We’re thankful for your time and the valuable insights you’ve shared. Your input has significantly advanced our project. In response to your feedback, we proposed a brand new perspective for volume rendering with a strong math foundation and validated it with experiments. We’ll address your concerns below.

The results show a performance drop of about ~2 PSNR, while the speed improvement is not significant. Since vanilla NeRF’s speed is bottlenecked by the coarse network, we showcase the significant improvement in speed using TensoRF in the general response and paste it below. Our method achieves comparable quality as the baseline while running almost real-time on a AMD Ryzen 9 5900HS CPU.

The motivation is not strong. Most state-of-the-art NeRF approaches use shallow MLPs or even no MLPs, making the evaluation less expensive. Reducing ray samples does not seem to address a core issue in radiance field research.

We’d like to highlight that the position of our paper is a brand new perspective for volume rendering, and its applicability towards the general NeRF pipeline that relies on volume rendering. Therefore, the experiments validate that it can be combined with any existing NeRF pipeline that relies on volume rendering regardless of the underlying representation. GL-NeRF is orthogonal to existing state-of-the-art work in the sense that all other works introduce additional representation such as neural networks and grids while GL-NeRF focuses only on the volume rendering integral itself. We believe it's a promising exploration of volume rendering and would bring insights into the radiance field research.

How the points are selected is unclear. The method requires approximating polynomial coefficients and resolving the roots for $x$. However, since $x$ is a highly non-linear function of $t$, finding the samples $t$ unavoidably requires root finding along the ray, which does not seem to actually improve accuracy or efficiency.

We’d like to point out that we do not need to compute the polynomial coefficients and resolving the roots. All the roots and coefficients needed are well defined and can be looked up online / in numpy. Please refer to the document for the function numpy.polynomial.laguerre.laggause for detail.

As for finding $t$ on the ray, traditional volume rendering uses a piecewise constant PDF for approximating the volume density, and so do we. Therefore, there is no need to find roots along the ray for a non-linear function, we use the same approximation as done in the hierarchical sampling strategy.

Why use a look-up table? Will this lead to worse performance?

The Gauss-Laguerre Quadrature is a well-known formula in numerical analysis, therefore there is no need to solve for the roots and coefficients on the run. Since using a look-up table is $O(1)$ in terms of time complexity, therefore it will not lead to worse performance.

Why did the performance not match TensoRF? Would increasing the number of point samples help?

We’d like to first point out that GL-NeRF uses fewer points than TensoRF (4 v.s. ~32), leading to a lower performance with no visual quality drop (our qualitative results are randomly chosen from the images, not cherry-picked). On the other hand, when increasing the number of points used for GL-NeRF, since the predefined Laguerre weights approach to zero and are smaller than machine precision(for 32-bit float, it’s $1.175494 × 10^{-38}$), the performance reaches a bottleneck. Therefore, we focus on the fact that GL-NeRF, with higher precision, can reduce the sample points by a large margin.

Thank you for dedicating your time and providing such perceptive feedback. Your recommendations have considerably enhanced our work. Based on your comments, we have provide a brand new general framework for computing volume rendering and validate its accessibility on 2 baselines. We will address your concerns below.

The proposed method results in dense sampling near the surface, but can it handle translucent objects? Intuitively, there seems to be a correlation between density solidity and rendering quality. It would be desirable to discuss for which scenes the proposed method is effective and for which scenes it is not.

Since volume rendering is originally derived for pure volumetric data, it is inherently not suitable for modeling translucent objects. To verify this argument, we conduct experiments using TensoRF on the DexNeRF dataset[5] and report the result here.

The result shows that our approach performs similarly to traditional volume rendering methods. However, it is ideal that the scenes are pure volumetric since TensoRF itself suffers at modeling translucent objects.

The effectiveness of the proposed method ... is certain. In general, however, volume rendering is used for both training and inference. It would be better to discuss whether the proposed method ... could be used for learning as well.

Since our focus is the training-free aspect of the proposed method, the results of network training with it are not reported in the paper. However, GL-NeRF can be used for learning as well. We conduct experiments combining GL-NeRF with Vanilla NeRF and show the results here. The training time is also reduced by $1.2\times$ to $2\times$.

Blender

LLFF

We expect the availability of the proposed method to be very broad. If possible, the application of the proposed method to NeRF backbones other than vanilla NeRF and TensoRF could be considered to further demonstrate the generality of the proposed method.

Please refer to the general response for the combination of GL-NeRF with Instant NGP.

The correspondence between the plots in Figure 4 is not clear. It would be easier to compare the results if the corresponding scenes of vanilla and the proposed method were connected by a line.

Small Comment: Related works -> Related work

Thank you for the construction comments! We’ll modify the figure and writing as suggested.

Why are the results for TensoRF not shown in Figure 4?

As mentioned in MCNeRF, NVIDIA GPUs have specialized components to accelerate the neural network inferences, therefore evaluating on these devices with more sample points may not cost that much. Therefore we instead implement a WebGL-based renderer as WebGL is a more accessible platform that is agnostic to the underlying hardware[1, 2, 3, 4]. Please find the result in the General Response section.

The sample points up to 8 are very small compared to the use of more than 100 sample points in vanilla NeRF. Could the performance of vanilla NeRF be exceeded with more sample points? In other words, vanilla NeRF also approximates integration by discretization, and this sampling density is constant. We believe that a comparison of vanilla NeRF and the proposed method in terms of sample points vs. image quality will more convincingly demonstrate the effectiveness of the proposed method.

Thanks for this very interesting point. When increasing the number of points used for GL-NeRF, since the predefined Laguerre weights approach to zero and are smaller than machine precision, the performance reaches a bottleneck. Therefore, we focus on the fact that GL-NeRF, with higher precision, can reduce the sample points by a large margin. An example of the Gauss Laguerre Quadrature look-up table with n=64 is presented here. Notice that when n gets bigger, the weights get extremely small and would become smaller than machine precision (for 32-bit float, it’s $1.175494 × 10^{-38}$). We do have a comparison in terms of sample points vs. image quality with a smaller number of points shown here.

[1] Gupta, Kunal, et al. "MCNeRF: Monte Carlo rendering and denoising for real-time NeRFs." SIGGRAPH Asia 2023 Conference Papers. 2023.

[2] Chen, Zhiqin, et al. "Mobilenerf: Exploiting the polygon rasterization pipeline for efficient neural field rendering on mobile architectures." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.

[3] Reiser, Christian, et al. "Merf: Memory-efficient radiance fields for real-time view synthesis in unbounded scenes." ACM Transactions on Graphics (TOG) 42.4 (2023): 1-12.

[4] Yariv, Lior, et al. "Bakedsdf: Meshing neural sdfs for real-time view synthesis." ACM SIGGRAPH 2023 Conference Proceedings. 2023.

[5] Ichnowski, Jeffrey, et al. "Dex-NeRF: Using a neural radiance field to grasp transparent objects." arXiv preprint arXiv:2110.14217 (2021).

We truly value your acknowledgment of our work and your recognition of our responses. Your insightful feedback and favorable recommendation are deeply appreciated.

We are grateful for your time and insightful comments. Your valuable suggestions have significantly elevated our work. In light of your comments, we have proposed a brand new mathematical perspective for volume rendering and proved its effectiveness on speedup using two baselines. We will address your concerns below.

The main issue with this paper is that the experiments do little to give an idea of how the proposed method would compare to the current state of the art.

We validated it with 3 baseline approaches (Vanilla NeRF, TensoRF and InstantNGP in the rebuttal) and show that our method has a plug-and-play attribute. In the experiments, we primarily aim to demonstrate GL-NeRF’s plug-and-play attribute, and can provide comparable performances as well as reducing the sample points, which results in speedup as a free lunch. Moreover, we’d like to point out that our paper’s main contribution is on the mathematical perspective itself.

proposal networks from Mip-NeRF 360 and Zip-NeRF.

While Mip-NeRF 360 focuses on anti-aliasing and Zip-NeRF is its combination with Instant NGP, we instead conduct experiments by combining GL-NeRF with InstantNGP since our focus is on the volume rendering integral itself. The results can be found in the general response and we paste it here for better reference. The result along with the results on Vanilla NeRF and TensoRF indicates that our method can be incorporated into any NeRF pipeline that relies on volume rendering regardless of the underlying representation.

On the other hand, our method needs no training, therefore can be plugged into any existing pipeline, which is the main advantage of our work compared to e.g. the proposal networks from Mip-NeRF 360 and Zip-NeRF.

It is strange that timings are reported for vanilla NeRF but not TensoRF, which would presumably see a larger gain. Is there a reason for this?

As mentioned in MCNeRF, NVIDIA GPUs have specialized components to accelerate the neural network inferences, therefore evaluating on these devices with more sample points may not cost that much. Therefore we instead implement a WebGL-based renderer as WebGL is a more accessible platform that is agnostic to the underlying hardware[1, 2, 3, 4]. The result is in the General Response section and we paste it here for better reference. As can be seen from the table, our method achieves comparable quality as TensoRF and achieves almost real-time on WebGL running on a AMD Ryzen 9 5900HS CPU by only reducing the sampling points.

It is mentioned, but I think the writing could make it a lot more clear that only the color samples are being reduced, as this significantly affects where the method would actually be expected to provide a speedup. As it is, this is quite easy to miss and could lead to misunderstanding if one does not read the method carefully.

Thank you for this construction feedback. We'll modify the manuscript to make the point clearer.

[1] Gupta, Kunal, et al. "MCNeRF: Monte Carlo rendering and denoising for real-time NeRFs." SIGGRAPH Asia 2023 Conference Papers. 2023.

[2] Chen, Zhiqin, et al. "Mobilenerf: Exploiting the polygon rasterization pipeline for efficient neural field rendering on mobile architectures." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.

[3] Reiser, Christian, et al. "Merf: Memory-efficient radiance fields for real-time view synthesis in unbounded scenes." ACM Transactions on Graphics (TOG) 42.4 (2023): 1-12.

[4] Yariv, Lior, et al. "Bakedsdf: Meshing neural sdfs for real-time view synthesis." ACM SIGGRAPH 2023 Conference Proceedings. 2023.

We appreciate the time and thoughtful feedback you've given. Your suggestions have greatly improved our work. Considering your insights, we have provided a highly theoretical framework for calculating volume rendering and validated it using 2 baselines. We will address your concerns below.

In the experiment, the authors validated the proposed method on two backbones (the original NeRF and the TensoRF). However, neither of these represents the current fastest method. It is suggested to compare the proposed method with Instant NGP, DVGO, or other faster alternatives. Such comparisons could not only better verify the plug-and-play capability but also significantly enhance the impact of the paper.

Since Instant NGP is a representative of grid-based representation for volume density, we combined GL-NeRF with Instant NGP as suggested and compared between the versions. We believe doing so would provide a more comprehensive perspective for the plug-and-play attribute of GL-NeRF. As can be seen from the table, our method achieves comparable performance with instant NGP using only $\frac{1}{8}$ number of color MLP calls. Since the experiment using TensoRF on WebGL has shown that reducing the color MLP calls can lead to significant speedup, here we simply showcase the color MLP calls needed in the table. The experiment indicates that as long as the underlying radiance fields pipeline relies on volume rendering, GL-NeRF can be an off-the-shelf replacement for dense sampling / hierarchical sampling, agnostic to the underlying representation.

We greatly appreciate your recognition of our work's novelty. We sincerely thank you for your thoughtful feedback and are grateful for your positive recommendation.