Thank you very much for your review! Let us address your points below:

"Fourier domain analysis … limit the practical applicability of the methods?”
“Why torus? Is there any topology constraint?”

In short: no, it does not limit applicability. This is because the periodic boundary conditions needed to leverage our Fourier methods apply to the bounding box inside of which the point cloud sits, not to the point cloud or reconstructed surface - the point cloud and surface do not need to be periodic in any way.

More specifically, to avoid periodic boundary conditions affecting reconstruction, we situate the point cloud sufficiently-far from the boundary to limit their effect: this mirrors what is done in ordinary PSR, which imposes other types of boundary conditions (say, Dirichlet or Neumann), and moves the point cloud far-enough away from the boundary so that it is largely unaffected by them.

From a mathematical standpoint, the main reason for using periodic boundary conditions - and, therefore, the torus - is because (a) its Fourier analysis is much simpler compared to alternatives and (b) it works. One could instead try to develop a method using, say, properties of the sphere, but this would necessitate computations in terms of spherical harmonics, which are more complicated than the sines and cosines we use. We therefore think extensions like this are better left to future work.

“The evaluation criteria in the current work are rather simplistic … I suggest to design more evaluations and compare with [25] comprehensively, including, accuracy, scalable etc”
“Since approximations are used during Gaussian process, the quantitative accuracies are expected in the experiments.”

We agree that improved evaluations would strengthen the paper (though with the caveat that we have followed the SPSR baseline paper’s lead and focused on qualitative evaluations because they are important in graphics applications), and to this end have completed a number of additional evaluations looking at things like the effect of hyperparameters on results, and timing experiments. We are additionally aiming to add a quantitative next-view planning comparison to the final manuscript draft. We describe these additional experiments to be added to the appendix - both the ones that are complete with results, and those still in progress - in detail in our response to Reviewer vrrd.

“Does the grid occupy too much space? What is the resolution?”

We use a grid of size $50^d$. As all of the experiments are done for $d = 3$, the grid size is ~500 KB of memory. More broadly, we agree with the importance of comprehensively evaluating how grid size affects performance, and have performed additional experiments showing a too-small amortization grid results in over-smoothing, which we will add to next manuscript version’s appendix.

“The experiments should valid the time and space on large point cloud comprehensively”

This is a good idea: we have added an experiment, which uses points sampled from the Stanford Dragon mesh, and examines runtime: we find that using 64 points takes a 15 seconds (much of it we suspect due to Python overhead), 4096 points takes about 30 seconds, and using 65536 points takes about 5 minutes. We will add a plot showing this to the next manuscript version.

“When Monte Carlo sampling is used to reduce memory consumption, how is the representativeness of the sampling results to the posterior distribution ensured? Is there statistical bias due to insufficient sampling times?”

Since our posterior is a Gaussian process, we can use direct Monte Carlo sampling, which is by definition unbiased (as opposed to, say, other settings which require Markov chain Monte Carlo methods or similar). However, one might be concerned about variance of posterior functionals: here, it is a good question to ask how many samples are enough in practice, as is requested by Reviewer vrrd. To address this, we will add additional comparisons which show how transmittance calculations and collision detection performance varies with the number of Monte Carlo samples to the next manuscript draft.

“It is possible to generalize the the method to screened Poisson surface reconstruction”

This is an interesting question! It seems highly plausible that an extension of our approach would work, but it would require analytically re-computing the relationship between the Karhunen-Loeve decompositions, which would be sufficiently-involved mathematically that we believe it is best to defer it to future work.

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Summary

Overall, your suggestions have led us to improve the writing - in particular, to emphasize that our Fourier formulation does not restrict the kinds of surfaces we can reconstruct - as well as the strength of evaluations used for this work. On behalf of these additions, we would gently like to ask whether you would consider increasing your score.

Thank you for your review! We appreciate that you mentioned our method “has the potential for broad applicability” and that we have a “mathematically principled approach” which was indeed part of our motivation for this work.

“I'm not convinced by the claim that the proposed method qualitatively matches the outputs of SPSR in Section 4.1”
“Despite the paper’s contribution to accelerating SPSR… diminished if the method does not maintain the original reconstruction quality”

These are very good points: but addressing them involves some nuance. First, let us draw attention to the following distinction between tasks (also mentioned in our response to Reviewer otmx):

1. “Surface reconstruction”: given a point cloud, produce a reconstructed surface
2. “Uncertainty quantification for surface reconstruction”: given an underlying surface reconstruction algorithm, and noisy or otherwise imperfect inputs, produce an estimate of uncertainty for the reconstructed surface

From this viewpoint, our proposed algorithm’s purpose is 2 - that is, uncertainty quantification for surface reconstruction, where the “underlying surface reconstruction algorithm” is classical PSR (or more precisely, a minor variant thereof, given our slightly-different boundary conditions).

With this framing, we agree improved evaluations would strengthen our paper, but where the focus should be on stochastic aspects to do with uncertainty as opposed to the mean reconstructed surface, which is essentially the same as in classical methods. Mirroring the standard in graphics and in the SPSR paper (our baseline), our focus has been on qualitative properties, though we agree quantitative evaluations would strengthen our paper further. To this end, we have decided to add additional experiments to the next manuscript’s draft in the form of an expanded appendix:

1. This includes a comprehensive examination of how hyperparameters affect results: we find that (a) too-small an amortization grid density leads to over-smoothing, (b) SGD and Cholesky perform similarly (assuming the latter succeeds), (c) for sufficiently-small length scales, the SPSR baseline can result in over-smoothing, whereas our approach works well.
2. We have also performed runtime comparisons, which show our approach to be faster than SPSR as long the length scale is not too large (note that the small length scale regime is the interesting one, as this allows the algorithm to capture fine surface details).
3. We are additionally working on a quantitative evaluation examining how the produced uncertainty affects next-view planning, as a way to test how our algorithm’s numerics affect situations where one needs to use the produced uncertainty for a downstream purpose.

We hope that these additions - of which the first two are complete - will help strengthen our evaluations and therefore alleviate your concerns.

“A key contribution of the paper is replacing the computation of means and covariances with Monte Carlo sampling from the posterior.”
“I would appreciate it if the authors could provide a more in-depth discussion on Monte Carlo sampling”

Thank you for raising these points: due to their importance, let us respond in two parts.

First, in situations where this is viable, we utilize analytic expressions derived from eqn. (6) to compute means and covariances rather than sampling: in hindsight, this point came out somewhat-hidden in our text, as we wanted to emphasize sampling as a new capability compared to baselines. We will therefore modify the next draft to make this clearer - thank you for drawing our attention here!

Second, in situations where sampling is needed, we agree that further evaluation of the number of Monte Carlo samples needed is appropriate. Since the random functions we are sampling do come from Gaussian process posteriors, we expect the number of samples needed, in most cases, to be similar to other situations where Gaussian process sample paths are used - such questions are explored to some degree in the pathwise conditioning papers, though for different purposes. To address this, we will add additional comparisons which show how transmittance calculations and collision detection performance varies with the number of Monte Carlo samples to the next manuscript draft.

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Summary

Overall, your suggestions have led us to significant improvements, both in terms clarity and especially the evaluations we present, which we believe, based on those parts we were able to complete so far, will significantly strengthen our results. On behalf of these additions, we would gently like to ask whether you would consider increasing your score.

Thank you for your review! We are very happy that you recognized our approach as “more efficient” than prior ones and that our “claims are supported relatively well by evidence.” We address your questions below:

“manipulation of Fourier coefficients … something like a linear solve.”
“Appendix B suggests … relatively expensive”

This is a fair point - but let us add a critical distinction: our Fourier coefficients depend only on the kernel and not on the point cloud, unlike the linear solves in SPSR. This property enables amortization, so one can precompute Fourier-coefficient manipulations once in advance, rather than on a per-point-cloud basis.

“how many samples … generating the queries in Figures 1 and 3”

These are done using analytic expressions derived from eqn. (6) rather than sampling: in hindsight, this point came out somewhat-hidden in our text, as we wanted to emphasize sampling as a new capability compared to baselines. We will modify the next draft to improve this.

“A first step in incorporating … one such scheme (Sellan and Jacobsen, Figure 15).”

We largely agree, with one tiny exception: we do introduce a modified score function in Sec. 3.4, which can be evaluated much faster than the one from the original SPSR paper, but is otherwise similar. Due to space constraints, we cut this in favor of future work: for completeness, we will add a comparison between the two in the appendix of the next draft, following the setup presented in Figure 15 of "Neural Stochastic Screened Poisson Reconstruction", Sellán and Jacobson, SIGGRAPH Asia 2023 (which studies the more-general screened Poisson surface reconstruction setting), and will modify our claims as you suggest.

“could benefit from additional quantitative analysis”

We acknowledge that our results are less-quantitative than many ML works: this is similar to prior work such as SPSR, and reflects norms in the graphics community. We nonetheless agree quantitative evals would strengthen the work, and will add a number of such comparisons to the appendix, described in further detail in our response to Reviewer vrrd.

“…unclear how the reconstruction reacts to changing $L$ … Figure 3”
“…role of $L$ on the reconstruction quality”

Thank you for this idea. We’ve done some preliminary tests, and found that setting $L$ too small results in loss of high-frequency details. The same holds for using too-coarse a grid for amortization. We will add this to the next draft’s appendix.

“Figure 2 shows … (induced?) hyperparameter for “f”.”

In both cases, this is a hyperparameter, specifically the number $\kappa$ distances are scaled by before going into the kernel, $k(x,x') = k(\frac{x-x'}{\kappa})$, which determines both $v$ and in turn $f$. We will make this clearer.

“… mechanism to calibrate hyperparameters … more convincing”

This is an excellent question. Since our kernel-matrix solve is identical to that of an ordinary GP, one way to do this would be to apply standard maximum marginal likelihood techniques. We will add discussion on this.

“Figure 10 … number of query points … Or have I misunderstood something?”

You are correct that $L$ is fixed and sufficiently large (we use $L = 100^3$ as mentioned in Appendix B). Here, allow us to again emphasize that our cross-covariance is independent of the input point cloud. While one may be able to optimize FEM meshes for specific point clouds, we can compute our cross covariance once and reuse it for any point cloud.

“Another criticism is that the proposed… either component isn't enough” and “Independent benchmark studies”

This is a good idea. We ran additional comparisons on the difference between SGD and Cholesky factorization, which show that performance (provided Cholesky works) is comparable, and will add them to the appendix. In the case of pathwise conditioning, we mainly view this as new functionality, since SPSR’s global nature limits the attainable resolution of samples.

“A more quantitative version… linear complexity gain”

Good idea! We will add this to the next draft’s appendix.

“Equation (11) .. clarification”

Here, $P(x \in \Omega) = P(f(x) \leq 0)$ as our implicit surface representation takes on negative values inside $\Omega$, zero on the boundary of $\Omega$, and positive values outside of $\Omega$. We will make this more clear in the camera ready.

“term "formal" .. consistently”
“term "problem-(in)dependent"

Thank you for these - we will fix them.

“How close is it … to non-stochastic [PSR]?”

This is a good point and might help readers get a sense of how expensive the method is - we will add numbers to the next draft.

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Summary

Your suggestions have led to improved clarity (via the many points - thank you for them!), as well as strength of evaluations of this work. Given these additions and above clarifications, we would gently like to ask whether you would consider increasing your score.

Thank you for your review! We appreciate your recognition that our approach “increase[s] computational efficiency” and that our paper has “little weakness” and are delighted by these comments! We address your key comments below:

“no quantitative evaluations and comparisons of reconstruction quality”
“However, as a method focused on reconstructing surfaces from point clouds, the paper does not directly compare the reconstruction quality and runtime with other approaches”
“Additionally, the paper only compares its approach with a single baseline, which appears to be somewhat limited”

These comments, while brief, are also quite important: let us address them in some detail. First, let us draw attention to the following subtle distinction between tasks:

1. “Surface reconstruction”: given a point cloud, produce a reconstructed surface
2. “Uncertainty quantification for surface reconstruction”: given an underlying surface reconstruction algorithm, and noisy or otherwise imperfect inputs, produce an estimate of uncertainty for the reconstructed surface

From this viewpoint, our proposed algorithm’s purpose is 2 - that is, uncertainty quantification for surface reconstruction, where the “underlying surface reconstruction algorithm” is classical PSR (or more precisely, a minor variant thereof, given our slightly-different boundary conditions).

As such - and given that algorithms of this kind are rather new - SPSR is the only appropriate baseline we are aware of. Mirroring the standard used in that work - and more broadly in graphics - our results focus on qualitative behavior, though we agree more quantitative comparisons would make the paper stronger.

To achieve this, in addition to the quantitative results that we currently have - such as runtime performance benchmarks - we have performed additional evaluations such as hyperparameter comparisons (as requested by other reviewers and described in our responses there). In addition, inspired by the need to quantitatively evaluate uncertainty in a manner fitting the above framing, we will also add a quantitative benchmark of our next-view planning heuristic (compared to SPSR baseline), following the setup in Figure 15 of "Neural Stochastic Screened Poisson Reconstruction", Sellán and Jacobson, SIGGRAPH Asia 2023 (which studies the more-general screened Poisson surface reconstruction setting). We anticipate to complete this by the next manuscript draft.

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Summary

Overall, your suggestions have led us to significant improvements, especially in clarifying appropriate positioning for this paper, but also in the right way to handle quantitative comparisons. On behalf of these additions - including extra experimental results described in our responses to other referees, which were inspired in part by points that became apparent to us through your review - we would gently like to ask whether you would consider increasing your score.