Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The paper proposes a shape-correspondence approach based on combined  spectral-domain filtering of the Laplace-Beltrami operators and the feature assignment in the corresponding spectral embedding space. The ideas per se are not new, but what I believe made a big difference is the adoption of the joint spectral filter $\omega(\lambda_i,\mu_j,e)$ that kills the signal associated to corresponding eigenvalues that are different from one another, thus not using noisy eigenfunctions in the spectral embedding.
strengths_and_weaknesses: Very strong experimental results
very nice idea of adopting the joint spectral filter which I really believe is the key to the performance.

The paper sees to want to reinvent spectral filtering on graphs and linear assignment on the spectral embedding space with slightly different names.
questions: It might be interesting to learn the joint spectral filter.
limitations: Negative societal impacts have not been mentioned, but apart from the use in a futuristic 3D surveillance system, I cannot see one myself.
The authors are honest about the extraction of non-smooth solutions, but I think the phenomenon warrants a more in-depth analysis/explanation.
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final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The paper presents a graph matching method using 2D graph convolution. In articular, the proposed method is designed to achieve better accuracy for extract shape matching. The method can be applied to both noisy non-isometric data and nearly-isometric shapes. Evaluations show that the proposed method achieves state of the art performance. 
strengths_and_weaknesses: Strengths:
- Shape matching in 3D has been well studied. The paper tackles the case for exact matching, in which previous methods don't perform well. 
- The quality of the evaluation is good. Comparisons to competing methods and self ablations are extensive. 
- The proposed algorithm include multiple steps involving various components, though the authors could well deliver the essential concepts. Notably, the relation between 2D-GEM and GRANMPA helps to understand the novelty.  

Weaknesses:
- Lack of theoretic foundations, it is not clear why the proposed method achieve excellent performance for exact matching on the selected datasets.
- Comparing to related works, the proposed pipeline has high similarity with GRAMPA [18], in terms of the use of cost function, 2-hop improvement, etc. LMD [73] is borrowed to adapt to near-isometric case.   
questions: - How sensitive is 2D-GEM to the correspondence initialization? 
- As mentioned, GRAMPA is normally used as initialization, how does 2D-GEM perform if initialized with GRAMPA, instead of using SHOT with GMWM?
- 2D-GEM could use point features, does this help on SCAPE or TOPKIDS? Intuitively, pointwise features are potentially more noisy than pairwise features? 
- The ablation on non-isometric shapes in Appendix E.2 shows that 2-Hop algorithm is as good as 2D-GEM. I found this disturbing as this effectively invalid the contributions of 2D-GEM. Similarly, in near-isometric case, it seems that LMD is the most important module for the performance gain.   
limitations: - Lack of theoretical support for good performance for exact shape matching. 
- The proposed method borrows various techniques from previous methods in order to tackle both isometric and non-isometric case. It achieves the state of the art performance, but it is clear that these performance gain are not due to the 2D Convolution.
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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: This paper proposes a new shape matching method called 2D-GEM, based on 2D graph convolution. The method is straight forward: at each iteration, (1) first find a set of reliable correspondences with small enough local smoothness error (called LMD); (2) use these reliable correspondences to compute/update a k-by-k graph spectrum; (3) use the updated graph spectrum to compute the correspondences for poorly matched points. 

The proposed method achieves the state-of-the-art performance on shape matching on SHREC'16 TOPKIDS dataset and original FAUST/TOSCA dataset; and on graph matching on Erdos Regny random graphs.
strengths_and_weaknesses: *Clarity*: 
- (++) the paper is well written and easy to follow. The provided code also helps to better understand the algorithm. 

*Originality*
- (++) The proposed method 2D-GEM is a generalization of the GRAMPA method: the mesh Laplacian is used instead of the graph adjacency; it can also incorporate per-vertex features; a different filter is used.  
- (--) comparison to GRAMPA: though the differences between 2D-GEM and GRAMPA are briefly discussed in Appendix B.3., the novelty seems to be limited. For example, though 2D-GEM is generalized to handle per-vertex features, in the real algorithm, no features are used. Specifically, as described in L205, the correspondence P is used as graph signals (instead of the pre-built descriptors XaXb). It sounds like the newly introduced feature is useless. Also according to the ablation study, the most important or performance-dominant part of the algorithm is the 2-hop initialization, which is proposed by Yu et. al. [77].

*Quality*
- (++) 2D-GEM achieves higher accuracy on TOPKIDS/FAUST/TOSCA with initializations computed from SHOT descriptors. 
- (++) 2D-GEM is faster than GRAMPA since in each iteration not all vertices are updated, which leads to less and less number of non-anchor points for linear assignment
- (--) It has been known that the SHOT descriptors can overfit to the uniform triangulation as used in the original FAUST/SCAPE dataset (For example, see discussion in "DiffusionNet: Discretization Agnostic Learning on Surfaces"). A more convincing justification would be testing the algorithm (and all the baselines) on remeshed or resampled dataset with different types of initializations (as done in ZoomOut [49]). 



questions: - As mentioned above, what's the performance of the proposed method on remeshed FAUST/SCAPE with SHOT initialization and other types of initializations (e.g., wave kernel signatures). 
- What's the key difference between 2D-GEM and "Fast Sinkhorn Filters": it seems both methods formulate the matching problem as a linear assignment in the spectral space. (of course, 2D-GEM uses a specially chosen filter and only update the correspondences between non-anchor points) it would be nice to know the critical differences in formulation and it might be better to include such a discussion, since: (1) the  2D graph spectrum computed in Eq.(9) seems to be equivalent to computing a functional map using the anchor points correspondences; (2) the correspondences updating in Eq. (10) seems similar to the linear assignment based on the computed functional map formulated in the Fast Sinkhorn Filters. 
- How are the thresholds in L272 chosen? The scales of the thresholds used for different datasets are very different. Is the algorithm sensitive to thresholds?
- What are the heuristics to choose the number of LBO? Why it is enough to use 10 LBO for TOPKIDS, but 500 LBO for FAUST/SCAPE? Does the algorithm work for 10 LBO for FAUST/SCAPE dataset? Also why 500 LBO is used for the ablation study on TOPKIDS while 10 LBO is used for generating results?
limitations: The limitations are adequately addressed. However, it might be worth mentioning the parameter tuning as well since it seems non-trivial.
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final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposes a novel method for exact shape correspondence based on 2D graph convolution. The main idea is to adapt graph convolution and signal denoising on graphs to the task of shape matching. The proposed pipeline exploits a specific filter inspired by isomorphic graph matching to achieve this goal. The matching results both in the case of near-isometric and non-isometric shapes are accurate. The timing is comparable to and generally faster than the most similar competitors. Furthermore, it reaches a more significant percentage of points for which the geodesic error is 0 compared to alternative approaches.
strengths_and_weaknesses: Strengths:
- The connection between 2D graph convolution and shape matching is interesting.
- The paper provides all the necessary background to understand the proposed idea.
- The performance on the datasets involved in the evaluation is good, and the proposed pipeline improves the state-of-the-art of the competitors considered without training and ground truth data.

Weaknesses:
- Section 4 is not easy to follow. An image of the pipeline with a visualization of the different steps can help improve the text's clarity.
- I am not sure about the novelty of the 2D graph convolution as a filter for shape matching. Section 3 should start with a more explicit and precise statement about the originality. Furthermore, a more comprehensive discussion about the connection with related works is missing in the current version. The paper can benefit from a specific paragraph on this discussion that highlights the novelty of the proposed method but in comparison to existing techniques.
- The pipeline proposed in Section 4 seems a bit incremental. It is given by SHOT+GMWM+[LMD]+refinement. The last part (the last step of the procedure from Algorithm 1) appears to be the main contribution of this paper. If I am not missing something, the first part of the complete pipeline can be composed of existing methods with some modifications on the choice of the parameters to target different settings. This discussion does not invalidate the submission, but the current version of the manuscript should be modified accordingly if correct. If I am wrong, the current version requires a more detailed motivation on the selected solution for step 1,2,3,4 in the Algorithm to clarify why all the procedure is a valid contribution from this work.

Some minor points:
- In line 113, adding a reference to the normalized symmetric Laplacian at the beginning of section 2 can make the paper self-contained.
- In line 167, \mu_i should be \mu_j.
- In Figure 4, fewer values on the x and y axes can make the figure clearer.
- Around line 178, a description of e should be given. From the caption of Figure 4 and section 5, it is possible to deduce that it is a real number (positive? integer?), but it could help to have it explicitly defined in the text.
- Reference [53] contains a typo: Rodolì should be Rodolà.
- The text in Figure 2 is too small. The quality of this figure should be improved by using a tikz file or other alternatives. Same for Figures 3 and 5. 
questions: 1 - Why, in line 164, is the text referring to z_{i,j} while the discussion is on a single graph and its filter g? It should be z_i, or am I missing anything?

2 - Is Figure 4 (c) generated on two isomorphic graphs? From the shape of the filter, it seems to be the case. If it is the case, it should be reported at least in the caption because this filter is dependent on the graphs (the spectra).

3 - The functional map framework [51] proposes a constraint for shape matching (commutativity with the Laplacian) that corresponds to a mask at least similar to the one suggested as a spectral filter in this paper. The recent work "Structured Regularization of Functional Map Computations" by J. Ren et al. (SGP 2019) provides more details on this constraint. Can be any further connection analyzed?

4 - What is the quality of the initialization provided by the proposed pipeline? Adding the evaluation of the initialization (both quantitatively to the curves and qualitatively to the images) will provide a general idea of its quality and highlight the contribution of the proposed pipeline.

5 - Selecting the threshold equal to 100 in the LMD for non-isometric pairs is similar to deactivating this step in the procedure. Is it correct? Or, with this threshold, is the LMD procedure removing any (probably few) matches with a huge error? This last claim seems confirmed by the ablation study in Figure 7 case (3), but this discussion should appear in the main paper.
limitations: I do not see any potential negative societal impact. The limitations listed by the authors give a complete analysis of the open problems in the proposed approach. One possible missing scenario is the one in which the two shapes have a completely different discretization. SHREC19 "Matching Humans with Different Connectivity" is a possible benchmark to evaluate the performance of the proposed method in this specific case.
Moreover, some more challenging settings can be discussed, such as partialities, point clouds, and noisy data from real acquisitions. In these cases, where geodesic constraints
review_title: null

Meta-review: metareview: All the reviewers agreed that the paper is a nice addition to the shape matching literature. While its contributions are a bit incremental, they highlighted the quality of the evaluation, and the introduction of ML methods in the graphics community should stimulate follow-up works. 
confidence: Certain
Decision: Accept