SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\mathbf{d}_1, \dots, \mathbf{d}_K$  are known viewing directions as the modes of the distribution. It is noteworthy that when  $c$  approaches zero, the probability density function becomes a uniform distribution. Thus, the uniform distribution used in the main paper is a special case of the mixture of vMF distribution. The concentration parameter  $c$  is a hyper-parameter. We vary it, and use the probability density function to estimate the color field of DTU scenes given the density field from trained DVGOs. The PSNR results of estimated color fields are reported in Table 5. It shows that a large concentration parameter decreases PSNR. Although  $c = 0.01$  is slightly better by 0.01 PSNR, we apply a uniform distribution  $(c = 0)$  in the main paper for its simplified and more efficient calculation process.
Table 5: Color estimation results on DTU scenes with different concentration parameters.
Concentration parameter	0	0.01	0.1	1	2
PSNR	29.44	29.45	29.43	28.93	27.97
B.1.2 Different SH Degrees
In Table 6, we report the PSNR of estimated color fields using different SH degrees. It shows a clear trend that the PSNR increases with the increase of the SH degree. The computational cost is not obviously affected by the SH degree, but the higher SH degree requires much memory.
Table 6: PSNR of estimated color fields using different SH degrees based on DVGO's density fields.
SH degree	0	1	2	3	4
SH basis	1	4	9	16	25
2563grid	26.13	26.99	27.89	28.30	28.62
5123grid	27.22	28.29	29.44	29.98	-
B.2 More Regularization Results
B.2.1 Visualization and Comparison of Ray-based Losses
We train the DVGO [15, 27] on a typical scene (DTU scan 40). After training, we calculate common ray-based losses, including the photometric loss, per-point RGB loss [27], background entropy loss
The existing ray-based losses are not aware about the geometry. The CF loss is more sensitive
False geometry can lead to low loss values.
Ground-truth of a training view
Trained color field
to the false geometry.
Closed-form color field
Background entropy loss
Photometric loss
Per-point RGB loss
CF loss
Ray-entropy loss
Figure 12: Visualization and comparison of common ray-based losses on a typical scene with false geometry. All losses are normalized into [0, 1]. The red rectangles mark the false geometry region, and the percentages of the losses in this region over the whole image are shown on the left-top corners.
Distortion loss
Depth trained with CF loss
[15, 27], ray-entropy loss [13], and distortion loss [1], of a training view based on the trained model. For ray-based losses, each pixel has a loss value, so we can visualize the losses as images as shown in Fig. 12. We marked a region with a red rectangle, where the top surface of the bottom rock is falsely reconstructed. On the left-top corners of the loss images, we list the percentages of the losses in the red rectangle over the whole image. These percentages reflect to what extent the corresponding losses focus on the geometric error. The photometric loss of the training view has a very low percentage of  $4.7\%$ . This demonstrates that minimizing the photometric loss cannot assure a correct geometry, which is caused by the shape-radiance ambiguity problem. For other ray-based losses, they are also not sensitive enough to the false geometry, with the largest percentage of  $12.4\%$ . In contrast, the CF loss has a significantly larger percentage of  $28\%$ . It focuses more on geometric errors than common ray-based losses. The errors can be reflected by the image rendered by the closed-form color field, as shown by the right-top image of Fig. 12. The red rectangle has a large closed-form photometric loss, and, more importantly, this loss can be used to improve the geometry of the scene.
B.2.2 Effect of the Weight Factor  $\lambda$  and Number of Rays  $|\mathcal{R}_{cf}|$
We vary the two hyper-parameters, including the weight factor and number of rays, on Plenoxels [7], and report the average PSNR and IMRC of the DTU dataset in Table 7. The PSNR and IMRC have an increasing trend when we increase the weight factor. Moreover, we showcase an example scene in Figs. 13-14. By increasing the weight factor or the number of rays, the problem of thick surface has been relieved. When the weight factor is too large ( $\lambda = 20$ ), however, the PSNR decreases. This is because the CF loss dominates the training. While it only provides the supervision signals to the density field, the color field cannot be well optimized. Therefore, the weight factor cannot be too large.
For the number of rays, increasing it helps improve the geometry of the scene. However, more rays mean higher computational cost. To be more specific, 17 minutes are needed on average to train the DTU scenes without CF loss. When we add the CF loss and use 10 rays in every iteration, 32 minutes are needed on average. The time cost further increases to 43 minutes if we use 25 rays. To ensure the training efficiency, we do not increase the number of rays further.
Table 7: Average PSNR↑/IMRC↑ of different weight factors  $\lambda$  and numbers of rays  $|\mathcal{R}_{cf}|$  on the DTU dataset.
|Rcf|	λ = 1	λ = 2	λ = 5	λ = 10
10	32.00 / 16.18	32.02 / 16.29	32.03 / 16.44	32.04 / 16.59
25	32.04 / 16.12	32.06 / 16.26	32.07 / 16.50	32.08 / 16.66
Figure 13: Effect of the weight factor on DTU scan 24.
Figure 14: Effect of the number of rays on DTU scan 24.