Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This work proposes to build neural field network using polynomial basis and a Fourier extension. Their goal is to provide sub-band frequency control for the output of the network, but for both upper and lower band frequencies, instead of only upper band frequencies per layer [1] . The authors propose to concatenate finite degree multivariate polynomials which facilitate sub-band decomposition. The final network operates with a controllable set of sub-band decompositions as it constitutes an ensemble across networks with different sub bands. The authors show the performance of the network on multiple types of tasks coordinate networks are used, including pixel reconstruction with different levels of noise, 3d shape reconstruction. The performance of the network is qualitatively better than previous work in these reconstructions for high frequency components, including [1], but quantitatively comparable.

[1] Lindel et al. 2022. BACON
strengths_and_weaknesses: This work leverages signal processing techniques for subband decomposition to build better coordinate networks. This in turn leads to an interpretable coordinate network where the layers are lower and upper banded. One interesting result from this work is Fig 
4, where the results show the network can isolate the image content and apply different filters at different layers.

The paper reads as a story on how to build a network Fig 1b, this in turn makes it hard to read unless you are familiar with all the component necessary for subband decomposition and coordinate networks. The paper would be accessible to a wider audience if it builds from the components/architecture necessary/standard for coordinate networks and the new components proposed by this work.
The results in Figs 2, 4 and, 5 show results that appear to be qualitatively superior than previous work including [1]; however, the quantitative results in Tables 1,2,3 do not provide any confidence intervals, and are .## significant digits. 
The 

[1] Lindel et al. 2022. BACON
questions: - It is unclear to me from the text, which parameters beyond the network weights need to be optimized? 
- What is the cost of scaling this approach with the dimensions of the image?
- The results in Figs 2, 4 and, 5 show results that appear to be qualitatively superior than previous work including [1]; however, the quantitative results in Tables 1,2,3 do not provide any confidence intervals, and are .## significant digits. What are the limitations of the metrics used in Tables 1,2 and 3?
limitations: N/A
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The authors propose a basis-encoded polynomial neural fields with a basic theoretical framework for implicit signal manipulation. Specifically, a set of PNFs (e.g., a Fourier PNF), with the outputs be decomposed in a fine-grained manner in the frequency domain. Experiments show that PNFs achieve comparable performance with other SOTA methods.
strengths_and_weaknesses: 1. An interpretable way for signal manipulation.
2. Multiple manipulation applications to verify the proposed ideas.
3. With several theories to support the proposed ideas.

questions: 1. Lack of efficiency analysis, how about the training and inference time with compared methods.
2. Why only perform experiments on Blender scenes for Neural Radiance Field?
3. More related works for Polynomial Neural Networks.
limitations: Please refer to the Questions section.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper describes a novel class of neural fields, which have explicit control over their frequency content. Inspired by classical wavelet decomposition of signals, the approach allows to construct neural fields that covers the Fourier spectrum in separate non-overlapping sections. Each section is controlled by both a lower and upper band-limited, as well as an angular sensitivity. Until now, NFs where only controllable via an upper frequency band-limit. The presented NFs thus have nice properties which could benefit a wide range of applications for which frequency content control is important (of which several are addressed in this paper), the NFs also seem to have generally good approximation quality (both in terms of quality of fit as well as convergence speed).
strengths_and_weaknesses: **Strengths**
* The paper is well-written
* The paper is timely
* The paper solves a relevant and so far unsolved problem for controlling frequency content in NFs
* The paper is sound and is thorough in its citations (even to classic works like Simoncelli-Freeman)

**Weaknesses**
* A discussion on parameter efficiency and computational efficiency would be appreciated

**Literature comment**
In terms of literature that band-limits NFs from above it only makes mention of BACON, however [R21], which is on controlling frequency content (band-limiting) via MFNs of the Gabor type, predates BACON (probably independently solved the same problem?)
[R21] Romero, David W., et al. "FlexConv: Continuous Kernel Convolutions With Differentiable Kernel Sizes." ICLR. 2021.

questions: None
limitations: I see no issues here.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper proposes an approach to allow subband manipulation in neural fields. Specifically, subbands are decomposed in the Fourier space and there is one network for each subband and the final results are combined. Subband manipulation is achieved via manipulating the corresponding neural network.
strengths_and_weaknesses: ## Strengths
1. The idea is interesting and it adds a subband manipulation capability to neural fields. I believe this is a valuable contribution.
2. The proposed approach yields marginally better results than existing approaches.
3. The subband manipulation capability is demonstrated via applications such as texture transfer.

## Weaknesses
1. In this approach, the subband decomposition has to be predetermined and having one network per subband increases the memory complexity of the overall approach. Due to this, the manipulation capability might be limited. Furthermore, how does the number of parameters of the proposed method compares to an existing approach like BACON?

## Post rebuttal
Thank you for providing the parameter efficiency analysis. I'm already positive and increasing my score.
questions: Line 99: how does $\gamma_i$ relate to $f$? The statement is not clear.
Line 110: What are $a_i$, $b_i$ and $I$? The statement is not clear.
limitations: Memory limitation is mentioned and the societal impact is adequately addressed.
review_title: null

Meta-review: metareview: There is a clear consensus for accepting the paper. The area chair agrees with the reviewer's comments and follows their recommendation.
confidence: Certain
Decision: Accept