Wormhole Loss for Partial Shape Matching
Authors: Amit Bracha
Thomas Dagès
Ron Kimmel
Venue: NeurIPS 2024 poster (2024)
Keywords: Partial surfaces
Shape correspondence
Geodesic distances
Primary area: machine_vision

Abstract: When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of whether distances measured between pairs of surface points should influence the matching process. The approach we propose treats surfaces as manifolds equipped with geodesic distances, and addresses the partial shape matching challenge by introducing a novel criterion to meticulously search for consistent distances between pairs of points. The new criterion explores the relation between intrinsic geodesic distances between the points, geodesic distances between the points and surface boundaries, and extrinsic distances between boundary points measured in the embedding space. It is shown to be less restrictive compared to previous measures and achieves state-of-the-art results when used as a loss function in training networks for partial shape matching.

Wormhole Loss for Partial Shape Matching
Amit Bracha' Thomas Dagès"
Ron Kimmel
Technion Israel Institute Technologg
Haifa, Israel
famit. bracha, thomas dages)@cs. technion.ac.
Abstract
When matching parts of surface to its whole, fundamental question arises: Which points should be included in the matching process? The issue intensified when using isometry measure similarity, as it requires the validation of whether distances measured between pairs surface points should influence the matching process. The approach propose treats surfaces manifolds equipped with geodesic distances, and addresses the partial shape matching challenge by intro- ducing novel criterion meticulously search for consistent distances between pairs of points. The new criterion explores the relation between intrinsic geodesic distances between the points, geodesic distances between the points and surface boundaries, and extrinsic distances between boundary points measured in the em- bedding space. shown be less restrictive compared to previous measures and achieves state-of-the-arr results when used as loss function in training networks for partial shape matching.
Introduction
Shape correspondence is a core challenge in computer graphics and computer vision, distinguished by its wide array applications extending from 3D modeling and animation to object recognition and beyond. The task involves establishing mappings between corresponding points across different surfaces that undergo non-rigid transformations. Shape correspondence grows particularly intricate when it involves partial surfaces, where the objective is to find correspondences between parts of surfaces or between a complete surface and a part of another. This sub-task is particularly difficult as it involves incomplete data and possible different topology of the matched surfaces. Consequently, training neural networks for the partial matching task is challenging, especially for unsupervised methods, there is no information about which parts missing and which should be matched.
The state the art in partial shape matching is performed with Functional Maps (FM), a popular framework for relating functions defined on surfaces via basis functions defined on the surfaces [2, 18, 22, 25, 36, 45, 34]. Recently, it was argued that the unused information from the full surface impairs the learning process when FM is incorporated in the learning pipeline [10], with better results obtained when bypassing FM altogether by direct estimation of correspondences. This method is primarily guided by loss function preserving the pairwise geodesic distances [25], which relies the fact that such distances are preserved under isometric transformations. When dealing with partial surfaces, not all distances are preserved, minimal geodesics the full surface may go through a region which is missing in the partial surface. Therefore, undesired biases are integrated into the learning procedure, which impair the quality of the learned correspondence. This applies to both FM-based unsupervised loss functions [48] and geodesic-based ones [25], where isometry between the surfaces assumed.
* Equal contribution
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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In this paper, we present a novel loss for partial shape correspondence that takes into account partial surfaces. It is constructed upon the notion of consistent pairs of points, where a pair said to be consistent if the geodesic distance between the points is the same on the partial and full surface. Our task to define a criterion that captures as much as possible consistent pairs, where these pairs are referred to as guaranteed pairs. Using such a criterion, we can limit our matching procedures to consider only guaranteed pairs, thereby avoiding distortions. Filtering out potentially inconsistent distances between pairs of points was first studied in [13, 12, 14, 46, 47J. In particular, in [12, 46] the authors showed that for non-Euclidean spaces the geodesic distance between two points on a manifold is preserved when this distance is smaller than the sum of the surface distances of each point to the boundary of the surface. We show that this criterion is too conservative and it filters out substantial number of consistent pairs of points. It can be improved by using the extrinsic information encapsulated in the manifold's embedding space. The new condition can be shown to provide significantly more guaranteed pairs of points.
Using this novel criterion to find consistent pairs of points, we present a new loss function for unsupervised shape correspondence tailored for partial surfaces. It is based on the ratio between the geodesic pairwise distance, that is calculated on the partial surface, and a measure defined by our criterion, which involves distances to points on the boundaries and the distance in the embedding space between the corresponding boundary points. The new loss was used for training an unsupervised shape correspondence neural network achieving faate-of-the-ass (SOTA) results the reference SHREC' 16 CUTS and HOLES datasets [20] and on the recent PFAUST dataset [10]. Our code can be found at https: //github. com/ ABracha/ SorhhaaWoomooe.....
Contributions
We introduce novel criterion for identifying consistent pairs, which pairs of points between which the geodesic distance is the same for the full and the partial surfaces.
Using this criterion, we create a novel unsupervised loss function tailored specifically for partial surfaces, achieving SOTA results on partial shape correspondence benchmarks.
2
Related Efforts
Early attempts to solve shape correspondence involved extracting hand-crafted features for each point [4, 55]. Leveraging the intrinsic nature of the Laplace- Beltrami operator (LBO), its eigenfunctions were often employed to define such features on the manifold, invariant to non-rigid isometry, thereby, removing the dependency the manifold's embedding in Euclidean space. Later, the Functional- Map (FM) framework was proposed [40]. It builds on the fact that the mapping of consistent feature representations in local bases is linear, even for complex non-rigid spatial deformations of the manifolds. In fact, the first neural network for solving shape correspondence was based on the FM framework [33], which introduced differentiable FM-layer to learn the correspondence map.
More recent methods suggested two alternative approaches for unsupervised shape matching. The first is based on the fact that isometries preserve geodesic distances [25], and the second is based on properties satisfied by the FM operator for isometric transformations [48]. Improved architectures followed [22, 53, 31, 54]. Alternatiee loss functions were suggested, such as supervised contrastive loss [30], or using the objective function from [2], which penalizes the difference between the FM output from the FM-layer and the FM estimated from the point-to to-point mapping
[18]. Other papers explored different metrics, such as the scale invariant metric [11, 11, 41], or anisotropic Riemannian metrics using the Finsler-based LBO [61].
Several efforts were made to tackle surface matching under partiality. Some approaches are based on a search strategy to find either the missing parts [7, 6, 43], or direct correspondence by exhaustive search [24]. Few methods learn to find the correspondence, one such example learned linear invariant bases suited for partial surfaces rather than the classical LBO eigenspaces [36]. Another supervised learning approach applied an attention layer [59] on extracted features, allowing feature interaction between the full and partial surfaces prior to the FM-layer [3]. An unsupervised learning method, based on preserving intrinsic distances, avoided the use of an FM-layer altogether [10]. The motivation was that FM introduces noise when used to match the whole a part of surface.
When dealing with partial surfaces, one needs to consider the fact that distances can be significantly different when measured between corresponding points in the partial and the whole surface. Thus,

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encouraging distance preservation between corresponding points can introduce undesired biases on the learned matching. Although [10] reported the current SOTA, it only tackles part of the problem. Here, we introduce novel loss function based on consistent pairs of points and geodesic distances, greatly reducing the effect of inconsistent distances on the learned correspondence.
To construct our loss function, we first revisit the notion of consistent pairs. The link between geodesic distances and distances to the boundaries of flat and non-flat surfaces was studied in [12, 14, 46, 47J. Validity conditions were suggested to predict whether distances between matching points are consistent between the partial and the full surface. Here, we revisit and refine these conditions, and introduce a more inclusive criterion that guarantees more consistent pairs by which the matching results improve.
3 On the Consistency of Distances between Pairs of Points in Partial Surfaces
3.1 Consistent and Guaranteed Pairs of Points
Px
dx(Px, 11
dy Rx) th dy Qy)
dy(P),
Qx
P
Whole X
Whole y
Partial y
Figure 1: For a distance preserving map between full surfaces X' and V also known as isometry. the minimal geodesics in the partial version y" may not correspond to those in the full surfaces y. That is, the geodesic distances between corresponding points may get larger.
In this section, we present the concept of consistent and guaranteed pairs of points.
Let X and y be isometric surfaces with bijective mapping p x Y. In partial shape matching. the surfaces may have missing parts, meaning that we only have access to X' C X and )" C y. For simplicity, we assume that we have access to the full surface X' but not the full surface y and so X" = X and )" * Y. We can easily generalise the discussion if X' + X. Consider two points Py and Qy on the partial surface y" and their corresponding points Px p p='(P) (Py) and Qx = (Q) on X. As the mapping p is isometric, the geodesic distances, reflecting the length of the shortest paths, on the full surfaces dx and dy are equal by definition, dx( Pxx, Qx) may be that the shortest path on the full surface ) between Py and Qy passes through missing part of )", such a case, computing the geodesic path between these two points could lead to longer path than the one on the full surface, implying that dy(Py. Py. Qy) * (Py: Qy) and that, in turn, dx (Px, Qx) * dy (Py, Qy). even if the original mapping P is isometric (see Figure 1). As such, methods relying transformation p are inappropriate when dealing with partial surfaces. They could incorrectly enforce equal distances when they should not. Although not all geodesic distances preserved for partial surfaces, many nevertheless are preserved, leading to the definition of consistent pairs.
dy Py, Qy). However,
enforcing the preservation of geodesic distances based on the isometry of the
Definition (Consistent Pair of Points). pair of points Py and Qy of a partial surface y" is said to be consistent if the geodesic distance on the partial surface is the same as that on the original full surface, i.e. (Py. Qy) dy (Py. Qy).
As there are many consistent pairs even in cases of cuts and partiality, methods enforcing distance preservation between all pairs, even inconsistent ones, perform fairly well. However, penalizing distance dissimilarity for inconsistent pairs is undesired and decreases the quality of the resulting mapping. Our goal is to filter the set of pairs by finding as many as possible consistent pairs, and then relying only these guaranteed pairs when matching surfaces.
3

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Definition (Guaranteed Pair of Points). A pair of points Py and Qy of partial surface )" is said be guaranteed with respect criterion C y y {0, 1}. C-guaranteed in short, the criterion proves that the pair is consistent.
Note, that the criterion is unsupervised as no oracle provides knowledge from the full surface y to compute it. The better the criterion, the more consistent pairs it finds, allowing for more consistent information be used for finding the matching, see Figure In the rest of this section, we focus on the search for consistent pairs in the partial surface )" as a preprocessing step and will return to the surface X when we perform partial shape matching.
Cuguaranteed pairs
CT guaranteed pairs
Consistent pairs
All pairs
Figure 2: Venn diagrams showing the relation between all pairs of points, consistent and guaranteed pairs for a surface with boundaries. All guaranteed pairs are consistent. Our criterion Cw is more inclusive than that of [12, 46], Cr. All guaranteed pairs by Cy are also guaranteed by Cw.
3.2 Expanding the Set of Guaranteed Pairs via Extrinsic Distances between Boundary Points
We introduce a criterion that finds as many consistent pairs as possible, which we then denote as guaranteed. Denote B C y" the boundary of y" For simplicity, treat all boundaries equally. In [12, 46], the criterion, Cr, was designed to reject pairs of points, Py and Qy on )" for which the sum of their distances on the boundary B is less than the geodesic distance between them,
[1 if (Py, Qy) S dy (Py, + dyr B)
Cr(Py,
to otherwise,
where is the minimal length of the minimal geodesic of Py to any boundary point,
dy (Py B) = min dy (Py, B).
BEB
This condition naturally follows from the fact that the length of a path between the points in the full surface y that passes through the boundary at least as long as the sum their distances to the boundary.
Note, that this criterion overly conservative and removes many consistent pairs. practically ignores the length of possible trajectories connecting the boundary points, see Figure 3. To mitigate this condition, we propose to include extrinsic information that bounds from below distances between pairs of boundary points. Note that when solving intrinsic problems, the practice of combining intrinsic and extrinsic information uncommon, yet has been shown to beneficial for other types of robustness [16]. For simplicity, assume from now that the manifold is embedded in Euclidean space. We then readily use the following straightforward relation between geodesic distances and Euclidean ones.
Theorem 1 (Euclidean bound). The geodesic distance between any points P and Q on a surface y E E, for some embedding Euclidean space R., larger than or equal to the Euclidean distance between the points measured in the embedding space.
dy(P de(P,Q),
where di (P,Q)) - P Q1
For proof see Appendix A. We propose to consider trajectories passing through any pairs of boundary points Bi and B2 in B. By acting like a wormhole connecting two boundary points, the length of the straight line in connecting B, and B2 can serve as lower bound of the geodesic distance on the full surface between the points. Geodesic distance measures the distance of P and Q

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to boundary points B1 and Bz, whereas the Euclidean distance dg (B,, B2) bounds from below the geodesic distance the full surface between B₁ and Bz. Our wormhole criterion Ey is given by
if dy (Py min dy (Py, dy (Qy, B2) dg (B2, B2), B.BB
Cw(Py,Qy) (
otherwise.
(3)
By design, this criterion can be used find consistent pairs.
Theorem (Cw guarantees). The wormhole criterion Cw yields guaranteed pairs.
Inconsistent pair
Consistent pairs
Consistent pairs
CT, Cw rejected
CT rejected
Cw guaranteed
Figure 3: Example of inconsistent and consistent pairs of points. The minimal geodesic paths are colored red, while paths to the boundary points are colored blue. Different boundary points are selected in CT [12, 46] and Cyv. The Euclidean lines connecting the boundary points selected by Cw are dashed blue. Both criteria correctly reject inconsistent pairs (left). Since CT ignores distance between boundary points, it discards many consistent pairs (middle). Criterion Cw finds more consistent pairs by including the extrinsic Euclidean distance between boundary points (right).
Figure 4: We plot in green the of points that together with the blue point satisfy CT [12, 46] (left) and Cw (right). The blue point and any of the green ones form guaranteed pairs. The cat surface is taken from the SHREC' 16 HOLES dataset [20].
See Appendix A.2 for proof. The new criterion for finding consistent pairs, generalizes the best known criterion CT as all Cr-guaranteed pairs are also Cy--uaranteed see, c.g. Figure 4.
So far, we implicitly assumed that the metric on the manifold is the one induced from the Euclidean metric of the embedding space, meaning that curves have the same length from the perspective of the manifold and the embedding space. We can generalize the wormhole criterion to handle general Riemannian metrics the manifold, by modulating the Euclidean distance of the straight line between boundary points by the minimal scaling relation between arclength measured by the embedding metric restricted to the manifold and one measured by Riemannian metric of the embedded manifold. We present our generalised metric-sensitive criterion in Appendix A.3
The definition consistent pairs has been in a continuous setting. We need adapt it to the discrete world. We describe in Appendix B.2 how to do so. In short, we only consider points on the surface or its boundary to belong finite sampled threshold and binary mask matrices K and M of size V x [V), containing at entry ij respectively the threshold of the criterion Cw and the binary criterion value Cw of the ij pair of vertices,
V of vertices the continuous surface. We construct the
Kij min dy (v; B1) + dy (ey, B2) + de (B1, B2) B,B
(4)
5

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and
Mj L(D
(5)
where the indicator function and (Dy) )is is the computed geodesic distance on the partial surface )" between eú and vz-
To evaluate the quality our criterion Cy, we searched empirically for consistent pairs using either our criterion Cy on the PFAUST-M and PFAUST-H [10] datasets, the latter comprising of shapes with more holes than the former. We provide in Table the percentage of consistent pairs, and among those the percentage of guaranteed pairs by the different criteria, along with the standard deviations. Our criterion Cy able to recover most pairs and it guarantees twice as many as the previous criterion Cr.
Table I: Empirical quantitative evaluation of the ability to recover consistent pairs. The numbers represent either the average percentage of consistent pairs out of the total number of pairs of points in discrete shape, or the average percentage of guaranteed pairs by criteria CT and Cw out of the number of consistent pairs. In parenthesis provide the standard deviation across different shapes. Averages and standard deviations are computed within the PFAUST-M or PFAUST-H [10] datasets, the latter possessing shapes with more holes.
Dataset
&Consistent
GGuaratteed (Cy) [12, 46]
%Guaranteed (Cw) (Ours)
PFAUST-M
78 (#16)
48 (+1
82 14)
PFAUST-H
53 (+16)
30 (+18
65
(=1
4 Applications
4.1
Multi-Dimensional Scaling
In this paper, we concentrate on partial shape matching. Additionally, we demonstrate that the proposed criterion for identifying consistent pairs can be beneficial for other tasks. Specifically, we show how it can be incorporated into multidimensional scaling (MDS) pipeline. The MDS goal is to embed curved manifolds into low dimensional Euclidean space " for some small m. such that the pairwise Euclidean distances of the new embedding are as close possible to the original geodesic distances between corresponding pairs of points. To obtain embeddings robust to missing parts, we handle boundary conditions on partial surfaces by relying consistent distances of guaranteed pairs. In [46, 47], the TCIE method minimises a masked distance preservation objective, with mask weights Wij Cy P3). Our MDS method, named wormhole constrained isometric embedding (WHCIE), adapts this approach by replacing the mask weights with our criterion Wij = Cw P, Pÿ) Mij-
We plot the resulting WHCIE planar embedding of toy surfaces in Figure 5. For each Swiss roll, we removed either a hole or full rectangular cut connected to the boundary. We compare with several reference manifold learning techniques: Isomap [57], MLLE [62], Laplacian eigenmaps [5], UMAP [37], and TCIE [46, 47J. Methods that do not check for distance consistency between pairs of points yield distorted embeddings. The wormhole criterion finds significantly more consistent pairs compared to the TCIE, resulting in an embedding similar to that of the actual surface. That is, a flat rectangle with hole or cut. For more details and comparison with related methods, see Appendix B.I.
4.2 Partial Shape Matching
We next return to our primary application matching shapes. To that end, we introduce the wormhole loss, a method based on guaranteed pairs specifically designed for partial shape matching.
4.2.1 Wormhole Loss: Unsupervised Partial Shape Matching using Consistent Pairs
Our mask M and threshold K matrices operate on minimal geodesic distances. A relevant unsuper- vised loss function using such distances for near isometric surfaces X' and is [25].
PDxP - -Dy D
Egen (P,Dx, Dy) = 1
(6)

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Laplacian Eigenmaps
WHCIE (Ours)
Input
Isomap
MLLE
UMAP
TCIE
Whole
Partial
Partial
Figure 5: MDS embedding of various manifold learning methods on whole Swiss roll (top) and on two versions of it with Gaussian noise, having either a rectangular hole (middle) or rectangular (bottom). Other methods are referred to in Figures 7 and 8 in Appendix B.1.
where I| j is the Frobenius norm, P is the estimated point-to-poinn soft correspondence matrix, Dx the geodesic distance matrix on surface X, and [ is the area of surface y. The stochastic matrix P of size |Vyl Vx takes values in P.; E [0, 1], and is wwwnoooalizedd such that Σ, P.j 1. This continuous setting accounts for possible different samplings of the surfaces, as the matched part in X' of each sampled vertex in V/, may not have corresponding sample point in Vx. If P is estimated to be the correct correspondence matrix, then, the distances are preserved and the loss is minimal. We incorporate mask M into this loss function for partial surfaces, ) as follows,
Lageo! (eesoPP.A Dy, = Σ My (PDx P - DD
where P, now a matrix of size [Vxl x |V5= is full-to-partial correspondence matrix, and Ay is diagonal matrix of the vertex areas of shape )" Thus, PDxP of size [Vyn X V/S maps the distances between pairs of vertices in X' to distances between pairs of vertices in )" The mask matrix M filters out inconsistent pairs, allowing the loss to minimized with distances transferred from the full surface matched with those on the partial surface. This loss generalises the one proposed in [10] by incorporating masking weights. We provide full details on how to derive this loss in the supplementary material Appendix B.3.
In our experiments, we found that the binary criterion can be relaxed better use the interaction between pairs. Many inconsistent pairs have distances only slightly increased by the presence of holes, thus, filtering them out removes valuable, though slightly noisy, information. We propose to regularize the binary mask matrix into a soft mask M as follows.
Kij
M = min
(8)
(Dy)
The weights of the soft mask matrix inhibit the contribution the loss of non-guaranteed pairs whose distance is significantly larger than the theoretical threshold of our criterion Cw- By the same token, it preserves the influence of distances of non- con-guaranteed pairs which near the threshold.
4.2.2 Implementation Considerations
We evaluate the new geometric- wormhole [10], replacing their primary loss component, first introduced in [25]. Our pipeline takes raw input
scometric-wormhole loss, Lges by integrating it into a pipeline adapted from
7

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features and then refines them independently for the surfaces using shared-weight neural network, DiffusionNet [54]. These refined features are then matched using Softmax similarity [7, 18] to directly compute a correspondence matrix P. which is then plugged into the geometric-wormhol loss. We keep the regularization loss [48] promoting the preservation of vertex area between corresponding vertices, which is computed with the functional map extracted from the correspondence matrix. We provide further details of the proposed pipeline and implementation details in the supplementary material Appendix B.4.
4.2.3 Evaluation
Table 2:
Quantitative analysis SHREC' 16 [20]. The numbers represent the average geodesic error (multiplied by 100) of the results following post-processing refinement. The best performance is highlighted in bold, and the second best is underlined. These results underscore the proposed method's 's dominance in unsupervised shape matching. It surpasses both supervised, unsupervised, and pretrained methods on the HOLES benchmark, known for its highly challenging scenarii.
Test-sct
CUTS
HOLES
Training-sct
CUTS
HOLES
CUTS
HOLES
Axiomatic methods
PFM [45] Zoomout FSP [34] + Zoomout
9.7 9.0
23.2 + 22.4
16.1 + 15.2
33.7 + 32.7
Supervised methods
GeomFMaps [22] Zoomout
12.8 10.4
19.8
16.7
20.6
17.4
15.3
13.0
DPFM [3] Zoomout
3.2 - 1.8
8.6 >> 7.4
15.8 -> 13.9
13.1 - 11.9
Pretrained using external datasets
RobustFMnet with Refinement [18]
3.2
5.6
13.5
8,2
Unsupervised methods
Unsupervised-DPPe DPFM [3] + Zoomout
11.8 + 12.8
19.5 - 18.7
19.1 + 18.3
17.5
16.2
RobustFMnet [18] - Refinement
16.9 10.6
22.7-+16.6
18.7 + 16.2
23.5 + 18.8
DireetMatehNNtt [10] - Refinement
6.9 + 5.6
12.2 -.2-+8.0
14.2 + 10.2
11.4
+ 7.9
DirectMatchNet LPF [10] -> Refinement
7.1 + 4.7
8.6 - 5.5
16.4 - 11.6
12.3
- 8.6
Wormhole (Ours) Refinement
6.9 > 4.3
10.8
7.2
17.1
10.9
10.9
6.6
Datasets. We evaluate method on the benchmarks SHREC'
SHREE [20] and PFAUST [10]. SHREC' includes two datasets, CUTS and HOLES. Both datasets contain processed figures from the TOSCA [15] dataset. In CUTS, the figures were cut using several 3D planes. HOLES, besides the cuts made by the planes, additional holes were added, resulting in partial surfaces with more topological changes and longer boundary, which makes this dataset particularly harder for shape correspondence. The second benchmark on which we evaluate our method is PFAÚST. recently created by [10]. It contains figures from the Reme-hee dataset [44] that were processed by creating holes in them. This benchmark is also divided into two datasets; in PFAUST-M, there are bigger but fewer holes, and in the PFAUST-H, there are smaller but more numerous holes. The latter presents harder task for shape correspondence, greater number of holes results in more significant changes in the topology.
Baselines. The baseline methods evaluated fall into three categories: Axiomatic methods PFM [45] and FSP [34], supervised methods - GeomFMaps [22] and DPFM [3], which is the current SOTA for the CUTS and PFAUST datasets, and unsupervised methods unsupervised DPFM [3], RobustFmaps [18], and DirectMatchNet [10]. which is the current SOTA for the HOLES and the unsupervised SOTA on the PFAUST datasets. We had re-implement the loss function for unsupervised DPFM due to source code unavailability. Recently, the authors of RobustFmaps updated their models to include a pretraining phase including four external datasets. For completeness, we include this pretrained version in our results table, albeit we separate it from the other methods not using any external datasets. For fairness, we also provide the results of RobustFFmaps when trained only on HOLES or CUTS without any pretraining, following the common protocol of the benchmarks. For post-processing refinement, we employed either Zoom-out [38] or test time adaptation refinement [18] depending what the original papers used.
8

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Results. The quantitative results shown in Tables 2 and 3 indicate that our method reaches state- of-the-art performance for unsupervised partial shape correspondence. We also display superior qualitative performance in Figure 6 and Appendix B.5, demonstrating our method's robustness to missing parts. Indeed, unlike other methods our mappings are hardly distorted at challenging locations, such as points close to boundaries. A major improvement over other approaches was in the more challenging datasets HOLES, PFAUST-H, and PFAUST-M, whereas in the CUTS dataset a relatively modest improvement is shown. The missing parts in the CUTS dataset were created by slicing figures with 3D planes. In practice, this procedure does not modify much the topology of the surface and it does not introduce many inconsistent pairs. As such, cuts mostly preserve all geodesic distances, thus, our novel loss function was almost identical to the one in Equation (6) [25]. On the more challenging datasets, a larger part of the inter-geodesic distances differ between the partial and full surfaces, enabling our method to demonstrate its strength, even in comparison to supervised methods pretrained models on external datasets. In fact, unsupervised approach demonstrates superior performance to the best supervised approaches on the HOLES and PFAUST-H benchmarks.
Source
RobustI'Mne Miet
DirectMatchNet
Wormhole (Ours)
RobustFMMeet
DirectMatichNet
Wormhole (Ours)
Qualitative results on the test set of PFAUST-H. Our method yields less distortions near boundaries demonstrating its robustness to missing parts.
Figure 6:
Table 3:
Quantitative results PFAUST [10]. The numbers indicate average geodesic error (x (x1)).. Our method surpasses previous unsupervised shape correspondence methods on the medium and hard datasets, and is on par with the supervised method the hard dataset. The largest improvement is on the hard dataset, showing robustness to topological changes due to missing parts of the surface.
PFAUST-M
PFAUST-H
Supervised
DPFM [3]
3.0
6.8
Ciappervised-DPFN [3]
9.3
12.7
RobustiMnet [18]
7.9
12.4
Unsupervised
DirectMatchNet [10]
5.1
7.9
Wormhole (Ours)
4.6
6.7
Ablation study. To evaluate which criterion, ours Cw or the more conservative Cr [12, 46], leads better masks for partial shape correspondence, we compare masking strategies on the HOLES dataset in the following setting: Binary mask from Cy [12, 46), non-binary mask from Cy [12, 46], our binary mask, and our non-binary mask. As can seen in Table 4, learning with masks derived from our wormhole criterion surpass the alternatives for partial shape correspondence, showing the importance of less restrictive criteria find consistent distances between pairs of points.
Conclusion
In this paper, we analyzed invariant properties of geodesic distances between partial and full surfaces. Following we developed an improved criterion for identifying consistent pairs pairs where the geodesic distance remains consistent between the partial and full surfaces. Our improved criterion

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Table 4: Ablation study on the criterion used to derive the mask in the proposed loss. We compare the criterion CT [12, 46] with the new wormhole Cw. Training and testing were conducted on the HOLES dataset. Numbers represent the average geodesic error (x 100).
Mask type
Binary
Non-binary
from
CT [12, 46]
Cw (Ours)
C [12, 46]
Cw (Ours)
HOLES
18.8
15.3
14.7
10.9
finds consistent pairs, or guarantees them, if the distance between two points on the partial surface is, for all couple of boundary points, smaller than the the sum of distances to the boundary points and of the distance in the embedding space between these boundary points, which is given by the straight line for Euclidean embeddings. We used this criterion to tackle the complex challenge of partial shape correspondence. by incorporating our guaranteed pairs novel unsupervised loss function specially designed to handle partial surfaces. With this loss function, our method demonstrated SOTA results on reference partial shape correspondence benchmarks, namely the SHREC' 16 CUTS and HOLES and the recent PFAUST dataset.
Limitations. Our paper presents novel criterion for finding consistent pairs in partial surfaces. However, it is still not able to recover all consistent pairs in general surfaces, meaning that some information that would be useful for partial shape matching is discarded. We suspect there could exist even less restrictive criteria for finding consistent geodesic distances in the presence of holes and cuts. The exploration of such criteria is left for future research. Additionally, finding boundaries in higher dimensions should be addressed with care, and operating in the embedding space in higher dimensions would not be as forgiving what we have done for 2D surfaces in 1e In this work, we focused on partial-to-ful shape matching. Although the wormhole criterion can be used to find consistent pairs on each shape independently, we would also need to find the shared consistent pairs for matching the shapes. Designing full pipeline for partial-to cistaal-t-ppartiii to-partial shape matching with our criterion is left for future work.
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A Further Theoretical Details and Proofs
A.1 Proof of Theorem
Proof. The geodesic curve between the two points a curve (in the embedding space) between the points restricted to the manifold Jr. The length of this curve is at least as long as that of the minimal geodesic (line in the Euclidean case) connecting the points in the embedding space. This trivial result holds since the metric on the manifold is the one induced by the metric of the (Euclidean) embedding space restricted to the embedded manifold.
A.2 Proof of Theorem 2
Proof. Consider the first case where the shortest paths on the full surface y from Py to B₁ and from Qy to B2 do not intersect the boundary B. In this case, we have that the computed geodesic distance on the partial surface to the boundary points is the same as that on the full surface. As such, dy (Pyy, B1) = dy B1) and dy (Qy, B,) = dy(Qy, B2). The shortest path on the full surface passing from py to Qy passing through B1 and B2 will thus have length dy B,) dy(Qy, B2) + dy dy,B, B₂) 2 dy B₂) + dy B2) + dg (B,, according to Theorem 1. Therefore if the pair satisfies the criterion, the geodesic distance computed on the partial surface is the geodesic distance on the full surface, meaning that the pair is consistent.
Consider now the second case where one or both of the shortest paths on the full surface y from Py to B3 and from Qy to B2 intersect the boundary B. Let Bg and B3 be the first intersection points on each of them. Any trajectory between them, such as the shortest path passing from Bo B1 followed by the Euclidean straight line from B1 to B3 and then the shortest path from B2 to B3. are at least as long as the Euclidean straight line between Bo and B3: dE (Bo B3) S dy-(Bo, B1)+ + dy B2,, Bs) + (B, B2). Since Bg and B3 are also closer to the original points, we then have dy (yy, Bo) dy (Qy, Ba) dg Bo. Bs) I dy (Py, B₁ + dy B2) + dg (B1, B2). Since the pair By and Bg satisfy the first case, the criterion will validate the pair Py and Qy only it is consistent.
A.3 Generalisation to Non-Standard Metrics
So far, we implicitly used the fact that we were working with the standard way of computing geodesic distances on surfaces. Formally, this means that we only considered manifolds equipped with the standard uniform isotropic Riemannian metric, where unit steps in the tangent plane have the same length as unit steps in the embedding space. Schematically, the manifold and the embedding space shared the same ruler. This allowed us to bound the Riemannian length of the geodesic curve with the Euclidean length of the straight line in the embedding space between points.
However, in some cases, is natural or beneficial consider other metrics on the manifold. As new metrics change the length concept, this means that same curve on the manifold will have a different length. With other metrics, the manifold and the embedding space no longer share the same ruler, and unit steps on the manifold tangent space are no longer of unit Euclidean length. We must thus adapt our criterion in consequence.
A Riemannian metric on the manifold is defined by a symmetric definite positive matrix M, such that at any point E X, the length of the tangent vector u at point given by a quadratic form (u) u M (x)u. u. The geodesic length of a curve on the surface is given by integrating the Riemannian length of all the infinitesimal tangent steps: f Re (t) where 7(t) parametrises the curve monotonically. The link between the manifold Riemannian length and the embedding Euclidean length of steps is given by the eigenvalues of the metric M. Let μ4 (x) < (2(2)) be its smallest and largest eigenvalues, with eigenvectors en (ar) and vz(x)) of unit Euclidean norm [ali (2) = [|02(x)]l2 2 = 1. Then Euclidean unit steps have length bounded in (x), / /u (x)). Under the assumption that min, /1 (x) CM > O, which systematically occurs for common metrics on natural surfaces, Fuclidean unit steps have at least yCar Riemannian length. As such, reparametrising the curve with the standard Fuclidean arclength, i.e. the arclength for the Riemannian metric with matrix I, we can bound each infinitesimal step along the curve, which has a fixed
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Euclidean length, to get bound on the geodesic length: dar (a, 2 VCMdd (z, y), where das is the geodesic distance on the Riemannian manifold associated the matrix M.
We return to partial surfaces, with y and y" full and partial versions of the same surface. The surfaces are associated with the intrinsic Riemannian metric M. As such, geodesic distances changed to dy = d and are no longer necessarily equal to dz. We assume that CM is known". We can now adapt our wormhole criterion the Riemannian metric M define the new wormhole metric-sensitive criterion
if dy (Py, - min (Py, + dy (Qy, + VCMdE(Br B2). B1,B2EB
(y)
otherwise.
Note, that in the degenerate case CMs = bound path lengths inside missing parts. By design, this generalised criterion to Riemannian metrics can be used to find consistent pairs.
0, our criterion Cw degenerates to CT as we are unable to
Theorem (Cw guarantees). The metric-sensitiv wormhole criterion Cw yields guaranteed pairs.
Proof. The proof generalises the one of Theorem 2. The difference comes when comparing the length of geodesic paths on the full surface y to length of curves in the embedding space. Name dj the curved M-based Euclidean distance of geodesic curves on the manifold for the metric M. The curved M-based Euclidean distance is greater than the Euclidean distance of straight lines between the points d, 2 dg (Theorem I). Since d 2 VCard,, i.e. the geodesic distance is at least vCM times larger than the curved M-based Euclidean distance, have the relationship between the geodesic and Euclidean distances dy 2 VCmdg. We can then conclude by applying the same reasoning as in the proof of Theorem 2.
B Further Experimental Details
B.1 Multi-Dimensional Scaling
Formally, the original MDS problem aims to minimise the quadratic stress function
X
argmin
Wiy (dgn (X,, - (;))),
(10)
X
where wiy E [0, 1] is a given optional weighting scheme, A common assumption in the theory of MDS is that the given manifold can indeed be isometrically mapped to low dimensional Euclidean space". If this assumption holds we could consider all pairs equally wij = 1, which yields the classical scaling approach [52, 57].
However, we would like embeddings of partial surfaces y" robust to missing parts, which implies that we need to handle the boundary conditions induced from partial surfaces. Enforcing the preservation of distances for inconsistent pairs could distort the embedding. To overcome the influence of holes, cuts, and boundaries in general, we resort to our criterion to filter out inconsistent pairs.
In [12, 14, 46, 47], the weights are set to wij = CT,,, Pj). leading to the TCIE method [46, 47J. To the best of our knowledge, [46, 47] provides the best weighting scheme based on consistent pairs of points without involving heuristics. Other approaches exist relying on heuristics, such as focusing on local pairs points due to local approximations of geodesic distances [51]. We propose to refine the TCIE idea by taking a more inclusive criterion instead Wij = Cw(P,, = M,,. Minimising Equation (10) can be efficiently done by any optimization method, for example, the SMACOF algorithm [9] was shown to be equivalent to a constant step size weighted gradient descent [17]. Classical scaling [52, 57] is a method of choice for initialization to avoid local minima.
Our toy surfaces are pointclouds of swiss rolls with 2000 randomly sampled vertices, stretched in width by factor of 1.5, to which either no noise (Figure 7) or very small Gaussian noise of
For full-to -arrtaal shape matching, then C's can be computed the full surface. For partial-to partial shape matching, we assume that Cs given by an oracle.
"This assumption is obviously violated for surfaces with effective Gaussian curvature, can be easily verified by the Gauss Gauss-Bonnet Bonnet theorem [8].
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standard deviation 0.2 is added* (Figure 8). For each surface, we removed either a hole or a missing part connected to the boundary. We use Dijkstra's s algorithm to compute geodesic distances on the pointclouds. Following [46, 47], we also set Waj = 1 for small geodesic distances (smaller than 3) to handle the boundaries. We here compare with more reference works in manifold learning than in the main manuscript, namely Isomap [57], LLE [49], Hessian LLE [23], MLLE [62], LTSA LLE [63], Laplacian eigenmaps [5], Diffusson maps [19]. t-SNE [58], UMAP [37] PCA [42, 26). Kernel PCA [50], and TCIE [46, 47J. We use either the Euclidean geodesic distance matrix for Diffusion Maps, I-SNE, and UMAP. We run t-SNE with perplexity parameter of 50 and UMAP with minimum distance parameter of 0.8. For both methods, these high parameter values are supposed to respect data topology rather than encourage clustering. Methods requiring nearest neighbor computations use k- 15 neighbors. For reference, we also show the computed embeddings on the full surfaces.
In the ideal noiseless case, the non-vanilla L.E. methods are able to handle the partial surface with a hole quite well. However, only a minor amount of noise breaks these methods completely. The 1-SNE and UMAP methods perform poorly, even on the full surfaces. These popular methods were indeed primarily designed to show nice clusters, but are not intended and should not be used for viewing non-clustered data manifolds in low dimensions as is too often the case. Both TCIE and our WHCIE are robust to small amounts of noise, yet our embeddings are superior as they are less distorted by the missing parts as we find more consistent pairs.
B.2 Discretization
The definition of consistent pairs is so far in continuous setting. We need be adapt it to the discrete world, as computational surfaces are themselves discrete. Let V be a finite set of vertices sampled from the continuous surface. We only compute guaranteed pairs between the vertices of V. To that end, we consider only the discrete set of boundary vertices B = V n B. Given this discretization, we calculated Cw-guaranteed pairs in naive manner. We first compute the geodesic distance matrix Dy, having (i entry (Dy') )sy as the geodesic distance on the partial surface y" between the i-th and j-th vertices of V. using a relevant method for computing geodesic distances on the discrete surface, e.g. Dijkstra's algorithm [21]. Fast Marching Method [28], MMP [56, 39], or deep learning methods for geodesic distance calculation [32, 27]. We then calculate the Euclidean distance between each pair of boundary vertices in B. We have thus computed all elements of the threshold in Equation (3), and all that is left to search for the two boundary vertices that give the minimal bound. Optionally, computations can be sped up using GPU, we used V100 and it took a few minutes to compute the masks for surfaces on the PFAUST datasets. We found that is best to compute the (([V)21B) 22) distances in batches due to the high spatial complexity. which for common surfaces becomes prohibitively large. This process computes a matrix K of size [V] x [V), containing the threshold of the criterion Cy calculated for each pair of vertices. The guaranteed pairs are the true elements of the binary mask M.j = A(D,y (D,K)...."
An alternative faster approach to the naive method above for computing the mask matrix M would be to recompute geodesic distances on a tweaked distance graph. The adjustment would be to create an edge between all boundary points with distance equal to the Euclidean distance between them. Shortest paths on this tweaked graph belong to one of two types. The first type of new shortest paths between any two points corresponds paths that do not pass through boundary points, which means that they are the same as the ones on the partial surface y": the pair is consistent and guaranteed by Cw. The other type of new shortest paths between any two points corresponds to paths using single new edge between boundary vertices, which means that their length is equal to the wormhole worst case distance bound between these two points: the pair is not guaranteed by Cw. This procedure thus computes the criterion matrix K defined as
K
)ij, Kish.
(1))
We can then compute the binary mask matrix by comparing the distances computed on the original and the tweaked graphs M,j- = 1 (D &K,, The total cost of this approach comes from computing twice the distance algorithm, once on graph and once on its modification with (( extra edges.
As such, the complexity of this method is ((IV((()) + 2 + [V] log((V)))) However, in practice, we usually have [ 0 VVVV As such, the complexity is usually ((IV((E)) (FF) + lo)g))()))))
WWe provide the code to generate the pointclouds in our code repository.
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Hessian LLE
LISA LLE
Kernel PCA
Isomap
LLE
MLLE
PCA
t-SNE Euclidean
t-SNE Goodesic
UMAP Euclidcan
UMAP
Input
Geodesic
Whole
Laplacian Eigenmaps
Diffusion Maps Euclidean
Diffusion Maps Geodesic
WHCIE
TCIE
(Ours)
Hcssian LLE
LTSA LLE
Kernel PCA
Isomap
LLE
MLLE
PCA
t-SNE Euclidean
t-SNE Geodesic
UMAP Euclidean
UMAP Geodesic
Input
Partial
Laplacian Eigenmaps
Diffusion Maps Euclidean
Diffusion Maps Geodesic
WHCIE
TCIE
(Ours)
Hessian LLE
LTSA LLE
Kernel PCA
Isomap
LLE
MLLE
PCA
t-SNE Euclidean
t-SNE Geodesic
UMAP Euclidean
UMAP Geodesic
Input
Partial
Laplacian Eigenmaps
Diffusion Maps Euclidean
Diffusion Maps Geodesic
WHCIE
TCIE
(Ours)
Figure 7: MDS embeddings of reference manifold learning methods on a whole swiss roll without noise and its variants with hole or cut.
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Hessian LLE
LTSA LLB
Kernel PCA
Isomap
LLE
MLLE
PCA
(-SNE Euclidean
t-SNE Geodesic
UMAP Euclidean
UMAP
Input
Geodesic
Whole
Laplacian Eigenmaps
Diffusion Maps Euclidean
Diffusion Maps Geodesic
WHCTE
TCIE
(Ours)
Ilessian
LTSA LLE
Kernel PCA
Isomap
LLE
MLLE
PCA
t-SNE Euclidean
I-SNE Geodesic
UMAP Euclidean
UMAP
Input
Geodesic
Partial
Laplacian Eigenmaps
Diffusion Maps Fuclidean
Diffusson Maps Gendesic
WHCIE
TCIE
(Ours)
Hessian LLE
LTSA LLE
Kernel PCA
Isomap
LLE
MLLE
PCA
t-SNE Euclidean
t-SNE Geodesic
UMAP Euclidean
UMAP Geodesic
Input
Partial
Laplacian Eigenmaps
Diffusion Maps Euclidean
Diffusion Maps Geodesic
WHCTE
TCIE
(Ours)
Figure 8: MDS embeddings of reference manifold learning methods whole swiss roll with small noise and its variants with hole cut.
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meaning that computing the mask criterion K has the same complexity as computing the shortest distances on the partial shape Dyr Note that for triangulations or kNN graphs, [E| (((V)) and then the complexity for computing the wormhole mask is simply O (2 log((V)) practice.
B.3 Derivation of the Masked Geodesic Distance Preservation Loss
Our discrete loss function Leo in Equation (7) based on the continuous loss presented in [I], see also [13]. Following their formulations we introduce a loss for the correspondence function between X' and y", defined by p: y x X + RR as,
2 f VFrc dx(m1, 12 1211, v1) da da - dy vb)) m(y 1/ day, days (12)
where dx and dy measure the distances between surface points, and m y" x 3" + (0, 1} is our binary masking function. We readily have the discrete version,
L(P, Dx, Dy, M)
IIVM @ (PAxDxARP Dy)lby
(13)
where |Bllyrg trace B' BAy and VM is the entry-wise square-root of M. Similar to previous papers [25] we assume that our learned correspondence matrix implicitly contains the area matrix Ax, thus, P PAx. Consequently, our loss function is defined as follows,
Lgo Dx, Dy, M)
Il @ (PDxP -Dy)l
(14)
For simplicity, we expressed it element-wise,
E My (PDxPP - -Dy)
gen ( D,Dx M)
(15)
The same derivation led to the non-masked loss in [10].
B.4 Detailed Implementation Considerations
Most partial shape matching pipelines [3, 18, 10] can be divided into four parts; feature refinement network, correspondence matrix creation, loss functions, and post processing. The feature refinement network gets raw input features of the surface, such as ryz coordinates, and additional geometric properties of the surface, e.g. the LBO cigendecomposition, and outputs refined feature for each vertex. Given the refined features from each surface, a correspondence matrix is built either from FM [48, 25], as in pipeline, directly from similarities between features on the different surfaces [2, 18, 10]. The loss functions operate on the FM matrix that is calculated directly from the the
feature matrix [48], on the FM which is calculated from the correspondence matrix [10]. correspondence matrix directly [25]. Post processing allows to refine test time results and usually consists in applying the methods Zoom-out[33) PMF [60], or refinement of the network weights via test time adaptation [18].
We use as input features the zyz coordinates of each vertex along with its estimated normal. We took DiffusionNet [54] to be the feature extraction network, with shared weights between the full and the partial surfaces. It outputs the refined features Fs E R/Vs|xd where feature dimensionality, and S E (X')") We follow [11]. by replacing the FM layer with direct computation of the correspondence matrix via Softmax similarity, similar to [7, 18),
256 is the
exp(Gig/T) Pū Σ, (xp(Gr//))
(16)
where G FT. Fx.
x, and 7 is temperature hyper- papper-rrramtte, where 7 = 0.07 for the HOLES dataset and 7 0.01 for the CUTS and PFAUST datasets. We point out that the correspondence matrix is row normalised Σ, P.j = 1 but not column normalised, as some vertices of X' in missing parts of y should not have corresponding vertices. The correspondence matrix is then fed into our loss function fages using the soft mask M unless stated otherwise (Equation (7)), with Lagrangian coefficient
Ageo = 10 Following [48], we additionally use regularization based on the FM matrix C calculated directly from the correspondence matrix P.
C= JJ Ay A
(17)
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where Dx and y are the LBO eigenfunctions of the surfaces A' and ) and Ay is diagonal matrix containing the area associated to each vertex in the discrete surface y" We plug the FM matrix into the regularisation loss Lontho with Lagrangian coefficient Aarth = 1, which promotes the preservation of vertex area under the estimated correspondence [48, 3].
Lotho (C) CC J,
(18)
where J, 1. 0] is the identity matrix until column and only zeros columns afterwards, and is number of eigenvalues of the LBO on yd that are smaller than the largest eigenvalue of the truncated LBO on the discrete surface X. We trained our network for 20000 iterations, with Adam optimizer [29], with learning rate of 10-3 and a cosine annealing scheduler [35] with minimum learning rate parameter 7imix 10-4 and maximum temperature of T max 300 steps. Lastly, for post-pree post-processing. we use the test time adaptation refinement method [18]. that refines the network weights separately for each pair of surfaces with iterations of gradient descent.
Compared to DirectMatchNet [10], our method only involves an additional preprocessing of the shape compute the mask matrices for the loss function. As such, from a time perspective, both methods are comparable and even identical at inference time without test time adaptation. For reference, inference time is 0.12s on average on our hardware for both DirectMatchhNtt and our method on the HOLES dataset.
B.5 Further Shape Correspondence Result
We here display further qualitative results the PFAUST M (Figure 9), (Figure 10), SHREC' 16 CUTS (Figure 11), and HOLES (Figure 12). The results support those presented in Figure 6. Methods based on the preservation of distances produce significant distortions compared those that do. Furthermore, by focusing on consistent pairs, our method yields less distortion near holes than similar approach [10] not taking them into account.
We also provide in Figure 13 Percentage Correct Keypoint (PCK) curves describing the percentage of correct predictions with respect to the geodesic error for the baseline methods UnsupDPFM [3], RobustFMnet [18], DirectMatchhet [10]. and our method, without refinement for all methods. In particular, our method yields clear improvement the challenging PFAUST-H benchmark, and moderate improvements on the other datasets, which corroborates the results of Tables 2 and 3.
20

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RAESS ARH
MatchrMatchNet
Wormhole (Ours)
HF
Figure 9: Additional qualitative results on the PFAUST-M dataset.
DireatMatchNet
Wornhole (Ours)
BFG
Figure 10: Additional qualitative results on the PFAUST-H dataset.
21

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DirectMatehNet
Wormhole (Ours)
MarchtMatchNet
Wormhole (Ours)
DirectMatchNet
Wormhole (Ours)
Figure 11: Additional qualitative results the SHREC'16 16 CUTS dataset.
22

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DirectMatchNet
Wormhole (Ours)
DirectMatchNet
Wormhole (Ours)
DirectMatchNet
Wormhole (Ours)
Figure 12: Additional qualitative results the SHREC' 16 HOLES dataset.
23

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PFAUSTTM
PFAUST-H
0.8
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Proposed
CirectMotchNet
DireclMetchleet
Mreet
Miret
Robusti Mnett
UnsugDPFM
UnsupDPFM
0.0
15
SHRECI6-CUTS
SHRECIB-HOLES
0.0
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Congspondences
0.6
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RobhucFFMett
UnSupDPFM
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0.0
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15
70
Cumulative Goodesic
Curnulative Geodesic Error
Figure 13: PCK curves of the geodesic error of the baselines and method on the SHREC' 16 CUTS and HOLES and the PFAUST M and H datasets.
24

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Reviews:
[{"review_id": "EGHYI0OEjK", "reviewer": "Reviewer_tftc", "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.", "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?\n- 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?\n- 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.", "limitations": "N/A", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "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\n- 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.\n- $\\mathbf{K}$ 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.\n- Line 21: \"Consequentially\" should be \"Consequently\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed comments and for recognizing the utility of the wormhole criterion in both MDS and partial-to-whole shape matching tasks. We are happy that the reviewer enjoyed the clarity, soundness, and contribution in our manuscript. We fixed the typos pointed out by the reviewer and added the definition of $\\boldsymbol{K}$ to the paper, $\\boldsymbol{K}\\_{ij} = \\min\\limits_{B_1,B_2\\in \\mathcal{B}} d_{\\mathcal{Y}'}(v_i, B_1) + d_{\\mathcal{Y}'}(v_j, B_2) + d_E(B_1, B_2)$, where $v_i$ and $v_j$ are $i$-th and $j$-th vertices. In the following we respond to the reviewer's remarks and questions.\n### Line 137\nRegarding the confusion in lines (137-139), we will rewrite this sentence to avoid confusion by taking a formulation similar to version (1) suggested by the reviewer.\n### When both shapes are partial\nRegarding our theoretical discussion about consistent pairs, we focused on the case when only one surface is partial. In our experiments, we then used the found consistent pairs to guide the learning process for partial-to-whole shape matching. If both shapes are partial, the theory allows to find consistent pairs on each partial surface independently. In practice though, we would need to find the shared consistent pairs between both partial surfaces should we want to learn the matching between both partial surfaces. This should indeed be done by taking some form of intersection of found consistent pairs. Designing a partial-to-partial shape matching pipeline using the proposed criterion is left to future work (we will emphasize it in Limitations section).\n### Boundaries\nRegarding boundaries, in practice, we treat all boundaries equally. We will remove Line 133 \"excluding the boundaries of $\\mathcal{Y}$\" and clarify in the paper that all boundaries are treated equally.\n### Double area scaling\nRegarding the double area scaling in Equation (5), our loss function is based on the continuous loss presented by Aflalo et al. IJCV 2016, see also [11]. Following their formulations we introduce a loss for the correspondence function between $\\mathcal{X}$ and $\\mathcal{Y}'$, defined by $\\tilde{p}:\\mathcal{Y'}\\times \\mathcal{X} \\rightarrow \\mathbb{R}^+$ as,\n$$\\int_\\mathcal{Y'\\times Y'} \\left(\\int_\\mathcal{X\\times X} d_\\mathcal{X}(x_1, x_2) \\tilde{p}(x_1, y'\\_1)\\tilde{p}(x_2, y'\\_2) da_{x_1} da_{x_2} - d_\\mathcal{Y'}(y'\\_1, y'\\_2)\\right)^2 m(y'\\_1,y'\\_2) da_{y'\\_1} da_{y'\\_2} $$where $d_{\\mathcal{X}}$ and $d_{\\mathcal{Y}'}$ measure the distances between surface points, and $m:\\mathcal{Y'}\\times \\mathcal{Y'} \\rightarrow \\\\{0,1\\\\}$ is our binary masking function. We readily have the discrete version,$$\\mathcal{L}(\\boldsymbol{\\tilde P}, \\boldsymbol{D}\\_\\mathcal{X}, \\boldsymbol{D}\\_\\mathcal{Y'},\\boldsymbol{M})=\\||\\boldsymbol{M}\\odot(\\boldsymbol{\\tilde P} \\boldsymbol{A}\\_\\mathcal{X} \\boldsymbol{D}\\_\\mathcal{X}\\boldsymbol{A}\\_\\mathcal{X}\\boldsymbol{\\tilde P}^\\top - \\boldsymbol{D}\\_\\mathcal{Y} )\\||\\_\\mathcal{Y'Y'},$$ where $\\||\\boldsymbol{B}\\||\\_\\mathcal{Y'Y'} = \\text{trace}(\\boldsymbol{B}^\\top\\boldsymbol{A}\\_\\mathcal{Y}\\boldsymbol{B}\\boldsymbol{A}\\_\\mathcal{Y})$. Similar to previous papers [21] we assume that our learned correspondence matrix implicitly contains the area matrix $\\boldsymbol{A}\\_\\mathcal{X}$, thus, $\\boldsymbol{P} = \\boldsymbol{\\tilde P} \\boldsymbol{A}\\_\\mathcal{X}$. Consequently, our loss function is defined as follows,$$\\mathcal{L}\\_{\\text{geo}}(\\boldsymbol{P}, \\boldsymbol{D}\\_{\\mathcal{X}}, \\boldsymbol{D}\\_{\\mathcal{Y}'}, \\boldsymbol{M})=\\||\\boldsymbol{M}\\odot(\\boldsymbol{PD}\\_\\mathcal{X}\\boldsymbol{P}^\\top - \\boldsymbol{D}\\_\\mathcal{Y'}) \\||\\_\\mathcal{Y'Y'}.$$For simplicity, we expressed it element-wise,$$\\mathcal{L}\\_{\\text{geo}}(\\boldsymbol{P}, \\boldsymbol{D}\\_{\\mathcal{X}}, \\boldsymbol{D}\\_{\\mathcal{Y}'}, \\boldsymbol{M})=\\sum_{ij} \\boldsymbol{M}\\_{ij} \\boldsymbol{A}\\_{\\mathcal{Y}'ii} \\boldsymbol{A}\\_{\\mathcal{Y}'jj} \\left( (\\boldsymbol{P}\\boldsymbol{D}\\_\\mathcal{X}\\boldsymbol{P}^\\top - \\boldsymbol{D}\\_{\\mathcal{Y}'})_{ij} \\right)^2.$$We will add a derivation of the loss from its continuous to the discrete versions.\n\nY. Aflalo et al. Spectral generalized multidimensional scaling. IJCV, 2016", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4wD1Up2Ck4", "reviewer": "Reviewer_1VSx", "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.", "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.\n\n- 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? \n\n- Are there specific scenarios where this regularization might fail?", "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.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The focus on partial shape matching and unsupervised shape matching for partial to full shapes is intriguing.\n- 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.\n-The authors could enhance evaluations by using other correspondence techniques, instead of only using DiffusionNet features, and demonstrate broader generalization capabilities with other signatures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the insightful comments and for the appreciation of the strengths of our method when applied to the challenging partial shape matching problem and our significant improvements for this task. We hereby respond to the reviewer's remarks and questions.\n### Reliance on the Euclidean embedding space\nRegarding the Euclidean term in our criterion, the Euclidean embedding space yields a lower bound on geodesic distances between boundary points. Naturally, this bound is less tight the more the surface is curved. Nevertheless, we demonstrate in both shape matching experiments with highly non-rigid transformations of shapes with complex partial topologies and flattening highly curved embeddings of the Swiss roll, that our criterion is greatly beneficial in non-Euclidean settings. We also ran in the rebuttal period new experiments to evaluate the amount of consistent pairs recovered by our criterion in the PFAUST benchmark and found that we are able to recover most pairs and find about twice as many as the previous criterion which does not rely on Euclidean distances. See the table below, that we will include in the paper, presenting the percentage of consistent pairs, and among those the percentage of guaranteed pairs by the different criteria, along with the standard deviations.\n||%Consistent|%Guaranteed ($\\mathcal{C}_\\mathcal{T}$) [42]|%Guaranteed ($\\mathcal{C}_\\mathcal{W}$) (OURS)|\n|-|:-:|:-:|:-:|\n|PFAUST-M|78 (+-16)|48 (+-18)|82 (+-14)|\n|PFAUST-H|53 (+-16)|30 (+-18)|65 (+-18)|\n### Generalization\nRegarding the generalization capabilities of our method, we showed the success of our criterion in two completely different tasks: partial shape matching and MDS flattening of partial surfaces. In the learning pipeline, we used the DiffusionNet as it is currently the prominent network for shape correspondence. Nowadays SOTA methods rely on DiffusionNet, and we show that we can improve them even further with our criterion. However, our method is complementary to DiffusionNet, and we could use the proposed criterion to train learnable feature extractors that would potentially replace DiffusionNet in the future.\n### Larger and more complex datasets\nRegarding the question on the performance of our method on larger datasets with more complex topologies, we evaluated our method on the reference datasets for partial shape matching. We not only evaluated on the leading benchmark in the field SHREC'16, we also tested on the recent benchmark PFAUST to show that our method generalises to other types of shapes with different types of shape partiality. Note that both benchmarks comprise complex topologies due to the way parts were removed. We did not find a larger dataset with more complex topologies for partial shape matching of surfaces undergoing non-rigid transformations. \n### Control measures\nRegarding control measures for analysing the effect of filtering inconsistent pairs, we point out Table 3 in our ablation study. There, we show how performance changes if we take either a binary or a soft criterion, based either on our wormhole one $\\mathcal{C}\\_\\mathcal{W}$ or $\\mathcal{C}\\_\\mathcal{T}$. In addition, our approach is built on top of DirectMatchNet. Thus, the comparative performance analysis in Tables 1 and 2 shows the effect of including our criterion.\n### Soft mask matrix and thresholding\nRegarding the soft mask $\\boldsymbol{M}^s$, there is actually no threshold parameter in its calculation. It is given directly form the distance matrix $\\boldsymbol{D}$ and the $\\boldsymbol{K}$ matrix by $\\boldsymbol{M}\\_{ij}^s = \\min(\\frac{\\boldsymbol{K}\\_{ij}}{\\boldsymbol{D}\\_{ij}},1)$, where $\\boldsymbol{K}$ is given by the criterion$$\\boldsymbol{K}\\_{ij} = \\min\\limits_{B_1,B_2\\in \\mathcal{B}} d_{\\mathcal{Y}'}(v_i, B_1) + d_{\\mathcal{Y}'}(v_j, B_2) + d_E(B_1, B_2),$$where $v_i$ and $v_j$ are the vertices at indices $i$ and $j$ respectively. We will clarify this by adding the definition of $\\boldsymbol{K}$. Also, since we want the soft mask matrix $\\boldsymbol{M}\\_{ij}^s$ to be 1 and equal to the binary mask matrix $\\boldsymbol{M}\\_{ij}$ for consistent pairs (when $\\boldsymbol{D}\\_{ij}\\le \\boldsymbol{K}_{ij}$), we cut off the ratio $\\frac{\\boldsymbol{K}\\_{ij}}{\\boldsymbol{D}\\_{ij}}$ to $1$ in the definition of the soft mask matrix.\n### Failure cases\nRegarding failure cases, our binary criterion cannot guarantee inconsistent pairs as proven in Theorem 2. Regarding failure cases due to the regularization, when we would get worse results by switching from the binary to the soft mask, we did not find such scenarios. Nevertheless, we do not claim that our regularized approach would never fail on unseen challenging contexts.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0QlSBPJyr0", "reviewer": "Reviewer_NPnj", "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.\n\nThe experiments are conducted for applications including multi-dimensional scaling and partial shape matching, in which the advantage of the proposed loss is demonstrated.", "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?", "limitations": "The limitations have been adequately discussed in the paper.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "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.\n\nHowever, 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful comments and their appreciation of the paper's clear contribution and the soundness and clear presentation of our framework. In the following we relate to the reviewer's remarks and questions.\n### Theoretical depth\nRegarding the theoretical depth of our paper, we respect the reviewer's perspective. However, we respectfully like to argue otherwise. The proposed criterion, which is indeed a simple deviation from a naive existing one, is less trivial than might seem at first glance. Most existing papers in shape analysis, and in particular in shape matching, focus on using intrinsic properties, such as geodesic distances, LBO eigendecompositions, functional maps, and interactions between non-regular metric spaces. Another philosophy in shape analysis, albeit less popular, is to consider only extrinsic properties, e.g. Wang et al. 2018, but such approaches are ill-suited for problems based on intrinsic properties like shape matching.\n\nIn our case, the proposed approach combines intrinsic and extrinsic non-differential quantities. In fact, we invoke an extrinsic measure (embedding distances in a Euclidean space) to analyze intrinsic quantities (geodesic distances). While maybe simple in appearance, such a perspective is far from trivial. In addition, extrinsic Euclidean distances are often ignored in the study of shapes undergoing non-rigid transformations. Finally, our criterion yields theoretically guaranteed pairs (Theorem 2), which is proven in the appendix. Bottom line - it is not a heuristic criterion, but rather a theoretically supported, and apparently practically useful one. Providing a simple approach that integrates intrinsic and extrinsic integral measures, that introduces the often ignored extrinsic Euclidean distances, and is well supported by a solid mathematical theory is a fundamental contribution. Moreover it leads to enhanced empirical performance.\n\nThat being said, we do not provide a theoretical guarantee on the measure of consistent pairs our criterion can find, and we are clear on that matter in the limitations paragraph. The same argument is true for other criteria, e.g. $\\mathcal{C}\\_\\mathcal{T}$. Nevertheless, the proposed criterion is theoretically guaranteed to recover all the pairs that the existing criterion $\\mathcal{C}\\_\\mathcal{T}$ finds. We also show in Figure 4 a qualitative difference between the consistent pairs both criteria guarantee.\n\nY. Wang et al. Steklov spectral geometry for extrinsic shape analysis. ACM TOG, 2018.\n### Amount of missed consistent pairs\nRegarding the gap between the number of guaranteed pairs via different criteria and the total number of consistent pairs, we conducted experiments during the rebuttal period to estimate it. We evaluated on the datasets PFAUST-H and PFAUST-M the amount of consistent pairs, guaranteed pairs by our criterion $\\mathcal{C}\\_\\mathcal{W}$, and guaranteed pairs by $\\mathcal{C}_\\mathcal{T}$. We provide in the table below the average percentage of consistent pairs, and among these, the average percentage of guaranteed pairs by each criteria. We also provide the standard deviations in parenthesis.\n||%Consistent|%Guaranteed ($\\mathcal{C}_\\mathcal{T}$) [42]|%Guaranteed ($\\mathcal{C}_\\mathcal{W}$) (OURS)|\n|-|:-:|:-:|:-:|\n|PFAUST-M|78 (+-16)|48 (+-18)|82 (+-14)|\n|PFAUST-H|53 (+-16)|30 (+-18)|65 (+-18)|\n\nWe will add this evaluation table to the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "f7wxu2jii3", "reviewer": "Reviewer_a2Wg", "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.", "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? \n2) How computationally expensive is the method?", "limitations": "The authors describe some limitations regarding future works but do not discuss failure cases of the proposed strategy.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "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\n- 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. \n- 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.\n- 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We  thank the reviewer for the detailed comments and for acknowledging the relevance and challenge of addressing partial shape correspondence in an unsupervised setting, as well as the applicability of  our method  to MDS embeddings. We will add to Figure 6 the ground-truths. We hereby respond to the reviewer's remarks and questions.\n\n**Clarification:** Regarding the reviewer's summary, it seems that there might have been some confusion and mix of concepts that we would like to address. As defined in the paper, consistent pairs are pairs of points for which the shortest path on the partial shape have rigorously the same length as the shortest path on the full shape. The lengths can be different only if the shortest path on the full shape passes through parts which are missing in the partial shape. We design a criterion that finds as many consistent pairs as possible. All pairs guaranteed by this criterion are definitely consistent. Our criterion is based on a lower bound estimation of the unknown geodesic distance on the full shape. It is realized by considering Euclidean distances between boundary points of the partial shape as potential shortcuts.\n### CUTS\nRegarding performance on CUTS, the reviewer suggests that the proposed criterion is restricted to simple cases, and would not apply to missing parts like in CUTS due to their large size. We respectfully disagree with these two claims as explained below.\n\n- Regarding simplicity, \"The missing parts in the CUTS dataset were created by slicing figures with 3D planes. In practice, this procedure does not modify much the topology of the surface and it does not introduce many inconsistent pairs. As such, cuts mostly preserve all geodesic distances, thus, our novel loss function was almost identical to the one in Equation (4)\" (lines 256-258). The implication is that cuts provide too simple, and not too complex, partial shapes where almost are pairs are consistent and found by our criterion, explaining why there our method does not lead to significant improvement.\n\n- Regarding size, it is the shape of the missing parts, and not their size, that dictates the complexity of the partial shape. For example, cutting off just a finger or both legs on a human shape leads to simple partial shapes with only consistent pairs but with very different size of missing parts.\n\nIn contrast, holes of various sizes present significantly harder topologies with many pairs that are inconsistent. In fact, our method leads to significant improvements when handling missing parts with holes by filtering out inconsistent pairs, as shown on the HOLES, PFAUST-M and PFAUST-H datasets.\n### Failure cases\nRegarding failure cases of our criterion. Our binary criterion cannot guarantee inconsistent pairs as proven in Theorem 2. Regarding failure cases of our methods, in all our experiments except on CUTS, our criterion managed to find a significant amount of consistent pairs, leading to improved performance in both shape matching and manifold flattening. In CUTS, our criterion does not fail, it is just not so relevant as almost all pairs are consistent, and we discuss this issue in the results section. At the other end, we do not claim that our regularized approach is guaranteed to always succeed. \n### Partiality magnitude and amount of missed consistent pairs\nThe reviewer suggests to analyse the impact of partiality magnitude on the ability to find consistent pairs. This question is interesting, however, we would like to emphasize that the amount of missing parts is not the main factor impacting consistent pairs, but more the topology of the missing parts. See our earlier discussion on CUTS for an example of a simple partial shape with large missing parts. Nevertheless, in the rebuttal period, we evaluated the gap between the total amount of consistent pairs and that found by different criteria on both the PFAUST-M and PFAUST-H datasets, where PFAUST-H is harder since it has many more holes (of smaller size). See the table below, that we will include in the paper, presenting the percentage of consistent pairs, and among those the percentage of guaranteed pairs by the different criteria, along with the standard deviations.\n||%Consistent|%Guaranteed ($\\mathcal{C}_\\mathcal{T}$) [42]|%Guaranteed ($\\mathcal{C}_\\mathcal{W}$) (OURS)|\n|-|:-:|:-:|:-:|\n|PFAUST-M|78 (+-16)|48 (+-18)|82 (+-14)|\n|PFAUST-H|53 (+-16)|30 (+-18)|65 (+-18)|\n### Several connected components\nRegarding several connected components, it is standard practice for partial shape matching to consider only a single connected component. By definition, consistent pairs can only belong to the same components. Our pipeline would simply generalise by computing the features of each component independently and then concatenating them before estimating the matching, just like other pipelines would. \n### Complexity\nRegarding the complexity of our method, we would like to point out to the reviewer that we have discussed the computational complexity of the method in appendix B.2. Note that this complexity occurs only during preprocessing, as then our methods does not induce any noticeable additional cost.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: All four reviewers unanimously recommend acceptance after the rebuttal. The paper is well-written and there is a clear contribution which has been demonstrated to work well in practice. The AC concurs and recommends accept as well.
Decision: Accept (poster)