Wormhole Loss for Partial Shape Matching

Summary

The paper proposes an unsupervised optimization for partial shape matching between (near-)isometric shapes, relying on distance similarity. Defined as the "Consistent Pairs" a pair of points that present a similar geodesic distance between the full and the partial shapes, this work relaxes this definition by assuming that the path on the partial shapes can traverse holes with straight lines. The paper shows results on CUTS and HOLES on par with the state-of-the-art and a slight improvement on PFAUST-M and PFAUST-H.

Strengths

- The paper addresses a relevant and challenging problem: solving for partial shape correspondence in an unsupervised setting is a difficult task, and shape matching community is active on this - The method shows interesting applicability in MDS embeddings, although the paper does not evaluate this aspect numerically.

Weaknesses

- The proposed principle seems sound but is also restricted to simple cases. The method shows better results on HOLES, where the assumption holds, while for CUTS, it seems to suffer. My intuition is that CUTS contains a large missing part, and this cannot be well approximated by straight lines on the boundary. - Some aspects of the presentation are not satisfactory. I appreciate the effort in grounding the formulation in a more systematic methodological explanation, but at the moment, it seemed cluttered. For example, I would say that Theorem 1 and its demonstration are unnecessary. Visualizations are not good for black-and-white printing. Figure 6 misses a Ground Truth to assess the quality of the final result. - The paper and its limitations do not offer a discussion nor analysis on the limit of the proposed strategy in terms of failure cases. A more in-depth analysis of the impact of partiality magnitude and its nature on the capability of finding guaranteed pairs would be beneficial and insightful for future work.

Questions

1) The paper analyzes the cases in which the partial shape is still a single component. How does the method perform in the presence of disconnected components? 2) How computationally expensive is the method?

Rating

5

Confidence

4

Soundness

3

Presentation

2

Contribution

2

Limitations

The authors describe some limitations regarding future works but do not discuss failure cases of the proposed strategy.

Summary

The paper investigates the problem of shape matching in an unsupervised and partial setting. In particular, the paper extends an existing criterion for filtering out potentially inconsistent point pairs. This is done by simply including extrinsic information that bounds from below the distances between pairs of boundary points, resulting in a new criterion that preserves more consistent pairs. The experiments are conducted for applications including multi-dimensional scaling and partial shape matching, in which the advantage of the proposed loss is demonstrated.

Strengths

The paper is in general well-motivated, and the idea is straightforward to follow. Also, the presentation is clear with sufficient figures to help understanding. the experiments include different applications.

Weaknesses

In general, the paper is well-written, explores a simple yet effective idea, and proves its effectiveness via experiments. However, the main weakness of this paper comes from the lack of theoretical depth. The proposed method is a modification of an existing idea, whereas new insights are limited. This is also discussed in Limitations of the paper, there are more consistent pairs that could be recovered, yet we have no idea how large the gap is. That being said, the contribution of the paper is still clear.

Questions

I have no particular critical question. But I am curious, how large the gap is between the preserved consistent pairs and all possibly existing consistent pairs?

Rating

6

Confidence

5

Soundness

3

Presentation

3

Contribution

2

Limitations

The limitations have been adequately discussed in the paper.

Summary

The paper presents a novel criterion for identifying consistent pairs based on geodesic distances between points on partial and full surfaces. This new loss function utilizes intrinsic geodesic distances, extrinsic distances between boundary points, and a consistency criteria to enhance the performance of partial shape matching. The authors validate the improvements in the loss function both theoretically and through experimental results.

Strengths

- The focus on partial shape matching and unsupervised shape matching for partial to full shapes is intriguing. - Both quantitative and qualitative evaluations demonstrate significant improvements over the state of the art.

Weaknesses

- The method fully relies on the Euclidean embedding space for boundary distances, which might not always reflect true geodesic distances on the surface, especially in highly non-Euclidean or complex topologies. -The authors could enhance evaluations by using other correspondence techniques, instead of only using DiffusionNet features, and demonstrate broader generalization capabilities with other signatures.

Questions

- How does the criterion perform on significantly larger datasets with more complex topologies? It would be interesting to integrate a control measure to analyze the effect of filtering inconsistent pairs. - Regularizing the binary mask into a soft mask M* involves a trade-off between retaining slightly noisy information and maintaining the integrity of the matching process. How is the threshold for the soft mask determined, and how sensitive is the model's performance to this threshold? - Are there specific scenarios where this regularization might fail?

Rating

6

Confidence

4

Soundness

3

Presentation

3

Contribution

2

Limitations

The method might fail in the presence of topological variations, leading to incorrect correspondences if the geodesic paths on the partial surface are substantially different from those on the full surface.

Summary

This paper introduces a refined criterion for detecting pairs of points whose geodesic distance on a partial surface is equal to that on the full surface. The "wormhole" criterion is sound and less conservative than previous such criteria. The refined criterion improves results in planar embedding with multidimensional scaling (MDS) as well as partial-to-whole shape matching.

Strengths

This paper is clear and well-written. It is focused on one core contribution, the wormhole criterion for guaranteed pairs of points. And the results on both MDS and partial matching validate the utility of the refined criterion.

Weaknesses

### Miscellaneous - This is confusing: "This condition naturally follows from the fact that the shortest path in the full surface $\mathcal{Y}$ is shorter than any path between the points passing through the boundary $\mathcal{B}$, and in particular, ones that pass through the closest boundary points" (137-139). I think what you mean is either that (1) in general, the length of a path between the points in the full surface that passes through the boundary is at least as long as the sum of their distances to the boundary; or that (2) for pairs of points that satisfy the condition $\mathcal{C}_{\mathcal{T}}$, the shortest path in the full surface is *no shorter* than the path in the partial surface. - $\mathbf{K}$ is only mentioned in passing on line 170 before being used in equation (6). It might be helpful to introduce more explicitly what you mean by the "threshold" matrix, though I think it is inferable from context. - Line 21: "Consequentially" should be "Consequently"

Questions

- You say you could "easily generalize the discussion" to the case of both surfaces being partial (103). Would this just amount to using the intersection of the criteria on both sides? - The boundary to which distances are measured is viewed as the boundary of $\mathcal{Y}'$ excluding the boundary of $\mathcal{Y}$. Given a partial surface, how do you know which parts of its boundary are part of the boundary of the full surface? - Why is the loss function scaled by the vertex area twice in equation (5)? It might be helpful to state this as an integral in the smooth setting before discretizing.

Rating

7

Confidence

4

Soundness

4

Presentation

3

Contribution

3

Limitations

N/A

Post-Rebuttal

I thank the authors for their clarifications and discussion. I appreciate their insights on CUTS, and I see that for the considered algorithm, it is probably less challenging than HOLES since it should generally preserve more of the geodesic distances of the remaining part. Concerning the Failure cases, I still find the discussion a bit confusing. Since "we do not claim that our regularized approach is always guaranteed to succeed," I would be curious to know more precisely which cases do not succeed. I see the complexity has been analyzed in Appendix B.2; it would also be interesting to know the wall-clock time of an inference. What is the required inference time for evaluating the HOLES test set compared to the DirectMatch baseline?

Responce to rebuttal

Dear authors, Thank you for submitting your rebuttal and including the additional experiment. I have reviewed the rebuttal and found that my questions have been addressed. 1VSx