SlowGuess/NeurIPS_2023_OCR huggingface dataset:
It is worth note that the proposed model may be interpreted as PSR because of (8). 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  $G$  transforms into the following Poisson problem:
$$
\left\{ \begin{array}{l l} \triangle u = \nabla \cdot G & \text {i n} \Omega \\ u = 0 & \text {o n} \Gamma . \end{array} \right. \tag {18}
$$
In the conventional PSR, the gradient field is set to surface normals. Thus, the auxiliary variable  $G$  can be regarded as playing a role of surface normals for PSR. 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
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 Thingi10K.
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 or even better surface restoration than SAP, even without using the given surface normal  $\mathbf{n}$ .
C.3 Additional quantitative results
We reported Chamfer distances and Hausdorff distances in Tables 1 and 2, but these two 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 Thingi10K datasets are summarized in Tables 5 and 6, 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 6 and 7 of Section 4.3 and summarized them in the Tables 7 and 8, respectively.
More results on effect of  $p$  Theoretically, an accurate SDF can be obtained as  $p$  grows infinitely. That the same story continues in practice is confirmed by the results shown in Figure 10. 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  $p = \infty$ , which we used in the implementation, gives a similar qualitative result to  $p = 100$ . These experimental results once again remind us how important it is to be able to use a large  $p$ .
Furthermore, we provide numerical verification for the use of  $p = \infty$  in Figure 11. For notational convenience, we use the subscript  $u_{p}$  to denote the dependence of the solution on the parameter  $p$ . Figure 11 depicts graphs of the mean squared error (MSE) of  $u_{p}$  and  $u_{\infty}$  over different  $p$ . MSEs are computed by discretizing the computational domain  $\Omega$  into a  $100 \times 100 \times 100$  uniform grid. The results show that the MSE decreases as  $p$  increases. In other words, it confirms that  $u_{p}$  is getting closer to  $u_{\infty}$  as  $p$  grows, which supports the justification for using  $p = \infty$ .
Table 7: Quantitative results on the effect of area loss on daratech.
Model	GT	Scans
dC	dH	dC	dH
PINC wo/ area loss	4.26	53.34	0.20	2.81
PINC w/ area loss	0.37	7.24	0.11	1.88
Table 8: Quantitative results on the ablation study of  $p$  .
Model	Gargoyle	Anchor
GT dC	dH	Scans d→C	dH	GT dC	dH	Scans d→C	dH
p = 2	3.96	43.13	0.51	6.31	4.32	46.66	0.77	14.58
p = 10	0.22	8.14	0.10	1.19	0.50	7.24	0.12	3.02
p = 100	0.17	4.90	0.10	0.82	0.31	7.20	0.13	1.80
p = ∞	0.16	4.78	0.05	0.80	0.29	7.19	0.11	1.17
$p = 2$
$p = 10$
$p = 100$
Figure 10: Quality of surface reconstruction with varying  $p$  from  $p = 2$  to  $p = \infty$ .
$p = \infty$
Figure 11: MSEs of  $u_{p}$  and  $u_{\infty}$  over different  $p$ .
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 Thingi10K discussed in Section 4.1.
Reconstruction of large point clouds We further provide qualitative results for surface reconstruction 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 on 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 9, we report the average training time per iteration and inference time at a resolution of  $32^{3}$  voxels. The proposed model requires more computational cost than baseline models because of the computation on curl using automatic differentiation.
Figure 13: Reconstructed surfaces of large point clouds from Thingi10K.
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