually lacks normal vectors or numerically estimated normal vectors [1, 18] 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 a 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 [47, 60, 22, 12] in the optimization literature. We then parameterize the auxiliary output to automatically satisfy the  $p$ -Poisson equation by reformulating the equation in a divergence-free form. The divergence-free 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 a 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, 55, 54], which define a surface as zero level-sets of neural networks, have been extensively studied. Early work requires the ground-truth signed implicit function [44, 16, 41], which is difficult to obtain in real-world scenarios. Considerable research [3, 4] 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 eikonal 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 [6] or the Hessian [63]. 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 [48], 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.
$p$ -Poisson equation The SDF is described by a solution of various PDEs. The existing work [23, 55, 6] 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 [55, 6] utilize the 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  $(p = 2)$ :
$$
\left\{ \begin{array}{l} - \triangle_ {p} u = - \nabla \cdot \left(\| \nabla u \| ^ {p - 2} \nabla u\right) = 1 \text {i n} \Omega \\ u = 0 \text {o n} \Gamma , \end{array} \right. \tag {1}
$$
where  $p \geq 2$ , the computation domain  $\Omega \subset \mathbb{R}^3$  is bounded, and  $\Gamma$  is embedded in  $\Omega$ .
The main advantage of using the  $p$ -Poisson equation is that the solution to (1) is unique in Sobolev space  $W^{1,p}(\Omega)$  [36]. The unique solution with  $p \geq 2$  brings a viscosity solution of the eikonal equation in the limit  $p \to \infty$ , 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.1. Moreover, in contrast to the eikonal equation, it is possible to describe a solution of (1) as a variational problem and compute an accurate approximation [5, 20]:
$$
\min  _ {u} \int_ {\Omega} \frac {\left\| \nabla u \right\| ^ {p}}{p} d \mathbf {x} - \int_ {\Omega} u d \mathbf {x}. \tag {2}
$$
As  $p \to \infty$ , it has been shown [11, 30] that the solution  $u$  of (1) converges to the SDF whose zero level set is  $\Gamma$ . As a result, increasing  $p$  gives a better approximation of the SDF, which is definitely helpful for surface reconstruction. However, it is still difficult to use a fairly large  $p$  in numerical computations and in this paper we will explain one of the possible solutions to the mentioned problem.
In this section, we propose a  $p$ -Poisson equation based Implicit Neural representation with Curl-free constraint (PINC). From an unorganized point cloud  $\mathcal{X} = \{\mathbf{x}_i : i = 1, 2, \dots, N\}$  sampled by a closed surface  $\Gamma$ , a SDF  $u: \mathbb{R}^3 \to \mathbb{R}$  whose zero level set is the surface  $\Gamma = \{\mathbf{x} \in \mathbb{R}^3 \mid u(\mathbf{x}) = 0\}$  is reconstructed by the proposed INR. There are two key elements in the proposed method: First, using a variable-splitting representation [45] of the network, an auxiliary output is used to learn the gradient of the SDF that satisfies the  $p$ -Poisson equation (1). Second, a curl-free constraint is enforced on an auxiliary variable to ensure that the differentiable vector identity is satisfied.
3.1  $p$ -Poisson equation
A loss function in the physics-informed framework [51] of the existing INRs for the  $p$ -Poisson equation (1) can be directly written:
$$
\min  _ {u} \int_ {\Gamma} | u | d \mathbf {x} + \lambda_ {0} \int_ {\Omega} \left| \nabla \cdot \left(\| \nabla u \| ^ {p - 2} \nabla u\right) + 1 \right| d \mathbf {x}, \tag {3}
$$
where  $\lambda_0 > 0$  is a 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 an auxiliary variable that satisfies the  $p$ -Poisson equation (1).
Variable-splitting strategy Unlike existing studies [23, 37, 6] 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, 12] and it has the advantage of decomposing a complex minimization into a sequence of relatively simple sub-problems. With the auxiliary variable  $G = \nabla u$  and the penalty method [13], the variational problem (3) is converted into an unconstrained problem:
$$
\min  _ {u, G} \int_ {\Gamma} | u | d \mathbf {x} + \lambda_ {0} \int_ {\Omega} \left| \nabla \cdot \left(\| G \| ^ {p - 2} G\right) + 1 \right| d \mathbf {x} + \lambda_ {1} \int_ {\Omega} \| \nabla u - G \| ^ {2} d \mathbf {x}, \tag {4}
$$
where  $\lambda_{1} > 0$  is a penalty parameter representing the relative importance of the loss terms.
$p$ -Poisson as a hard constraint Looking more closely at the minimization (4), if  $G$  is already a gradient to satisfy (1), then the second term in (4) is no longer needed and it brings the simplicity of one less parameter. Now, for a function  $F: \Omega \to \mathbb{R}^3$  such that  $\nabla \cdot F = 1$ , for example  $F(\mathbf{x}) = \frac{1}{3}\mathbf{x}$ , the  $p$ -Poisson equation (1) is reformulated by the divergence-free form:
$$
\nabla \cdot \left(\| \nabla u \| ^ {p - 2} \nabla u + F\right) = 0. \tag {5}
$$
Then, there exists a vector potential  $\Psi : \mathbb{R}^3 \to \mathbb{R}^3$  satisfying
$$
\| G \| ^ {p - 2} G + F = \nabla \times \Psi , \tag {6}
$$
where  $G = \nabla u$ . Note that a similar idea is used in the neural conservation law [52] to construct a divergence-free vector field built on the Helmholtz decomposition [33, 57]. From the condition (6), we have  $\| G\|^{p - 1} = \| \nabla \times \Psi -F\|$  and  $G$  is parallel to  $\nabla \times \Psi -F$ , then the auxiliary output  $G$  is explicitly written:
$$
G = \frac {\nabla \times \Psi - F}{\| \nabla \times \Psi - F \| ^ {\frac {p - 2}{p - 1}}}. \tag {7}
$$
This confirms that the minimization problem (4) does not require finding  $G$  directly, but rather that it can be obtained from the vector potential  $\Psi$ . Therefore, the second loss term in (4) can be eliminated by approximating the potential function  $\Psi$  by a neural network and defining the auxiliary output  $G$  as a hard constraint (7). To sum up, we use a loss function of the form
$$
\mathcal {L} _ {p - \text {P o i s s o n}} = \int_ {\Gamma} | u | d \mathbf {x} + \lambda_ {1} \int_ {\Omega} \| \nabla u - G \| ^ {2} d \mathbf {x}, \tag {8}
$$
where  $G$  is obtained by (7), the first term is responsible for imposing the boundary condition of (1), and the second term enforces the constraint  $G = \nabla u$  between primary and auxiliary outputs. It is worth mentioning that  $G$  in (7) is designed to exactly satisfy the  $p$ -Poisson equation (1).
An advantage of the proposed loss function (8) and the hard constraint (7) is that (1) can be solved for sufficiently large  $p$ , which is critical to make a better approximation of the SDF. It is not straightforward in (3) or (4) because the numeric value of  $(p - 2)$ -power with a large  $p$  easily exceeds the limit of floating precision. On the other hand, in (7) we use  $(p - 2) / (p - 1)$ -power, which allows stable computation even when  $p$  becomes arbitrarily large. The surface reconstruction with varying  $p$  in Figure 7 shows that using a 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, it is worth noting that the proposed representation expresses the second-order PDE (1) with first-order derivatives only. By reducing the order of the derivatives, the computational graph is simplified than (3) or (4).
Note that one can think of an approach to directly solve the eikonal equation  $\| \nabla u\| = 1$  with an auxiliary variable  $H = \nabla u$  as an output of the neural network:
$$
\min  _ {u, \| H \| = 1} \int_ {\Gamma} | u | d \mathbf {x} + \eta \int_ {\Omega} \| \nabla u - H \| ^ {2} d \mathbf {x}, \tag {9}
$$
where  $\eta > 0$ . The above loss function may produce a non-unique solution of the eikonal equation, which causes numerical instability and undesirable estimation of the surface reconstruction; see Figure 1. To alleviate such an issue, the vanishing viscosity method is used in [37, 49] to approximate the SDF by  $u_{\sigma}$  as  $\sigma \to 0$ , a solution of  $-\sigma \triangle u_{\sigma} + \mathrm{sign}(u_{\sigma})(|\nabla u_{\sigma}| - 1) = 0$ . However, the results are dependent on the hyper-parameter  $\sigma > 0$  related to the resolution of the discretized computational domain and the order of the numerical scheme [17, 24].
3.2 Curl-free constraint
In the penalty method, we have to compute more strictly to ensure that  $G = \nabla u$  by using progressively larger values of  $\lambda_{1}$  in (8), but in practice we cannot make the value of  $\lambda_{1}$  infinitely large. Now, we can think of yet another condition for enforcing the constraint  $G = \nabla u$  from a differential vector identity which says a conservative vector field is curl-free:
$$
\nabla \times G = 0 \Longleftrightarrow G = \nabla u \tag {10}
$$
for some scalar potential function  $u$ . While it may seem straightforward, adding a penalty term  $\int_{\Omega} \| \nabla \times G \|^2 d\mathbf{x}$  at the top of (8) is fraught with problems. Since  $G$  is calculated by using a 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]. In order to relax the mentioned issue, we augment another auxiliary variable  $\tilde{G}$ , where  $G = \tilde{G}$  and  $\nabla \times \tilde{G} = 0$  are constrained.
By incorporating the new auxiliary variable  $\tilde{G}$  and its curl-free constraint, we have the following loss function:
$$
\mathcal {L} _ {\mathrm {P I N C}} = \mathcal {L} _ {p - \text {P o i s s o n}} + \lambda_ {2} \int_ {\Omega} \left\| G - \tilde {G} \right\| ^ {2} d \mathbf {x} + \lambda_ {3} \int_ {\Omega} \left\| \nabla \times \tilde {G} \right\| ^ {2} d \mathbf {x}. \tag {11}
$$
Figure 2: The visualization of the augmented network structure with two auxiliary variables.
Note that the optimal  $\tilde{G}$  should have a unit norm according to the eikonal equation. To facilitate training, we relax this nonconvex equality condition into a convex constraint  $\| \tilde{G} \| \leq 1$ . To this end, we parameterize the second network auxiliary output  $\tilde{\Psi}$  and define  $\tilde{G}$  by
$$
\tilde {G} = \mathcal {P} (\tilde {\Psi}) := \frac {\tilde {\Psi}}{\max  \left\{1 , \| \tilde {\Psi} \| \right\}}, \tag {12}
$$
where  $\mathcal{P}$  is the projection operator to the three-dimensional unit ball. Appendix A provides further discussion on the importance of the curl-free term to learn a conservative vector field.
Figure 2 illustrates the proposed network architecture. The primary and the auxiliary variables are trained in 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 a real point cloud to estimate a closed surface by range scanners, it is inevitable to have occluded parts of 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 to check the quality of  $\mathcal{L}_{\mathrm{PINC}}$  (11) in this paper, we choose a simple rule to minimize the area of zero level set of  $u$ :
$$
\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\mathrm {P I N C}} + \lambda_ {4} \int_ {\Omega} \delta_ {\epsilon} (u) \| \nabla u \| d \mathbf {x}, \tag {13}
$$
where  $\lambda_4 > 0$  and  $\delta_{\epsilon}(x) = 1 - \tanh^2\left(\frac{x}{\epsilon}\right)$  is a smeared Dirac delta function with  $\epsilon > 0$ . The minimization of the area is used in [21, 49] and the advanced models [15, 27, 63] on missing parts of the point cloud to provide better performance of the reconstruction.
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 to existing INRs? (ii) Is 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.
Implementation As in previous studies [44, 23, 37], we use an 8-layer network with 512 neurons and a 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 (13), we empirically set the loss coefficients to  $\lambda_1 = 0.1$ ,  $\lambda_2 = 0.0001$ .  $\lambda_3 = 0.0005$ , and  $\lambda_4 = 0.1$  and use  $p = \infty$  in (7) for numerical simplicity. We implement all numerical experiments on a single NVIDIA RTX 3090 GPU. In all experiments, we use the Adam optimizer [32] with learning rate  $10^{-3}$  decayed by 0.99 every 2000 iterations.
Figure 3: 3D Reconstruction results for SRB and Thingi10K 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, we 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  $(d_{\mathcal{C}})$  and Hausdorff  $(d_H)$  distances between the ground-truth point clouds and the reconstructed surfaces. Moreover, we report one-sided distances  $d_{\vec{\mathcal{C}}}$  and  $d_{\vec{H}}$  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 the baseline models that were trained without a normal prior. Table 1 summarizes 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 is on par with the leading INRs, and we achieved state-or-the-art performance for Chamfer distances.
Table 1: Results on surface reconstruction of SRB.
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
IGR	0.45	7.45	0.17	4.55	4.9	42.15	0.7	3.68	0.63	10.35	0.14	3.44	0.77	17.46	0.18	2.04	0.16	4.22	0.08	1.14
SIREN	0.72	10.98	0.11	1.27	0.21	4.37	0.09	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	0.11	1.17	0.20	3.72	0.09	1.80	0.15	1.70	0.07	2.75	0.17	4.10	0.09	0.92	0.12	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 Thingi10K.
Model	Squirrel	Buser head	Screwstar	Frogrock	Pumpkin
dC	dH	dC	dH	dC	dH	dC	dH	dC	dH
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 Thingi10K, we reproduce the results of IGR, SIREN, and DiGS without normal vectors using the official codes. Results on Thingi10K 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.