GL-NeRF: Gauss-Laguerre Quadrature Enables Training-Free NeRF Acceleration

Summary

Volume rendering requires numerical integration for estimating output colors. This work proposes using the Gauss-Laguerre quadrature to reduce the number of samples and improve integration accuracy. The paper demonstrates that this method can be a plug-and-play module for any NeRF model. Experimental results show that with a limited drop in performance, the GL-NeRF can significantly reduce the number of ray samples and MLP calls.

Strengths

1. The perspective of improving quadrature for NeRF is new and interesting. 2. The formulation of using the Gauss-Laguerre quadrature looks good, and the mathematical formulation appears rigorous.

Weaknesses

1. The results show a performance drop of about ~2 PSNR, while the speed improvement is not significant. 2. 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. 3. 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.

Questions

1. Why use a look-up table? Will this lead to worse performance? 2. Why did the performance not match TensoRF? Would increasing the number of point samples help?

Rating

5

Confidence

4

Soundness

3

Presentation

4

Contribution

2

Limitations

The paper mentions that it has a theoretical guarantee of the highest precision. However, there is no evidence that the current precision matches previous works.

Summary

This paper proposes a computational method for volume rendering using Gauss-Laguerre quadrature. In the context of NeRF, volume rendering is performed by evaluating MLPs (or other data structures) at a sequence of query points on a ray and integrating the weighted results. The proposed method reduces the number of evaluations without much performance degradation by computing the integrations using Gauss-Laguerre quadrature. The proposed method has been embedded in vanilla NeRF and TensoRF for validation, and the reviewer reports reductions in computation time and memory usage. The reviewer acknowledges and appreciates the effectiveness of the proposed method and looks forward to future discussions on the generality of the proposed method (other backbones and learning time applications).

Strengths

- As emphasized in the paper, the approach proposed in this paper is a replacement of integral computations, not a learning of sampling or a change in data structure. Therefore, it is applicable and highly available for various NeRF variants that use volume rendering. - The position of the proposed method in NeRF research is clearly stated, especially the related work is very clearly described as an introduction to the proposed method. - In the introduction of the method, the reviewer's first question was l.177, and the paper answers the question. This helps the understanding of the paper and increases the credibility of the proposed method.

Weaknesses

- 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. - The effectiveness of the proposed method as an integration method for oracle or trained models is certain. In general, however, volume rendering is used for both training and inference. It would be better to discuss whether the proposed method is specialized for inference or whether it could be used for learning as well. - 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. - 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

Questions

- Can the proposed method be used for learning? If so, the proposed method could be a very powerful tool. (identical to the description in the Weaknesses section). - Why are the results for TensoRF not shown in Figure 4? - 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.

Rating

8

Confidence

4

Soundness

3

Presentation

3

Contribution

3

Limitations

- I agree with the limitations described in the conclusion. This reviewer believes that the proposed method contributes to real-time rendering, but it better to be validated to make this claim in the future.

Summary

This paper focuses on accelerating novel view synthesis using neural radiance fields (NeRF). Unlike previous works that concentrate on designing lightweight networks, this study is motivated by the specific volume rendering formula, which includes a negative exponential term in the integration function. By employing Gauss-Laguerre quadrature, the authors approximate this complex integral operation, thus improving the rendering speed of existing NeRFs. This approach is validated on two backbones: the original NeRF and TensoRF, demonstrating speed improvements ranging from 1.2X to 2X.

Strengths

1. The idea is intriguing and represents a promising exploration originating from the specific volume rendering formula. 2. This paper is highly theoretical.

Weaknesses

see Questions.

Questions

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.

Rating

7

Confidence

5

Soundness

3

Presentation

3

Contribution

3

Limitations

n/a.

Summary

This paper presents a method for reducing the number of color samples needed for volume rendering in Neural Radiance Fields. The method works by applying Gauss-Laguerre quadrature as a replacement for the importance sampling used by some NeRF methods to reduce the number of calls needed for their fine color MLPs. This approach is shown to yield some efficiency improvements on plain NeRF as well as TensoRF models.

Strengths

The mathematical aspects of the paper are detailed and well supported. It is very clear what the method is trying to achieve and how the Gauss-Laguerre formulation is being applied. Generally, the quality of explanation is good and not too hard to follow, even though the math is fairly dense. Overall, I think this is an original idea with some likely applications in volume rendering.

Weaknesses

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, which has progressed quite significantly beyond the baselines shown here. While it is quite believable that the GL method outperforms the naive hierarchical sampling of the original NeRF, I have significant doubts about that holding when applied to something more recent like the proposal networks from Mip-NeRF 360 and Zip-NeRF. Given the weak evaluation, I would lean towards rejecting.

Questions

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? I would suggest that the authors try to show quantifiable improvement on a more recent baseline. If there were a substantial run-time speed up for TensoRF, that would be good, but something lime Mip-NeRF 360 or Zip-NeRF would be even better as that would show how GL compares to another method which tries to draw samples near the surface.

Rating

6

Confidence

4

Soundness

3

Presentation

3

Contribution

2

Limitations

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. I don't think there are any notable concerns with the paper regarding societal impact.

Response by Reviewer PEjS

Thank you for your explanation. Most of my initial concerns have been satisfactorily addressed, and I now recognize the novelty in the proposed method. Although the current approach may seem somewhat limited in its impact, I believe it holds potential for broader applications in other areas. Given these considerations, I would like to raise my score.

We sincerely appreciate your time and valuable feedback

Dear Reviewer GERG, We sincerely appreciate the time and effort you have dedicated to providing valuable feedback. As the discussion period concludes on Tuesday, August 13, please let us know if there are any remaining questions or if further clarifications are needed. We would be more than happy to provide any additional details. Best Regards, Authors