SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\frac {\partial \Phi}{\partial \theta_ {m}} = D \left[ \begin{array}{c} \left(\frac {\partial \hat {d}}{\partial \theta_ {m}} \sin \theta_ {m} + \left(\hat {d} + \frac {d _ {0}}{D}\right) \cos \theta_ {m}\right) \cos \phi_ {m} \\ \left(\frac {\partial \hat {d}}{\partial \theta_ {m}} \sin \theta_ {m} + \left(\hat {d} + \frac {d _ {0}}{D}\right) \cos \theta_ {m}\right) \sin \phi_ {m} \\ \frac {\partial \hat {d}}{\partial \theta_ {m}} \cos \theta_ {m} - \left(\hat {d} + \frac {d _ {0}}{D}\right) \sin \theta_ {m} \end{array} \right] \tag {9}
$$
Here, we can replace  $\frac{\partial\hat{d}}{\partial\theta_m} = \frac{\partial\hat{d}}{\partial\boldsymbol{r}}\frac{\partial\boldsymbol{r}}{\partial\theta_m}$  and  $\frac{\partial\hat{d}}{\partial\phi_m} = \frac{\partial\hat{d}}{\partial\boldsymbol{r}}\frac{\partial\boldsymbol{r}}{\partial\phi_m}$ , so the surface normal  $\boldsymbol{n}$  can be denoted as a function  $Q(\frac{\partial\hat{d}}{\partial\boldsymbol{r}},\boldsymbol {r},D,\hat{d})$
A.3 Additional Implementation Details
In this section, we provide details of implementation and some minor adaptations of baselines, as well as the workflow for 3D shape reconstruction.
A.3.1 Baselines
OF/DeepSDF/NDF: Pre-processing on mesh before training is required for OF/DeepSDF/NDF. For fair comparisons, we use all depth images from training views to reconstruct the training mesh, which is then used for sampling occupancies/signed distances/unsigned distances of a  $256^{3}$  voxel grid for training. We follow the official settings of hyperparameters and network architecture to train on the three datasets. For inference, we apply
sphere tracing for DeepSDF and NDF to render depth images from testing views. The sphere tracing will stop after 100 iterations or when the predicted distance is smaller than a threshold at 0.005 meters. To avoid missing the intersections, we follow the damped sphere tracing [14] from NDF, i.e., marching with  $\alpha \cdot f(\pmb{p})$  with  $\alpha = 0.6$ . Since sphere tracing for OF can only take a fixed step size (e.g., 0.005 meters), it requires about 1000 iterations to march the surface, which is time-consuming. We thus construct a testing mesh by querying a value grid of occupancies and use ray-mesh intersections to render depth images for evaluation.
NeuS/DS-NeRF: In general, we follow the same network architecture, training schedule, and color supervision from NeuS and DS-NeRF. The vanilla DS-NeRF adopts sparse 3D points from structure-from-motion (SFM) and uses the reprojection error for depth supervision. For a fair comparison, we provide the same dense depth supervision from training views as ours on all three datasets for both
Figure 6: Sphere for surface normal computation.
NeuS and DS-NeRF. For the experiments of Group 1, i.e., 3D shape representation only, we remove the color supervision of NeuS and DS-NeRF and leave depth supervision only.
LFN/PRIF: We adopt the same network architecture and hyperparameters from the official settings of LFN/PRIF. In particular, we train LFN/PRIF for 10 epochs on each dataset, which is the same as ours. Since the proposed depth computation from LFN makes it difficult to obtain dense depths, we modify the output of the last MLP layer to predict depth value with supervision. For the experiments of Group 2, i.e., 3D Shape and Appearance Representations, we extend a 2-layer color branch both for LFN and PRIF with appearance supervision. Note that, the LFN/PRIF results are obtained without our dual-ray visibility classifier. We additionally conduct experiments on LFN/PRIF with our own classifier for comparison. Figure 7 and Table 10 show that our method still achieves the best performance.
Figure 7: Qualitative results of LFN/PRIF with our visibility classifier.
A.3.2 3D Shape Reconstruction for Evaluation
For a fair comparison, we initially render depth images of testing views for all baselines and ours. For OF/DeepSDF/NDF, we set the depth value to 0 for those unconverged rays (i.e., no intersection during sphere tracing). For PRIF and ours, we apply specific outlier removal strategies. Subsequently, we use TSDF fusion on the processed depth images to obtain complete meshes for further evaluation.
Table 10: Quantitative results of LFN/PRIF with our visibility classifier.
ADE↓	CD (×10-3) ↓ mean / median
LFN + Our Visibility Classifier	90.82	121.289 / 44.564
PRIF + Our Visibility Classifier	9.97	4.187 / 0.891
RayDF	7.97	3.388 / 0.663
A.4 Additional Ablation Studies
In this section, we show more details of ablation studies, including ablations on dual-ray visibility classifier, ray-surface distance network, training strategies, and post-processing.
A.4.1 Ablation Studies on the Dual-ray Visibility Classifier
The key issue of a naive ray based distance function is the lack of generalization ability across novel views during testing, fundamentally because training a single ray based network (without classifier) tends to fit all ray distance data pairs of the training set, but cannot guarantee the consistency of surface distances between seen (training) rays and unseen (testing) rays. As demonstrated in Tables 11, 12, 13 and Figure 8, the absence of the classifier significantly impacts testing performance, resulting in notably inferior results compared to those optimized with the well-trained classifier. To provide better clarity, we also present visualizations of the visibility predictions in Figure 9. Moreover, the results of our classifier with noises in Tables 14, 15, 16 and Figure 10 demonstrate that our visibility classifier is reliable for multi-view consistency.
Figure 8: Qualitative results of RayDF without visibility classifier and one-stage trained RayDF.
Figure 9: Visualization of visibility predictions,  $w$  denotes visibility,  $d$  denotes distance difference.
Table 11: Ablation study on the visibility classifier. Accuracy  $(\%)\uparrow$  and F1-Score  $(\%)\uparrow$  on 8 scenes and the average scores of the Blender dataset.
Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1
(1) w/o classifier hΦ	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
(2) use d in function k	91.72	91.82	82.45	81.87	65.14	60.00	93.11	93.73	80.06	74.98	75.62	51.96	81.53	81.92	82.81	85.06	81.56	77.67
(3) remove function k	91.37	91.55	80.06	78.51	64.10	48.07	93.33	94.44	81.55	75.86	80.55	60.55	82.77	82.88	81.74	82.32	81.93	76.77
(4) use g(r1) ⊕ g(r2)	97.56	64.57	93.96	54.01	95.53	23.75	96.31	51.63	93.82	52.37	96.04	31.77	97.56	64.04	89.67	59.24	95.06	50.17
Dual-ray Visibility Classifier	92.92	93.22	84.98	84.73	70.34	61.83	93.61	94.67	87.74	85.24	89.95	81.08	87.00	87.78	84.16	85.26	86.34	84.23
Table 12: Ablation study on the visibility classifier. ADE  $\downarrow$  on 8 scenes and the average score of the Blender dataset.
Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
(1) w/o classifier hΦ	30.39	66.12	54.93	23.92	32.19	84.10	52.54	32.46	47.08
(2) use d in function k	6.14	12.57	15.43	4.15	13.44	12.49	9.19	13.15	10.82
(3) remove function k	6.24	11.57	15.39	4.08	12.08	10.42	8.64	13.45	10.23
(4) use g(r1) ⊕ g(r2)	5.89	11.69	14.29	4.70	11.01	7.91	8.84	13.85	9.77
Full RayDF	4.59	9.24	12.56	3.06	7.98	5.68	7.55	13.07	7.97
Table 13: Ablation study on the visibility classifier. CD  $(\times 10^{-3})\downarrow$  (mean / median) on 8 scenes and the average score of the Blender dataset.
Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
(1) w/o classifier h0	892.472 / 856.377	1170.313 / 1064.195	2077.365 / 2038.962	1052.000 / 541.289	1088.382 / 822.446	1879.326 / 1565.162	1394.594 / 1277.556	1077.019 / 423.644	1328.934 / 1073.704
(2) use d in function k	3.286 / 0.426	6.368 / 2.021	7.390 / 0.789	7.020 / 0.768	3.114 / 0.858	3.618 / 1.118	3.730 / 1.110	5.349 / 0.863	4.984 / 0.994
(3) remove function k	4.312 / 0.437	5.480 / 1.637	7.497 / 0.588	7.527 / 0.778	3.010 / 0.848	2.967 / 0.956	2.613 / 1.035	4.614 / 0.837	4.753 / 0.890
(4) use g(r1) ⊕ g(r2)	7.362 / 0.424	4.930 / 1.513	5.473 / 0.647	7.111 / 0.782	1.719 / 0.745	1.832 / 0.793	2.417 / 0.829	7.078 / 0.866	4.740 / 0.825
Full RayDF	3.807 / 0.312	3.310 / 0.958	2.914 / 0.543	7.255 / 0.733	1.095 / 0.702	1.604 / 0.617	3.571 / 0.644	3.544 / 0.792	3.388 / 0.663
Table 14: Ablation study on adding noises to the well-trained visibility classifier. Accuracy  $(\%)\uparrow$  and F1-Score  $(\%)\uparrow$  on 8 scenes and the average scores of the Blender dataset.
Noise level (σ2)	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1	Acc.	F1
1.0	65.04	66.57	61.75	63.31	55.19	51.58	65.91	70.55	62.56	59.06	63.79	50.11	62.92	65.24	62.92	65.24	62.51	61.46
0.5	70.46	71.61	65.99	66.82	57.04	51.57	71.64	75.67	63.99	67.76	68.75	53.99	67.59	69.42	67.59	69.42	66.63	65.78
0.1	85.02	85.37	77.31	76.32	61.78	49.78	86.96	88.96	68.01	72.59	81.98	66.75	80.23	81.01	75.44	76.08	77.09	74.61
0 (RayDF)	92.92	93.22	84.98	84.73	70.34	61.83	93.61	94.67	87.74	85.24	89.95	81.08	87.00	87.78	84.16	85.26	86.34	84.23
Table 15: Ablation study on adding noises to the well-trained visibility classifier. ADE $\downarrow$  on 8 scenes and the average score of the Blender dataset.
Noise level (σ2)	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
1.0	10.31	17.08	17.07	18.21	23.09	19.02	11.33	20.09	17.03
0.5	9.11	15.67	16.28	16.33	21.16	15.92	10.60	19.24	15.54
0.1	7.05	12.91	15.04	11.92	16.83	10.51	9.46	17.39	12.64
0 (RayDF)	4.59	9.24	12.56	3.06	7.98	5.68	7.55	13.07	7.97
Table 16: Ablation study on adding noises to the well-trained visibility classifier. CD  $(\times 10^{-3})\downarrow$  (mean / median) on 8 scenes and the average score of the Blender dataset.
Noise level (σ2)	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
1.0	3.954 / 0.329	3.353 / 1.299	3.068 / 0.630	7.474 / 0.963	2.844 / 0.705	3.310 / 1.045	2.841 / 1.564	5.415 / 0.943	4.032 / 0.935
0.5	3.815 / 0.287	3.187 / 1.221	2.793 / 0.533	7.080 / 0.933	2.313 / 0.658	2.923 / 0.959	2.817 / 1.429	4.796 / 0.933	3.716 / 0.869
0.1	3.808 / 0.276	3.483 / 1.103	2.395 / 0.548	6.730 / 0.857	1.739 / 0.622	2.840 / 1.012	2.883 / 1.510	3.751 / 0.875	3.454 / 0.850
0 (RayDF)	3.807 / 0.312	3.310 / 0.958	2.914 / 0.543	7.255 / 0.733	1.095 / 0.702	1.604 / 0.617	3.571 / 0.644	3.544 / 0.792	3.388 / 0.663
A.4.2 Ablation Studies on the Ray-surface Distance Network
We perform the following ablation studies on the ray-surface distance network.
(1) Multi-view Rays Sampling: Here we report the detailed quantitative results of each scene on Blender dataset in Tables 17 and 18.
(2) Network Architecture: We conduct additional experiments on the Blender dataset as shown in Table 19. We can see that our framework tends to favor a wider or deeper network. However, to extensively explore an optimal network architecture is non-trivial and we leave it for our future work.
(3) Multi-view Consistency Loss  $\ell_{mv}$ : In our multi-view consistency loss function  $\ell_{mv}(\mathrm{Eq.~3})$ , we simply change it from  $\ell_1$  to  $\ell_2$  optimization, and conduct an additional experiment on Reception of DMSR dataset. From Table 20 and Figure 12, we can see that  $\ell_2$  based loss function can better reconstruct thin structures but with more noisy results than  $\ell_1$  based loss. In this regard, it is of great interest to fully address the recovering task of thin structures and we leave it for future exploration.
Figure 10: Qualitative results of RayDF using visibility classifier with noises.
Figure 11: Qualitative results of using sparse depth values in supervision.
Figure 12: Qualitative results of RayDF on Reception in DM-SR dataset. Red: RayDF-  $l_{1}$  Green: RayDF-  $l_{2}$
Figure 13: Qualitative results of the predicted mesh by RayDF with and without post-processing.
Table 17: Ablation study on the number of multi-view rays sampling. ADE $\downarrow$  on 8 scenes and the average score of the Blender dataset.
M	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
10	4.89	9.82	13.27	3.14	8.27	6.44	7.71	13.29	10.12
20 (RayDF)	4.59	9.24	12.56	3.06	7.98	5.68	7.55	13.07	7.97
40	4.46	8.84	12.05	2.90	7.78	5.37	7.48	12.91	7.72
Table 18: Ablation study on the number of multi-view rays sampling. CD  $(\times 10^{-3})\downarrow$  (mean / median) on 8 scenes and the average score of the Blender dataset.
M	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
10	6.341 / 0.328	5.680 / 1.057	5.789 / 0.630	7.052 / 0.697	1.235 / 0.734	1.743 / 0.675	3.724 / 0.789	4.190 / 0.792	4.469 / 0.713
20 (RayDF)	3.807 / 0.312	3.310 / 0.958	2.914 / 0.543	7.255 / 0.733	1.095 / 0.702	1.604 / 0.617	3.571 / 0.644	3.544 / 0.792	3.388 / 0.663
40	3.445 / 0.314	2.395 / 0.865	2.616 / 0.508	7.264 / 0.742	1.080 / 0.672	0.918 / 0.531	1.508 / 0.579	2.463 / 0.758	2.711 / 0.621
Table 19: Ablation study on the network architecture of ray-surface distance network.
ADE↓	CD (×10-3) ↓ mean / median
8 layers, 512 units	9.13	4.311 / 0.954
8 layers, 1024 units	8.52	4.010 / 0.805
13 layers, 512 units	8.80	4.085 / 0.854
13 layers, 1024 units (RayDF)	7.97	3.388 / 0.663
Table 20: Ablation study on the multi-view consistency loss function on Reception in DM-SR dataset.
ADE↓	CD (×10-3) ↓ mean / median
l2	6.56	12.141 / 6.853
l1 (RayDF)	6.96	10.632 / 5.714
A.4.3 Ablation Studies on Training Strategies
(1) Sparse Depth Supervision: We additionally provide experimental results of our method using sparse depth values in supervision in Tables 21, 22 and Figure 11.
(2) One-stage RayDF: We present detailed quantitative and qualitative results of one-stage training of our pipeline, the performance drops noticeably as shown in Tables 23, 24 and Figure 8.
A.4.4 Ablation Studies on Ray Parameterization
(1) Choices of Ray Parameterization: We discuss our ray parameterization in detail in Section A.1.3. In fact, our pipeline supports various ray parameterizations. To verify this, we conduct ablation studies by replacing our spherical coordinate with the one used in LFN/PRIF. As shown in Table 25 and Figure 14, PRIF parameterization can also achieve satisfactory performance, while LFN parameterization is inferior in our pipeline.
Table 21: Ablation study on sparse depth supervision. ADE $\downarrow$  on 8 scenes and the average score of the Blender dataset.
Depth Sparsity	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
1%	4.29	11.06	15.06	3.21	8.69	4.84	8.33	12.87	8.54
5%	4.83	10.63	14.44	3.65	8.55	4.58	8.44	14.48	8.70
10%	4.85	10.15	14.36	3.99	8.06	4.78	8.22	15.01	8.68
100% (RayDF)	4.59	9.24	12.56	3.06	7.98	5.68	7.55	13.07	7.97
Table 22: Ablation study on sparse depth supervision. CD  $(\times 10^{-3})\downarrow$  (mean / median) on 8 scenes and the average score of the Blender dataset.
Depth Sparsity	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
1%	3.625 / 0.498	3.276 / 1.080	2.365 / 0.544	6.844 / 0.912	1.517 / 0.937	2.446 / 1.048	2.922 / 1.556	3.415 / 0.919	3.301 / 0.937
5%	3.536 / 0.348	3.067 / 1.078	2.323 / 0.578	6.577 / 0.875	1.522 / 0.928	2.422 / 1.091	2.927 / 1.532	3.623 / 0.930	3.250 / 0.920
10%	3.486 / 0.341	3.068 / 1.088	2.307 / 0.593	6.867 / 0.871	1.503 / 0.952	2.381 / 1.076	2.922 / 1.533	3.969 / 0.934	3.313 / 0.924
100% (RayDF)	3.807 / 0.312	3.310 / 0.958	2.914 / 0.543	7.255 / 0.733	1.095 / 0.702	1.604 / 0.617	3.571 / 0.644	3.544 / 0.792	3.388 / 0.663
Table 23: Ablation study on one/two-stage training. ADE $\downarrow$  on 8 scenes and the average score of the Blender dataset.
Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
one-stage	6.69	12.35	17.37	10.94	15.88	9.20	9.27	17.61	12.41
two-stage (RayDF)	4.59	9.24	12.56	3.06	7.98	5.68	7.55	13.07	7.97
Table 24: Ablation study on one/two-stage training. CD  $(\times 10^{-3})\downarrow$  (mean / median) on 8 scenes and the average score of the Blender dataset.
Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
one-stage	4.771 / 0.201	3.459 / 1.202	2.956 / 0.310	10.068 / 0.557	1.515 / 0.616	2.875 / 1.030	2.881 / 1.450	3.727 / 0.700	4.032 / 0.659
two-stage (RayDF)	3.807 / 0.312	3.310 / 0.958	2.914 / 0.543	7.255 / 0.733	1.095 / 0.702	1.604 / 0.617	3.571 / 0.644	3.544 / 0.792	3.388 / 0.663
Table 25: Ablation studies on ray parameterization on Blender dataset.
ADE↓	CD (×10-3) ↓ mean / median
LFN param. + (Ray-surface Distance Network + Visibility Classifier)	89.53	82.102 / 10.298
PRIF param. + (Ray-surface Distance Network + Visibility Classifier)	8.42	3.409 / 0.621
(LFN param. + Ray-surface Distance Network) + (Our param. + Visibility Classifier)	91.30	142.570 / 74.345
(PRIF param. + Ray-surface Distance Network) + (Our param. + Visibility Classifier)	8.44	3.526 / 0.653
RayDF	7.97	3.388 / 0.663
Figure 14: Qualitative results of using different ray parameterizations.
(2) Artifacts from Spherical Coordinate: The artifacts of fixed patterns in the predicted distance maps (as shown in Figure 15 (red)) are caused by the choice of spherical coordinate. We conduct additional experiments given two different choices of ray parameterizations. As shown in Table 25, our applied spherical coordinate can achieve the best ADE scores, and the caused artifacts can be easily fixed by using PRIF's ray parameterization as shown in Figure 15 (green). We leave it for our future work to fully address such an issue.
Table 26: Ablation study on post-processing. CD  $(\times 10^{-3})\downarrow$  (mean / median) on 8 scenes and the average score of the Blender dataset.
Post-processing	Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
4.378 / 0.203	3.414 / 1.058	4.539 / 0.236	9.846 / 0.508	1.749 / 0.832	2.546 / 1.159	2.955 / 1.540	3.829 / 0.678	4.157 / 0.777
✓	3.807 / 0.312	3.310 / 0.958	2.914 / 0.543	7.255 / 0.733	1.095 / 0.702	1.604 / 0.617	3.571 / 0.644	3.544 / 0.792	3.388 / 0.663
A.4.5 Ablation Studies on Post-processing
We follow the outlier removal step in Section 3.4 as post-processing to obtain clean 3D point clouds for further mesh reconstruction and evaluation. In all experiments, we set the outlier threshold as 5 for post-processing. We conduct additional experiments w/ and w/o the outlier removal step on the Blender dataset. In Table 26, the experiment without post-processing yields higher CD errors compared to the one with post-processing. The qualitative results are shown in Figure 13.
Ours + PRIF Param.
Figure 15: Visualizations of distance prediction using different ray parameterization.
A.5 Limitations of RayDF
Since our RayDF takes the conventional spherical parameterization for the input ray, i.e., a sphere is predefined to bound the target 3D scene, our RayDF with such a parameterization cannot represent the 3D scene with rays starting from a position inside the sphere. However, the proposed distance field with a visibility classifier of our RayDF is flexible to different ray parameterizations. How to integrate distinct parameterizations of input rays for different types of 3D scenes with our ray-surface distance field and the dual-ray visibility classifier can be a future exploration direction.
A.6 Additional Quantitative Results
In this section, we report the detailed quantitative results of each scene on the three datasets.
Blender: In Tables 28 and 29, we show the quantitative results of the experiments in Group 1. Our RayDF achieves significantly better performance than the other baselines on most scenes, especially on complex scenes (e.g., Drums, Ficus, and Ship). In Tables 30, 31, 32, 33 and 34, we report the geometry and appearance metrics in Group 2. Our method achieves comparable performance with DS-NeRF for novel view synthesis while recovering the 3D shape with fewer outliers.
DM-SR: In Tables 35 and 36, we report the quantitative comparisons in Group 1 and our RayDF performs better than the other baselines on those indoor scenes with more complex objects. As shown in Tables 37, 38, 39, 40 and 41, our method still achieves high-quality results for novel view synthesis on all scenes in Group 2 experiments.
ScanNet: In Tables 42 and 43, our RayDF performs significantly better than the other baselines on the real-world scenes. To represent these real-world 3D scenes with much more noise and complex geometry, our method outperforms all baselines while obtaining photo-realistic novel view synthesis
results, as shown in Tables 44, 45, 46, 47 and 48.
Moreover, we provide comparisons of optimization time with baselines in Table 27. We compare the average training time of our method and all baselines on each scene of Blender dataset, using a single NVIDIA RTX 3090 GPU card and a CPU of AMD Ryzen 7. Since we apply a two-stage training strategy, our method is not as fast as baselines during optimization. Nevertheless, once our ray-surface distance network is optimized, it can achieve superior efficiency in rendering novel views, as shown in Table 1.
Table 27: Optimization time consumption (hours).
Time
OF [46]	0.2
DeepSDF [54]	0.6
NDF [14]	2.1
NeuS [77]	10.2
DS-NeRF [19]	22.8
LFN [64]	0.9
PRIF [23]	2.2
RayDF (Ours)	24.9
A.7 Additional Qualitative Results
We provide additional qualitative results on the three datasets as follows, including distance map, mesh from Group 1, and radiance map, textured mesh from Group 2. We also report the visualizations of our surface normal, which is computed following Section A.2.
Blender: In Figures 16 and 17, we show the qualitative results on Materials and Drums. In particular, Materials is a scene that contains empty spaces between multiple objects. Our RayDF predicts
distance with fewer outliers near the object surfaces. Our surface normal indicates that our method recovers the 3D scene with a smoother surface while preserving more details of the complex geometry.
DM-SR: In Figures 18 and 19, we report the qualitative results on Reception and Study. Our RayDF predicts accurate distances and hence is capable of reconstructing more detailed structures.
ScanNet: In Figures 20 and 21, we show the qualitative results on Scene0005_00 and Scene0030_00, which are challenging real-world scenes. Our method recovers the 3D real-world scenes with reasonable details while predicting realistic radiance results.
Figure 16: Qualitative results of all methods on Materials in Blender dataset.
Figure 17: Qualitative results of all methods on Drums in Blender dataset.
Figure 18: Qualitative results of all methods on Reception in DM-SR dataset.
Figure 19: Qualitative results of all methods on Study in DM-SR dataset.
Figure 20: Qualitative results of all methods on Scene0005_00 in ScanNet dataset.
DeepSDF
NDF
Distance
DS-NeRF
LFN
PRIF
OF
Mesh
DS-NeRF
Normal
DeepSDF
Figure 21: Qualitative results of all methods on Scene0030_00 in ScanNet dataset.
Table 28: ADE↓ scores of all baselines and our method in experiments of Group 1 on the 8 scenes of Blender dataset.
Chair	Drums	Ficus	Hotdog	Lego	Materials	Mic	Ship	Avg.
OF [46]	6.52	14.37	19.88	4.35	9.85	6.06	8.36	15.20	10.57
DeepSDF [54]	6.84	16.55	22.80	8.37	12.18	9.45	9.31	18.07	12.95
NDF [14]	7.83	14.93	17.26	7.92	11.42	11.61	9.77	16.39	12.14
NeuS [77]	6.87	10.61	19.02	3.36	8.43	7.43	13.92	25.37	11.88
DS-NeRF [19]	11.73	19.42	18.75	2.22	11.01	5.77	19.20	17.66	13.22
LFN [64]	16.33	28.09	31.56	14.02	20.27	36.93	20.89	27.65	24.47
PRIF [23]	7.65	19.57	25.73	5.83	12.86	15.69	13.35	16.72	14.68
RayDF (Ours)	4.59	9.24	12.56	3.06	7.98	5.68	7.55	13.07	7.97