HOPE: Shape Matching Via Aligning Different K-hop Neighbourhoods
Authors: Barakeel Fanseu Kamhoua
Huamin Qu
Venue: NeurIPS 2024 poster (2024)
Keywords: Shape analysis
correspondences
registration
Primary area: active_learning
TL;DR: We propose to use different k-hop neighboorhoods of vertices (nodes) for refining initialized maps  for shape matching.

Abstract: Accurate and smooth shape matching is very hard to achieve. This is because for accuracy, one needs unique descriptors (signatures) on shapes that distinguish different vertices on a mesh accurately while at the same time being invariant to deformations. However, most existing unique shape descriptors are generally not smooth on the shape and are not noise-robust thus leading to non-smooth matches. On the other hand, for smoothness, one needs descriptors that are smooth and continuous on the shape. However, existing smooth descriptors are generally not unique and as such lose accuracy as they match neighborhoods (for smoothness) rather than exact vertices (for accuracy). In this work, we propose to use different k-hop neighborhoods of vertices as pairwise descriptors for shape matching. We use these descriptors in conjunction with local map distortion (LMD) to refine an initialized map for shape matching. We validate the effectiveness of our pipeline on benchmark datasets such as SCAPE, TOSCA, TOPKIDS, and others.

HOPE: Shape Matching Via Aligning Different K-hop Neighbourhoods
Barakeel Fanseu Kamhoua Huamin Qu'; 'The Hong Kong University of Science and Technology
Abstract
Accurate and smooth shape matching is very hard to achieve. This is because for accuracy, one needs unique descriptors (signatures) shapes that distinguish different vertices on a mesh accurately while at the same time being invariant to deformations. However, most existing unique shape descriptors are generally not smooth on the shape and are not noise-robust thus leading to non-smooth matches. On the other hand, for smoothness, one needs descriptors that are smooth and continuous on the shape. However, existing smooth descriptors are generally not unique and as such lose accuracy as they match neighborhoods (for smoothness) rather than exact vertices (for accuracy). In this work, we propose to use different k-hop neighborhoods of vertices as pairwise descriptors for shape matching. We use these descriptors in conjunction with local map distortion (LMD) to refine an initialized map for shape matching. We validate the effectiveness of our pipeline on benchmark datasets such as SCAPE, TOSCA, TOPKIDS, and others.
Introduction
Shape matching is a very important task and has been increasingly so with the increase in the
availability and affordability of 3D scans 31 It has important applications including but not limited
to shape registration [2] 37]. comparison
[ recognition [6], and retrieval [50].
Shapes can undergo different types of transformations which will group as isometric and non- isometric transformations this work. Nonrigid isometric transformations preserve most of the geometric properties of the shape i.e., after the transformation, its angles, geodesie distances, scale, connectivity, and other geometric properties are mostly preserved. On the other hand, most geometric properties not preserved under nonrigid non-isometric transformations. Examples of isometric transformations are rotations and translations (though minute scaling may also be included in this category for nonrigid cases). While examples of non-isometric transformations are splitting, scaling, dilating, and others which change the geometric properties of the shapes.
Several works have been proposed to address shape matching. However, most of these methods will 34
either aim for smoothness [7 19 [3]
31 20) or for accuracy [45), but not both. Moreover, most of these methods will focus specific types of deformations undergone by the shape
39
37
44
either isometric 12 19
34 39 37
20 49 or non-isometric 14 22 52]. Some methods
such 2D-GEN 22 have also tried to address both accuracy and smoothness in shapes undergoing both isometric and non-isonen -nommettt deformations. However, 2D-GEM is very using different user-defined parameters for isometric shapes than for non-isometric shapes which in real life is not practical as one usually does not know whether one is dealing with isometric or non-isometric deformations.
Given these obsevations, there is a need for pipeline that aims for accuracy and smoothness across different settings without significantly changing the pipeline. In this light, propose HOPE, k-hop
*Corresponding author: huamin @ ust.kk
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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neighborhood- refinement technique. The rest of the paper is organized into: (a) preliminaries, (b) related work, (c) HOPE, and (d) experiments, (e) limitations and remarks, and (f) conclusion.
2 Preliminaries
3D shapes are usually represented by their coordinates X' E R and their triangulations (meshes) which can be used to build an adjacency R"x", with A(i if vertices and are connected in the shape and 0 otherwise.
Given two direc-dimensional shapes M and N with n.M and n. vertices respectively. Though our proposed pipeline (HOPE) and some other baselines work with n.M # n.W, we will assume n.M n2,V" n, and for simplicity except when specified. The goal of nonrigid shape matching to find a meaningful correspondence T M - N, where T is (a) bijective, (b) continuous (smooth). and (c) similar vertices should be matched captures both the smoothness and the accuracy of the map is the geodesic error. The geodesic error measures how far map T, maps vertex from its matching position given by the ground truth map higher proportion of vertices having geodesic
each other [26] 35 22 A measure that
T. where a higher map accuracy will correspond error O. and smoother map will correspond to huge increase in the geodesic error curve as we move slightly away from error 0 (implying that several vertices though not accurate are mapped to their intended neighborhoods).
The two main components of shape matching involve: (1) initializing a map, and (2) refining the initialized map.
First, the map is initialized either by using landmarks vertices (i.e., vertices with known correspon- dences) or by using robust descriptors. These descriptors can be based on: (a) the spectrum of the shape laplacian 88 [33], (b) The face normals, vertex location, and triangulation [45] 41], geodesic
distances 44 or others [4 46 [5] [T 43 23). this work, we will assume that one such initialization approach has been utilized and we have initial map at hand.
Second, the initialized map To can then be refined by either using pair- descriptors 48
36 26
52
14
38 22
and solving:
T arg min [ (T', T
Wxll,
(1)
or by using vertex-wise descriptors
37
31
26
28 and solving:
T arg min (M Qv (x
T
Where in equation WM TT-1) E (R))xn the pair-wise descriptors of vertices for shape M with its rows and columns aligned using the map T and the previous iterations map T'- respectively. Wy is the pair-wise descriptor for shape N. Here for conciseness for aligning rows columns will use T (while in reality if the row map is T argmaz(P, dim=-1) 1) the column should be T orgmar((P, dim=-2) where P is the permutation matrix such that PQM = QN and PWMPT = -WA
While in equation 2 (T)(T".:) Raxd is the vertex-wist descriptors for shape M with feature dimension d for each vertex and with its rows aligned according to the map T f() is a function (such as functional map 34)) to transfer the previous iteration's map to the descriptor space, and QN E R Fond is the vertex-wise descriptors for shape A.
For the map T refined by equations Torl 2 to have high accuracy (in the ideal case), the pair-wise descriptors (i, E I R or vertex- wise descriptors Qv(i,:) :) Ild will need to be: (a) unique for each vertex shape N, and (b) identical to its ground truth corresponding pair-wise feature WM((((), R map. Morcover, for smoothness, vertices should be mapped such that they remain in their relative neighborhoods before and after the mapping.
vertex- wise feature Q. :) on shape M. Where i is the ground truth
3 Related Work
ZoomOut
3 and other related
26 35 all aim for smoothness of the
map and focus on the settings in which the shapes have undergone isometric deformation, such

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uniform laplacian 2
LBO 2
2-hop n-100
6-hop of n=100
SHOT 1-ring geodesic n=2
For sample shapes from the TOPKIDS (first row) and TOSCA (second row), this figure shows the second eigenvector of the mesh laplacian (LBO 2), the second eigenvector of the uniform shape laplacian from the triangulation adjacency (uniform shape laplacian 2), the 2-hop and 6-hop neighborhoods of vertex 100, the second SHOT descriptor and the LMD of vertex 2.
Figure I:
that geometric shape properties are preserved. These works use vertex-wise descriptors usually based on some truncated (reduced) basis of specific shape properties, generally leading to some loss of information such as uniqueness. For example, consider the d dimensional vector (i,;:) E TRd
as the vertex-wise descriptors of vertex i, supposing that the matrix U E i@axd contains the first d eigenvectors of the uniform shape laplacian built from A, it can be seen that U(i;:) is the soft cluster assignment of vertex i, and as such most vertices in the same cluster as é will have similar vertex- -iie descriptors. This can be seen in figure where the second eigenvector U(: shown for example shapes from TOSCA and TOPKIDS. One can see that (4(;, 2) E R" indeed a soft cluster assignment grouping some vertices in the same cluster together. This explains the success in achieving smoothness by some methods that use the d dimensional spectrum of the shape laplacian.
R" is
Given the shape descriptor U(i,:) E Rd as the vertex-wise descriptors of vertex i, supposing that the matrix E nxd contains the first d eigenvectors or left singular vectors of some unique shape pairwise descriptor W. Using W for the map refinement via Functional maps helps group nearby clusters logether assuming the functional map is perfectly accurate.
Theorem 3.1
Proof 3.1 Recall that given a map T and two vertex-wise descriptors UM and UN both E Rrad the functional map C E Raxd is obtained as:
C'= -3) uxc
This functional map can then be used to converi functions in the busis of shape M into those of shape N and vice versa. This is then used to refine the mup via solving equation] 2 where [(T+) = C². One see that Ux converts the soft clusters (left singular vectors first eigenvectors) of shape Ux into the basis space of the descriptors (soft clusters) of shape Uy. and so solving is basically aligning the soft clusters in the same basis space. Thus solving equation is matching aligned soft clusters.
which iteratively refine the map T by using an increasing number of cigenvectors of the shape laplacian can be said to be refining the map by increasingly aligning more fine-grained clusters the shapes. recovered maps T on the isometric SCAPE(20 dataset are generally smooth (seen by the rapid increase of the geodesic curve), they are generally not accurate due to the non-uniqueness of the descriptors it uses to refine the initialized map (seen by the proportion of vertices at geodesic error 0).
In fact, methods such as ZoomOut [31 20
shows that though ZoomOut's
Moreover, though geometric properties may be preserved even for some mild non-isometric deforma- tions, they usually are not preserved when the transformation deviates significantly from isometry. For example, consider the shape Laplace Beltrami Operator (LBO) on which many vertex-wist and wise descriptors are built:
L= L=
3

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Source
GEM
GRAMPA
ZoomOut
HOPE
SHOT
GEM GRAMPA
DAR
6:56
6.15
gredree
Source
GEM
GRAMPA
DIR
ZoomOut
HОPE
SHOT
10-0EM
HOPE SHOT
Figure 2: Showing mating of sample shapes from the isometric SCAPE dataset (first row) and the
non-isometric TOPKIDS (second row), this figure shows the performance of different baselines.
where V is the diagonal mass matrix with V(i, i) entry along the diagonal being the vertex area of vertex i, and the matrix S being defined as:
(coloss + Bog Exeneigh(i) St.k
jE neigh(i) i=k
S
(4)
otherwise
where neigh(() denotes the neighborhood of i. It can be observed from the way the LBO is defined that non-isometric deformation which changes angles, areas, even merges part of the shape (such as gluing hands to face for example) will change the LBO and by implication its spectrum, hence changing the vertex-wise and pair-wise descriptors based on the LBO. Figure shows the poor performance of DIR 49] and ZoomOut []] (which use the spectrum of the LBO as vertex-wist descriptors) pair of sample shapes from the non-isometric dataset TOPKIDS 26 (which contains shapes with topological noise).
Other methods aim for accuracy generally not smooth since they generally do not capture neighborhoods and as such do not match neighborhoods. Figure shows the first SHOT 45) descriptor and figure 2 show that while generally provides an accurate map 45 (seen by the proportion of vertices at geodesic error 0) the map is not smooth (as seen by the lack rapid increase the geodesic curve). That why methods aiming for smoothness [31 37 [22 [7] only use these unique descriptors for initialization and then use the LBO or other smooth basis for refinement.
unique descriptors. However, these descriptors
Others have used the 1-hop 14 or 2-hop 52 22 neighborhoods of vertices as witnesses to im- prove initialized map. Though these methods aim for both accuracy (via uniqueness of vertex neighborhoods), and smoothness via using neighborhoods as witnesses to improve the map (match- ing neighborhoods that agree best), they are nonetheless affected by the fact that 1-hop 2-hop neighborhoods may not be unique enough and distant vertices may have similar 1-hop and or 2-hop neighborhoods especially in the presence of symmetries A. For example, consider GRAMPA [14] which initializes the map by using a graph matching kernel in the full graph spectrum and then uses the I-hop (or 2-hop as this figure) neighborhood to refine the map. Figure shows GRAMPA [14] performing well on non-isometric shapes from TOPKIDS where there are few symmetries (and other graph properties) which normally cause the neighborhoods of the vertices to be non-unique. However, figure shows that GRAMPA struggles on a pair of shapes from the nearly isometric dataset SCAPE where shapes have symmetries (and other graph properties) that normally cause the neighborhoods of the vertices to be non- non-unique. To address this challenge, 2D-GEM[22 proposed to use the concept local map distortion LMD laplacian. However, though 2D-GEM seems perform well on non-isometric and isometric shapes (figure 2D-GEM does not generalize, but rather significantly changed their model (depending on user-defined parameters) in order to handle isometric cases discussed in Section
48 and the spectrum of the uniform shape

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4 HOPE
To address both accuracy and smoothness for correlated meshes undergoing isometric and non- isometric deformations we propose HOPE (k-HOP niEghborhood matching). HOPE is based on the iterative refinement of an initialized map by: (I) using the local map distortion (LMD) 49 S to identify poorly matched vertices, and (2) using noise robust k-hop neighborhood based descriptors for refinement of the maps of these poorly matched vertices.
This section is organized as: (a) the local map distortion, (b) k-hop pairwise descriptors, and (c) iterative refinement pipeline and algorithm.
4.1 Local Map Distortion (LMD)
Let : .M- MM-N W be a map between two shapes. The LMD 49 22 given follows:
the map T at the vertex z; is
(x(T)( EmjeB,(as) VM(i)DET(Ss.
(5)
Enje8,(a,) VM(i)
where (x:) (x; M d S the 7-geodesic ball of z4. VM is the area element of the mesh of shape M, and (E;( = (x; - dy wise distance distortion of mapping nearby vertices and Tj to T (x:)) and T T(zy). A smaller value of (T)(() means a better map continuity of T at the vertex I,, in other words, the local distance at the point z; is well preserved. Based on the above definition of LMD, one can check that if T is an isometric map, then D. (z;) = O, Va, E M, > O. Conversely, if D., (Z) = 0, Va, E M the LMD find well-matcher pairs (Imks) i...,
(xx)))// represents a pair-
for some 7 > 0, then T is isometric. We Imks = (z,, (P, S <), where e a threshold, We fix e to be the same values for all dataset for all our experiments. We then call the rest of the vertices shape M of lmks as non lon-landmsrks Nlmks (i.e., poorly mapped vertices).
4.2 K-Hop Pairwise Descriptors
Given an initial map To matching fraction 33 of the vertices n correctly and the fraction - B incorrectly, has been shown that one can refine the map by (a) using the I-hop (29 neighborhood contained in the graph adjacencies of the triangulations AM and Av. (b) using k-hop [52] neighborhoods contained binary matrices AM,E and Av, Rax* indicating whether vertices
connected at path of length k on the shape mesh adjacencies (note that & = 1 is the adjacency). The refinement is done by solving:
T arg mag T
T ())
(6)
the main assumption being that 2 the meshes M and N are correlated i.e., their adjacencies are assumed come from the same parent graph, with their correlation ratio being s. Specifically, given ((n, p) Erdos Regny parent graph and the correlation s, 1 is the probability of independently randomly deleting edges from this graph to either obtain the adjacency of M or that of N. For k
and k 2, rigorous analysis and tight bounds were given for recovering the ground truth map 7 (for
given values of s, p and (3) by solving equation 6 (see (0, 52 29
for exact bounds).
Theorem 4.1 Given the k-hop based descriptors AM,K and AN,s solving equation 6 is matching vertices whose neighborhoods agree best under T
Proof 4.1 Notice that:
columns of A & andAN &s are descriptors for the vertices (rows), and these columns are the k-hop neighborhoods of vertices (rows)
Tt-1 is used in equation first aligning these descriptors i.e, AM, TT-1) rearranges the columns of AM, thus realigning the descriptors of each vertex (the rows),
each entry J) in the product == AMal;, Tt-1)AN will be the number of vertices that are common in the k-hop neighborhoods of vertices and j. after the alignment of the k-hop neighborhoods of vertices in M by AM, T
5

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M (2-ring gendestes)
M ho-hop)
M (HOT))
Build Features
Initialize Map
Re Map
N (suur)
ie hap)
N (2-rinn geodostes)
Figure 3: HOPE (a pipeline) for shape matching.
finding T via equation 6 matches the vertices i, and j whose k-hop neighhorhood have most vertices in common based the alignment T
However, the presence symmetries the next I-hop and 2-hop neighborhoods may not unique enough as even distant nodes may have the same 1 and or 2-hop connectivity. To address this non- uniqueness, we follow 2D-GEM, and use the LMD to first detect Nimks vertices. Unlike 2D-GEM which used the laplacian spectrum to refine the map T T(Nlmks) for Nimks vertices, we instead propose to iteratively use different k-hop neighborhoods for refining the map T(Nlmks) for Nlmks vertices, i.e., iteratively using nodes at different lengths & to refine T(N (Nmi.s). We propose this strategy because: (b) first, 2D-GEM's strategy of using the spectrum will suffer from non-uniqueness according theorm 3.1 and will also not adapt non-isometric shapes discussed in Sections a.4 (b) Second, according to theorem 4.1 our strategy will ensure that T is consistent across different neighborhoods. We do for 1 S K kinax where kmaz is large. This strategy is inspired by the fact that it was shown that large neighborhood statistics are essential in graph matching
[51 30 52 29
10 32
and other graph tasks [18 47 21 22 53
4.3
Pipeline for HOPE
Here we show the general pipeline for HOPE which consists of the aforementioned steps, namely:
(a) Map initialization: use any robust map initializations e.g., the map initialized from 45 in our experiments
(b) map refinement for iteration steps consisting of:
Using the LMD 22
49 48 to detect the poorly matched Nlmks vertices
refine the map for the Nlmks vertices via enforcing different noise robust k-hop neighborhoods consistency (for S k S -man)) by solving:
T'(lmkss) arg max Tr TT(AAM T ( AN,,),
(7)
(c) returning the final map T
4.4 HOPE vs 2D-GEM
this section, we briefly introduce 2D-GEM 22 and highlight some key differ- ences between 2D-GEM and HOPE. Like us, 2D-GEM initializes its map via 4HOT

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Source
and then at each iteration t 2D-GEM refined the map by:
2D-GEM
HOPE
(1) updating the map T by solving one iter- ation of equation 6 using the 2-hop neigh- borhoods, meaning
vedeesic Sivor 0.3
HOPE
2D-GEM
2 in equation (2) finding the imks well-matched pairs. and poorly matched pairs Nlmks pairs us- ing the LMD.
Figure Comparison on TOPKIDS showing HOPE using e in 5.33 for its L.M) threshold (same as in the isometric cases), against 2D-GEM with set as in the isomet- ric case (i.e., using the isometric parameters in 22 rather than = 100 as in the non- isometric case).
(3) using the (mks vertices to update the map of the Nlmks vertices in the spectrum by using the GMWM [52] on cost matrix built from the spectrum of the laplacian of the two shapes (see their paper for more details). They showed similarity between this approach and the functional map 34 approach.
The main two differences between 2D-GEM and HOPE are: (a) HOPE completely removes their step (1), and (b) they used the laplacian (LBO) spectrum via their proposed 2D-graph convolution to refine the map for the Nimks in step (3). However, mentioned in section and shown in figure using LBO descriptors loses uniqueness since vertices in similar clusters will be grouped together, and in addition, these descriptors not robust non-isometric deformations as discussed in Section 3
A consequence of this is that, 2D-GEM 22 needed to deactivate step (2) and (3) of their algorithms (the Nlmks refinement step) by setting 100 as their LMD threshold when dealing with non- isometric shapes. Hence though effective, their algorithm can be seen as two disjoint algorithms that are used for isometric or non-isometric shapes based on the setting of But in real life, in some cases, it is not obvious whether the deformation used is isometric propose refining the Nlmks map based on the preservation different k-hop connectivity which is a constraint that holds for isometric and non-isometric shapes with correlated triangulations [5] 30 52]
not. On the other hand,
Figure shows comparison between HOPE and 2D-GEM on a non-isometric pair shapes from TOPKIDS using the 2D-GEM parameters proposed for isometric shapes. It can be seen that when we use the isometric parameters of 2D-GEM on non-isometric shapes, 2D-GEM fails to perform well (the same holds when using their non-isometric parameter 100 on isometric shapes.
4.5 Time complexity analysis
See the algorithm for HOPE in Appendix A Given that for step (1) HOPE follows DIR D9R[49 and [22] in using LMD, checking the LMD takes O(n). The GMWM used to solve step (2) of HOPE takes O(INimss (Pogn).. Hence the total time complexity of HOPE for all iterations is
[tt[NNikks[/loge + n)).
5
Experiments
We report experimental results that validate the effectiveness, efficiency, and generalization ability of HOPE in the matching of nearly and non-isometric 3D shapes.
5.1
Experimental Set-up
All experiments are conducted Matlab 2023 on a Windows 11 system with 32GB RAM and Intel(R) i5 13500 CPU @ 2.50-4.8GHz.
5.2
Datasets
We evaluate the performance of HOPE on two nearly isometric benchmark datasets TOSCA 77]
and SCAPE (3), as well as on the non-isometric dataset SHREC' 16 (TOPKIDS) (26). TOPKIDS
7

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contains 25 shapes of the same class with up to 12K vertices, undergoing near-isometric deformations in addition large topological noise (such as merging hands to thighs) which results in n.M F n.y- TOSCA consists of 80 shapes in 8 different categories (human and animal shapes) with vertex numbers ranging from 4k 50k. SCAPE has 71 shapes (12,500 vertices for each) of the same person with different poses.
Furthermore, to see the generalization abilities of HOPE, we equally used datasets SCAPE_rD ( FAUST_r(9), and TOSCA_ (_r which are remeshed shape datasets. The SCAPE consists of the same 71 shapes from the SCAPE dataset, but remeshed, while the FAUST (TOSCA r) likewise contains the same FAUST (TOSCA) datasets but remeshed. We used the same 71 test pairs SCAPE_r as for SCAPE, and for FAUST r and TOSCA we followed [
5.3 Evaluation Metries
We use the geodesic error as our error metric [4] 22 Given that the map of algorithm maps a; M to Iy E E,, and the true map maps 2; to z.. the geodesic error is defined as e(x;) = dw(z,,2)) where dy denotes the geodesic distance N,, and diom(N) is the geodesic diameter of .V.
)Vam(V))
5.4 Baselines
Following 2D-GEM 12 we compare HOPE with the following methods: EM
42 GE 25 RF
40 PFM [39], FSPM (28), Kernel-Matching (26), and GRAMPA [4] [22] SGMDS[], FM B4].
24) Mobius 27). Best- Dest-Conformal [24). Kernel-Matching (26]. DIR-500
44 which uses 500
cigenvectors, DIR- 1000 49
which uses 1000 eigenvectors.
5.5 Parameter Settings
On ZoomOut, we start with functional map using the first 20 eigenvectors of the LBO, then we iteratively add an eigenvector until we reach the 120th eigenvector after which we stop.
On all other Baselines, we follow the settings from [22].
On HOPE, all datasets set the LMD threshold staring from 100 and 10 equally spaced values to c <= 0.2 i.e., we use c = linespace(100) 0.2, 10) and we set f = 60. When the last value of is reached, it is maintained for the rest of the iterations. We equaly set Kmax for all datasets. For the LMD, we used the second ring neighborhood following (49 (2).
5.6
Performance Analysis
100
% Correponndncess
80
Correspondencer
80
SGNDS
FM
% Correspodences
30
SEMCS
60
RF
FSPM
PFM
60
Best Conformal
Mobius
60
BM
Vobies
40
20-GEM
KermiMatch
40
20- 20-GEM
KamelMaach
40
240-1030
GRAMPA
249-00
HOPE
20
DR-530
20
ZoomOut
ZeceOut
DIR.500
ZoomOut
X00EN
0
0.05
0.1
0.15
0.2
0.25
0.02
0.04
0.06
0.08
0
8:02
0.04
0.06
0.08
0.1
geodesic error
geodesic error
geodesic error
(a) TOPKIDS
(b) SCAPE
(c) TOSCA
Figure 5: Performance comparison on TOPKIDS 5(a)
SCAPE 5(b) and TOSCA 5(c)
Comparison on non-isometric shapes. On the dataset with topological noise TOPKIDS (Figure ((a) the top 3 methods are the noise-robust methods 2D-GEM with 96. 96.7% vertices correctly matched, followed by HOPE with 94.9%, and GRAMPA with accuracy 84.5%. This shows that 2D-GEM, HOPE, and GRAMPA are indeed adapted for non-isometric shapes. Moreover, discussed in

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22]. GRAMPA is very time-consuming as it needs the full basis of the graph adjacencies of the shapes, while HOPE and 2D-GEM do not. For example on shape with 12500 vertices from SCAPE, GRAMPA takes roughly 118 seconds on our set up while 2D-GEM and HOPE take 56 and 52 seconds respectively. Better still HOPE a general model as it can be seen that with the same parameter of (see Section 5.5) for its LMD, it performs well on all datasets.
76 0oComeependeencco
20 GEM ZoomOut
% Cocorepeoeeeecco
% Coomsspondeeee
GUVG
XD-GEM
(49-50)
HOPE
Toorruur
48-
48-
HOFE
28-
0.95
1.2
025
goodesic arror
grodesic arror
geodesic
(a) FAUST_
(b) SCAPE_r
(c) TOSCA_r
Figure 6: Performance comparison on FAUST 6(a)
SCAPE_ 6(b) and TOSCA 6(C)
Comparison on isometric shapes. isometric shapes, methods that enforce stronger geometric constraints on their refinement pipelines such as 2D-GEM (with appropriate parameters), DIR, and HOPE. HOPE outperform all other baselines. 2D-GEM achieves 92.54% accuracy at geodesic error O on the TOSCA dataset, well 74.8% accuracy the SCAPE dataset, while HOPE achieves 80.11% on SCAPE and 92.54% on TOSCA using no matrix decomposition. Third is DIR- [000[49] with 1000 eigenvectors which achieves around 69.5% accuracy at geodesic error 0 on SCAPE, and 59.8% accuracy at geodesic error 0 on TOSCA. Moreover, unlike DIR (and 2D-GEM.), HOPE generalizes to this isometric setting using the same parameters (see Section 5.5).
Comparison on remeshed shapes. On the remeshed datasets FAUST SCAPE r and TOSCA in figure methods that enforce stronger geometric constraints such DIR performed poorly, while those that are more robust to noise such as 2D-GEM (with appropriate parameters), ZoomOut and HOPE perform relatively well. Though 2D-GEM does not outperform ZoomOut in this setting (probably due to differences the mesh connectivity), HOPE still outperforms ZoomOut though by smaller margin than datasets with similar mesh connectivity amongst pairs.
3500
3003
HOPE
time (ecomdde)
ZoomOat
2000
1500
1000
500
0
shape number
Figure 7: Time Comparison on TOSCA showing HOPE, 2D-GEM, and ZoomOut.
Time Comparison. Here we compare the time usage per shape for both 2D-GEM, HOPF, and ZoomOut on the TOSCA dataset. It can be seen from figure 7 that HOPE is relatively faster than 2D-GEM, and even ZoomoOut (for the given parameters of ZoomOut that we used for our experiments).
6 Limitations
The main limitation of this work the reliance on correlated triangulations between the pair of shapes to be matched as outlined in Section 4.2 One can see this drawback on figures which show

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that though HOPE still performs well on remeshed shapes (shapes where the triangulations are not very correlated), and even outperforms other baselines, it nonetheless does not perform as well as on figures ((b) ((a) and [(c) where the triangulations of the pairs matched are strongly correlated, This a major drawback because triangulations are often hard and expensive to get, especially consistent triangulations between shapes.
7 Conclusion
We introduced an effective and easy-to- -osplee-mmpeeeet map refinement strategy consisting of; (a) detecting poorly matched vertices (nodes) using the concept of local map distortion (LMD), and (b) improving the map of these poorly matched vertices via noise robust k-hop pairwise descriptors. We then conducted a series of experiments show that our framework is effective and generalizable to different shape datasets. We also discussed the main limitation of our work (the fact that it is reliant on the triangulation consistency between the shapes matched).
Broader Impact
This work proposes map refinement strategy for shape matching that is based on matching different k-hop neighborhoods of vertices. It then validates the effectiveness of this strategy on several shape matching datasets. This can spark new research on the importance of large neighborhood statistics for shape matching and other related tasks. As we focus solely proposing a framework for map refinement for shape matching, we not see clear negative impact of this work.
Acknowledgements
This work partially supported by Hong Kong Research Grants Council under the Areas of Excel- lence Scheme grant AoE/P-601/23-N.
References
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EHOPE: Shape Matching Via Aligning Different K-hop Neighbourhoods Appendices:
A HOPE Algorithm
In this section, we present the algorithm for HOPE. This is given in algorthm 1
Algorithm HOPE
0. input An initial map T distance matrices dı and dnc LMD threshold e, maximum iteration and the maximum hop kimaz and while 0 < i S do
1. build AM,A and AM,
2. set K
if kmaz then
3. k-
end if
4. Given T', use the LMD locate well-matched points Imks and poorly- motchy- points
Nlmks,
5. Use these Nlmks pairs to update the (Nllmss by using the GMWM solve equation end while return T
B Ablation and Parameter Sensitivity Studies
In this section, we conduct parameter and ablation studies. We use the following settings:
HOPE: the hope algortithm as algorithm ! with t = 60 and c (1nespace(100,, 0.2, 10) in the experiments in Section in the main paper,
HOPE-M: where we reduce the number of iterations to / = 20 in algorithm
HOPE-th: where set linespace 0.2, 10) in algorithm |
HOPE-fixhop: where we simply solve equation 6 with & and alternatively iteration as algorithm
HOPE-L.DDvaryhopp where use kmar = 2 in algorithm 1
HOPE-varyhop: where we simply solve equation 6 with k = [1, (1.,,..., ,,, alternatively per
iteration, in algorithm 1
It can be observed from figure that the best model overall is HOPE-th which is best isometric (figure ((b) shapes and non-isometric shapes (figure ((a) and performaing comparatively to other variants remeshed shapes (figure [(c)). This is tollowed by HOPE. This indicates that the we used for our main experiments in Section 5 is sub-optimal since we randomly selected the range to be linspacce (100, 0.2, 10) without any parameter tuning demonstrate the effectiveness and generalizability of HOPE. We notice that the variants without the L.D (HOPF-varyhop and HOPE- fixhop) all struggled the isometric shape (figure 8(b)) validating observation that the k-hop neighborhood of nodes may not be very unique especially when there are symmetries, and as such using stronger constraint like the LMD beneficial in such settings (Section and
C Different Initializations
In this section, we conduct studies on the effects of different initializations on HOPE, We use the following settings:
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HOPE-SHOT: where we use SHOT 45
descriptors to initialize t in algorithm |
HOPE-HKS: where we use Heat Kernel Signatures (HKS)[8) descriptors to initialize 1 in algorithm [
HOPE-WKS: where we use Wave Kernel Signatures (WKS) 4 descriptors to initialize ( in algorithm !
Figure shows that indeed SHOT 45 which is commonly used in practice is robust and good descriptor to use as initialization. On the other hand while both the HKS [8 and WKS 44] provided an initialization that could be enhanced by HOPE relatively well on the isometric shape (figure((()). only WKS and SHOT were suitable for the remeshed shape (figure [c)). while only SHOT initializations where relatively okay in terms of accuracy the non-sometric shape (figure (a))
D Partial Shape Matching
In this section, we show the performance of HOPE on partial shape matching where we match the full shapes to the partial shapes. Here we used the SHREC16 HOLES and SHREC16 CUTS following Cao et al. [9]. Figure T0 shows that although HOPE was not designed specifically for the partial shape setting, nonetheless performs relatively well on average seen by the average geodesic curve.
E Non-Isometrie Shape Matching
In this section, we show the performance of HOPE on another non-isometric shape dataset SMAL_ Cao et al. 19], where perform intra-class matching of 298 pairs of shapes. Figure [1] shows that HOPE again outperforms other baselines even when the mesh triangulations are not strongly correlated (due to noise).
F
Visual Comparisons
In this section, we show the visual comparisons between HOPE and other baselines, on SCAPE (figure SHREC16 (figure [3). and TOPKIDS (figure 14).
15

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Source
HOPE-M
HOPE4h
HOPE-fixhop
HOPE-varyhop
HОPE
HOPE-LMDvaryhop
100
% Correspondencess
80
HOPE-th
60
HOPE-fixhop
HOPE-waryhop
40
HOPE
HOPE-LMDvaryhhp
20
0.1
0.15
geodesic error
(a) TOPKIDS
Source
HOPE-M
HOPE-th
HOPE HoPh-fixhop
HOPE-waryhop
НОРE
HovE-LMDvaryhop
% Correspondences
80
HOPE-M
HOPE-th
HOPE-fixhop
HOPE-varyhop
40
HOPE
HOPE-LMDvaryhop
20
0.05
0.1
0.15
0.2
0.25
geodesic error
(b) SCAPE
Source
HOPE-varyhop
HOPE
HOPE-LMDvaryhhp
193
HOPE4h
HOPE-SSMR
HOPE
HOPE
grodeste orrer
(c) T
Figure 8: Ablation and sensitivity studies TOPKIDS 8(a)
SCAPE 8(b)
and TOSCA 8(c)
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HOPE-HKS
HOPE-WKS
HOPE-SHOT
100
Source
% Correspondences
80
HOPE-WKS HOPE-SHOT
60
0.05
0.1
0.15
0.25
geodesic error
(a) TOPKIDS
HOPE-HKS
HOPE-WKS
HOPE-SHOT
Correspondencess
80
HOPE-HKS HOPE-SHOT
60
40
0.05
0.1
0.25
geodesic error
(b) SCAPE
HOPE-HKS
HOPE-WKS
HOPE-SHOT
Comesponndncee
HOPE-HKS HOPE-WWKS HOPE-SHOT
Source
0.15
0.25
geodesic error
(c) TOSCA_r
Figure 9: Different initialzations on TOPKIDS (a)
SCAPE 9
and TOSCA_1 (c)
% Correspondences
% Correspondeencess
20-GEM
DIR-500
ZoomOut
DIR-500
ZoomDut
HOPE
0
0.05
0.1
0.15
0.2
0.25
0.05
geodesic error
geodesic error
(a) SHREC16 HOLES
(b) SHRECI6 CUTS
Figure 10: Performance comparisons on SHREC16 HOLES 10(a) SHRECI6 CUTS 9(b) and
T 10(b)
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100
% Correspondences
80
60
20-0EM
DIR-500
40
20
0.05
0.1
0.15
0.2
0.25
geodesic error
Figure 11: Comparison on intra-class matching on SMAL using 298 pairs of shape. Figure showing HОPE, 2D-GEM, ZoomOut, and DIR.
ROOR
GEM
Figure 12: Sample shape from SCAPE_r. Figure showing HOPE, 2D-GEM, ZoomOut, and DIR.
Source
GEM
GRAMPA
ZoomOut
HOPE
SHOT
Figure 13: Sample shape from SHREC16-cuts. Figure showing HOPE, 2D-GEM, ZoomOut, and DIR.
GEM
GRAMPA
DIR
ZoomOut
HOPE
SHOT
Figure 14: Sample shape from TOPKIDS, Figure showing HOPE, 2D-GEM, ZoomOut, and DIR.
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Reviews:
[{"review_id": "HbbN8KoNNw", "reviewer": "Reviewer_dry5", "summary": "The paper focuses on the shape-matching tasks and proposes a shape-matching refinement technique based on the K-hop neighborhood descriptor and local map distortions. The experiments show improved results compared to existing approaches in isometric and non-isometric shape registration.", "questions": "The paper suggests using geodesic error as a loss function to improve smoothness and accuracy. Given that other methods also utilize geodesic error, what aspects of the HOPE approach allow it to leverage geodesic error more effectively and potentially yield superior results? How does this approach manage the balance between local and global geometric consistency?", "limitations": "The method might face challenges in handling topological variations between shapes.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper presents the background, theory, and methodology very well.\n- The methodology is simple and sound and can positively impact shape-matching refinement.\n- The performance, runtime, and generalizations are promising.", "weaknesses": "- While there is existing work on deep and modern feature extraction, the experiments in this paper are limited to the SHOT descriptor. Incorporating other classical or learned signatures could demonstrate broader generalizable performance. Additionally, evaluating the impact of the quality of initial matches could provide further insights into the robustness of the proposed method.\n\n- The paper is motivated by the observation that existing approaches often prioritize smoothness over accuracy. Although this paper emphasizes improving accuracy, it raises the question of whether smoothness might be compromised as a result. Detailed analysis and results are needed to assess the balance between these two aspects.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the Reviewer for the insightful and constructive comments. We address some of the Reviewer’s comments and suggestions below:\n\n# Weaknesses:\n- W1: While there is existing work on deep and modern feature extraction, the experiments in this paper are limited to the SHOT descriptor. Incorporating other classical or learned signatures could demonstrate broader generalizable performance. Additionally, evaluating the impact of the quality of initial matches could provide further insights into the robustness of the proposed method. \n- RW1: We refer the reviewer to Appendix C where we equally use the HKS and WKS as initializations. We added more qualitative results in the attached PDF to the global response and will add them to the appendix as well.\n- W2: The paper is motivated by the observation that existing approaches often prioritize smoothness over accuracy. Although this paper emphasizes improving accuracy, it raises the question of whether smoothness might be compromised as a result. Detailed analysis and results are needed to assess the balance between these two aspects. \n- RW2: Mathematically, an accurate map will equally be smooth (in case the map recovered is same as the ground truth for example). But as the Reviewer points out, since most models may not perfectly recover this ground truth map, there will be a tradeoff between aiming for smoothness and accuracy as discussed in Section 3 which shows the difficulty to maintain smoothness while aiming for accuracy by methods such as such SHOT, since it does not strictly enforce smoothness. A more precise and general response will be that the level of smoothness of the final map obtained by an algorithm (aiming for accuracy) will depend on how it enforces this smoothness constraint as while aiming for accuracy. For example, in the case of HOPE, one can observe that enforcing local and global neighborhood consistency in order to aim for accuracy also enforces the smoothness constraint since points will be matched if their local and global neighborhoods are similar. We will add this discussion in the Appendix. \n\n# Questions:\n- Q1: The paper suggests using geodesic error as a loss function to improve smoothness and accuracy. Given that other methods also utilize geodesic error, what aspects of the HOPE approach allow it to leverage geodesic error more effectively and potentially yield superior results? How does this approach manage the balance between local and global geometric consistency? \n- RQ1: We used the geodesic error as a measure of accuracy (via measuring the %Correspondences at geodesic error 0) and smoothness (by seeing how rapidly the %Correspondence increase as we move away from geodesic error 0).  In order to achieve this accuracy we employed the concept of neighborhoods to refine an initialized map. We used both local (lower hops) neighborhoods and global (higher hops) neighborhoods to match points. When the local and global neighborhoods of points were similar (based on some initial map) we matched them together. The reviewer indeed raise an interesting future research question on how to balance the importance of local consistency with that of global consistency. We briefly discussed this in Section 3 where we highlighted the reasons on why GRAMPA failed in matching isometric shapes where the local neighborhoods of the meshes are similar. It has been studied in graph isomorphism (with models such as the Wesfeiler Lehman), and even in the field of Graph Neural Networks (in papers such as “How powerful are graph neural Networks” or “Topological graph Neural Networks”) that local neighborhoods being similar does not necessarily imply a match. For example, if in the neighborhood of node A, two of its neighbors are white and two are black, and in the neighborhood if node B the same class distribution is observed, even if the connectivity in these neighborhoods are the same, it does not necessarily mean that these two nodes are the same as their 2 hop neighbors may be significantly different. We will add discussion on this in the main paper or the appendix as well.\n\n# Limitations:\n- L1: The method might face challenges in handling topological variations between shapes. \n- RL1: Indeed the main limitation of our work is that we rely on the mesh connectivity as mentioned in Section~6", "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": "XW9OtlY4WN", "reviewer": "Reviewer_E6Vo", "summary": "This submission proposes a descriptor utilizing k-hop neighborhoods for non-rigid 3D shape matching. The descriptor is used jointly with local map distortion for map refinement. \n\nOverall, I am highly frustrated during reviewing this submission. While I have tried my best to parse and understand the technical details, the confusing formulations and derivations, together with missing definitions have prevented me from doing so. Overall, I would encourage the authors to revise carefully the submission, so that it is readable.", "questions": "1. I would disagree with the argument in L27. Shape matching methods would rarely aim for solely smoothness without accuracy -- the latter is the most important, while the former can serve as good regularization. \n\n2. Should niegh(i) in Eqn.(4) be neigh(i)? \n\n3. In Fig.4, I can not see the curve regarding HOPE. It would be useful to report geodesic error in the legend.\n\n4. If I understood correctly, the appendix should be behind checklist.", "limitations": "It looks fine to me.", "rating": 2, "confidence": 5, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "To be honest, I have trouble reading out the paper, therefore I can hardly conclude any strength for real.", "weaknesses": "The paper lacks readability, especially on the technical part. An incomplete list of frustrations I have encountered is as follows: \n\n1. The presentation is poor. Starting from Eqn.(1), I have got lost. There has never been an explicit formula or at least a clear reference to the prior works for W_M(T^t, T^{t-1}). Similarly Q_M, Q_N are not defined. I have tried my best to parse the statements, but have to give up in the end. \n\n2. Theorem 3.1 and its proof are also confusing. First of all, Q_M and Q_N in proof 3.1 are *not* mentioned in the theorem. Second, functional maps by its definition is defined regarding the basis, instead of some arbitrary vertex-wise descriptor. However, I did not anything around L95 saying Qs are eigenbasis (or any kind of basis regarding C). \n\n3. In Eqn.(6), the definition of A_M, k(T, T^{k-1}) is again undefined. \n\n4. The proof of Theorem 4.1 is pointless. Please use the standard mathematical derivation, rather than piling up a series of unverified statements.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the Reviewer for the clarification seeking comments. We sincerely apologize to the reviewer for any confusion or misunderstanding. We will strive to the best of our ability to address all concerns. However, considering the good scores of other Reviewers, we strongly plead with the Reviewer to please reconsider the reviews and to please ask us any further questions that may help in this regard:\n\n# Weaknesses:\n- W1: The presentation is poor. Starting from Eqn.(1), I have got lost. There has never been an explicit formula or at least a clear reference to the prior works for W_M(T^t, T^{t-1}). Similarly Q_M, Q_N are not defined. I have tried my best to parse the statements, but have to give up in the end. W_M(T^t, T^{t-1}), Q_M, and Q_N are all defined in the paragraph below equations 1 and 2. See lines 64-76.  \n- RW1:  Equations 1 and 2 are general formulations of the iterative refinement strategy of methods such as Kernel Matching[1], 2D-GEM[2] and GRAMPA[3], and others. For example  the minimization in equation 1 is equivalent to the maximization  argmax_{T^t} Tr(W_M(T^t, T^{t-1}) W_N), where T^t is the map we are trying to obtain at iteration t and the map T^{t-1} is the map that was previously obtained in iteration t-1 (which is done by all 3 papers namely; Kernel Matching , 2D-GEM and GRAMPA for example).\n- W2: Theorem 3.1 and its proof are also confusing. First of all, Q_M and Q_N in proof 3.1 are not mentioned in the theorem. Second, functional maps by its definition is defined regarding the basis, instead of some arbitrary vertex-wise descriptor. However, I did not anything around L95 saying Qs are eigenbasis (or any kind of basis regarding C). \n- RW2: First, we want to agree with the reviewer here that a functional map can be computed in any basis and not only in the eigen basis (see section~4 of the original functional map paper). As such, given any basis (e.g., first k eigen-vectors i.e., a truncated basis,  or any other shape basis which may also be used as descriptors), a functional map can be computed. So theorem 3.1 will still hold. For example, as mentioned in this theorem, if Q_M and Q_N are singular vectors (eigenvectors) of some descriptor and act as a basis, then the first singular vectors will act as soft clusters for the points in that basis (figure 1), as such the iterative functional map refinement will be matching soft clusters in that basis which is what Theorem 3.1 says.  However, we thank the Reviewer for mentioning this to us, it is a typo and we apologize. This should really be the Us mentioned in the introduction of Theorem 3.1 rather than the Qs used in the proof. We will fix this typo in the final version should our paper be accepted as this in no ways changes theorem.\n- W3: In Eqn.(6), the definition of A_M, k(T, T^{k-1}) is again undefined. \n- RW3: We please refer the reviewer to lines 167-172 and 173-178 where  we defined A_M, A_N, A_{M,k} and A_{N,k }.The (T, T^{k-1}) in A_{M,k}(T, T^{k-1}) is trying to recover the new map T^t that aligns the columns given the previous map T^t-1 that aligned the rows. We apologize that since we already mentioned in lines 64-70 we did not see the need to repeat thjs. We will edit to make it explicit here in equation 6 too then. \n- W4: The proof of Theorem 4.1 is pointless. Please use the standard mathematical derivation, rather than piling up a series of unverified statements. \n- RW4: We are sorry to disagree with the Reviewer’s remarks that these statements are unverified. All these statements are mathematically sound and are verifiable to the best of our knowledge. Else, we would never make them for a conference such as Neurips. Statement 1 gives a sound interpretation of the rows of the neighborhood matrices A_{M,k} and A_{N,k} as descriptors of the nodes (neighborhoods as descriptors). Statement 2, states mathematically the role of T^t-1 in A_{M,k}( :, T^t-1 ), that is to align its columns (that is reorder the neighborhood/k-hop descriptor of each node). Statement 3 gives the mathematically verifiable statement that each entry K(i,j) in the product K=A_{M,k}(:,Tt−1)A_{N,k}, will be the number of vertices that are common in the k-hop neighborhoods of vertices i and j, after the alignment of the k-hop neighborhoods as mentioned in statement 2. Thus Statement 4 logically concludes that finding T via equation 6 is equivalent to seeking to match all the vertices i, and j whose k-hop neighborhoods have most vertices in common based on the alignment T^t−1.\n\n# Questions:\n- Q1: I would disagree with the argument in L27. Shape matching methods would rarely aim for solely smoothness without accuracy -- the latter is the most important, while the former can serve as good regularization. \n- RQ1: We refer the Reviewer to our definition of accuracy as the %Correspondences as geodesic error 0. To the best of our knowledge, besides 2D-GEM, Dual Iterative Refinement and GRAMPA, most previous works do not aim for 100% Correspondences at Geodesic error 0 but rather to have this 100% Correspondences at a reasonable geodesic error as seen by even the low % correspondence curves at geodesic error 0 of most of these works. We please refer the Reviewer to 2D-GEM[1] for example who discussed this point extensively.\n- Q2: Should niegh(i) in Eqn.(4) be neigh(i)?\n- RQ2: We apologize for this typing error (typo). We will correct this.\n- Q3: In Fig.4, I can not see the curve regarding HOPE. It would be useful to report geodesic error in the legend. \n- RQ3: This is because HOPE is overlaid by 2D-GEM (zooming in one can see the pink curve of HOPE). We refer the reviewer to lines 297-302 which states “HOPE… achieves 92.54% accuracy at geodesic error 0 on the TOSCA dataset…”. Please see a modified figure in the attached PDF \n- Q4: If I understood correctly, the appendix should be behind checklist.\n- RQ4: We please refer the reviewer to the Neurips paper Checklist requirement at the link https://neurips.cc/public/guides/PaperChecklist.", "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": "dGm8NATauJ", "reviewer": "Reviewer_NAeg", "summary": "This paper introduces HOPE, a method that leverages k-hop neighborhoods as pairwise descriptors to achieve both accuracy and smoothness in shape matching. This approach aims to overcome the limitations of existing methods that often struggle to balance these two crucial aspects. The k-hop neighborhood concept involves considering the vertices within a certain distance (k hops) from a given vertex, providing a local context that can be used to compute pairwise descriptors. The authors validate their approach using several benchmark datasets, including SCAPE, TOSCA, and TOPKIDS.", "questions": "1. What is the impact of the initialization map quality on this method, and how does it compare in tolerance level with other baselines?\n2. How much does the method improve over the baseline in specific quantitative results on the SHREC16 dataset? There are no qualitative results or specific quantitative results, including geodesic error.\n3. There are no quantitative metrics provided to demonstrate the smoothness of the map.\n4. What is the value of kmax set in the experiments, and is it consistently optimal across different datasets?", "limitations": "Please refer to the Weaknesses and Questions above.", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper is well-structured, with a clear organization.\n2. The method section provides a theoretical argument result. The theoretical results demonstrate the validity of the method.\n3. The authors provide a comparative explanation with the main baseline, 2D-GEM.\n4. The method achieves good results in partial shape matching.", "weaknesses": "1. Novelty: The methods in the paper lack overall novelty, as the proposed approach involves minor optimizations and modifications based on existing work.\n2. Parameter Sensitivity: Although not explicitly discussed in the provided sections, methods that involve iterative refinement and multiple descriptors often require careful tuning of parameters. The performance of HOPE may depend on the selection of parameters such as the number of hops (k) and the weights used in the LMD optimization. A thorough analysis of the method's sensitivity to different parameters would be beneficial, possibly including guidelines or heuristics for parameter selection.\n3. Experimental Results: In the compared datasets, this method does not show a significant improvement over the baseline. The advantages of the technique are not evident from the provided qualitative and quantitative results. The paper mentions other works that address smoothness and accuracy but could benefit from a more detailed comparative analysis, showing direct performance metrics against the most recent and relevant methods.\n4. Benchmark: There are relatively few results on non-isometric datasets in the comparison. It is suggested to supplement results from other datasets, such as SMAL_r [a], DT4D-H [b], or other data in SHREC07 [c].\n[a] Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching.\n[b] Smooth Non-Rigid Shape Matching via Effective Dirichlet Energy Optimization.\n[c] http://partial.ge.imati.cnr.it.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the Reviewer for the constructive and insightful comments. We will strive to the best of our ability to address all concerns in hopes that the Reviewer can please reconcider the reviews . \n\n# Weaknesses:\n- W1: Novelty: The methods in the paper lack overall novelty, as the proposed approach involves minor optimizations and modifications based on existing work. \n- RW1: To the best of our knowledge, we are the first and only work that use different k-hop neighborhoods as pairwise descriptors to refine the map in shape matching.\n- W2: Parameter Sensitivity: Although not explicitly discussed in the provided sections, methods that involve iterative refinement and multiple descriptors often require careful tuning of parameters. The performance of HOPE may depend on the selection of parameters such as the number of hops (k) and the weights used in the LMD optimization. A thorough analysis of the method's sensitivity to different parameters would be beneficial, possibly including guidelines or heuristics for parameter selection.\n- RW2: We please refer the Reviewer to lines 163-164 and lines 284-286 where we mention that all our parameters are fixed for all datasets and to appendix B were we conduct the parameter sensitivity which show that HOPE is not sensitive to parameters chosen.\n- W3: Experimental Results: In the compared datasets, this method does not show a significant improvement over the baseline. The advantages of the technique are not evident from the provided qualitative and quantitative results. The paper mentions other works that address smoothness and accuracy but could benefit from a more detailed comparative analysis, showing direct performance metrics against the most recent and relevant methods. \n- RW3: We please recall with the Reviewer that as shown in Section 5.6 and in appendix B, HOPE performs significantly better than all baselines on 8 different datasets while using the same set of parameters with no need of tweaking.\n- W4: Benchmark: There are relatively few results on non-isometric datasets in the comparison. It is suggested to supplement results from other datasets, such as SMAL_r [a], DT4D-H [b], or other data in SHREC07 [c]. [a] Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching. [b] Smooth Non-Rigid Shape Matching via Effective Dirichlet Energy Optimization. [c] http://partial.ge.imati.cnr.it. \n- RW4: We will add and briefly expound the suggested works in the related work section. We conducted interclass matching on SMAL_r as suggested givin a total of 298 pairs averaged on the curve. Please see the results in the attached PDF.  Moreover,  we also believe that we have provided a wide range of experiments on several datasets to illustrate our point. We plead with the reviewer to look at the variety of datasets we used (up to 8 different datasets, 6 in the main paper in Section 5, and 2 more in Appendix D). So we plead with the reviewer to Review the scores again as we believe our work can indeed be very beneficial to the research community. Finally, for any other tests, we please refer the reviewer to our open-source code provided to test any custom datasets in mind (we will appreciate feedback on how our model performs on those as well).\n\n# Questions:\n- Q1: What is the impact of the initialization map quality on this method, and how does it compare in tolerance level with other baselines? We please refer the Reviewer to Appendices B, C where we discuss in detail with more experiments the parameter sensitivity, ablation studies, and different initializations. We equally please refer the reviewer to Section 5 and Appendix D where on all 6 datasets (in Section 5) and 2 datasets (in Appendix D) with the same parameters HOPE outperforms all other baselines.\n- Q2: How much does the method improve over the baseline in specific quantitative results on the SHREC16 dataset? There are no qualitative results or specific quantitative results, including geodesic error. \n- R2: For the Shrec16 dataset in Appendix D we provided the %correspondences with geodesic error curves on page 17. We apologize that it is on this page. We will move this to a better location.\n- Q3: There are no quantitative metrics provided to demonstrate the smoothness of the map.\n- RQ3: Though we did not use an explicit metric for the smoothness, as noted in lines 50-53, one can infer this by observing the rapid increase of the %Correspondence as it moves away from geodesic error 0 on the plots. We will welcome any smoothness metric suggested by the reviewer and add it to our work, since the ones we are familiar with such as conformal distortion, bijectivity, chamfer distance and so on all do not measure smoothness explicitly. Hence we resorted to observing how fast 100% Correspondence accuracy is reached as one moves away from geodesic error 0 closest one to the best of our knowledge.\n- Q5: What is the value of kmax set in the experiments, and is it consistently optimal across different datasets? Kmax is fixed to 8 for all experiments in the paper.\n- RQ5: We apologize to the Reviewer for not making this explicit in lines 284-286 (where we mention that all our parameters are fixed for all datasets). We will add this in the final version of 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": "MUmUbIOd1v", "reviewer": "Reviewer_hBzn", "summary": "The paper presents a new method for shape correspondence that extracts descriptors that are both smooth over the manifold and distinctive enough to find high precision point matching. One can view the presented method as an improvement over the 2D-GEM method, that, as opposed to the latter, does not use the eigenfunction of the laplacian and interacts with more the 2-hop neighborhood.  It is extensively tested on various benchmarks in the field, near-isometric and non-isometric ones, and presented promising results.", "questions": "see weaknesses", "limitations": "see weaknesses", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "It presented a descriptor that shows improvement over previous ones. The method is well explained. The related work section is written well, and it adds an important context to the paper. Additionally it extensively tested and presented superior results in comparison to previous methods.", "weaknesses": "The new method to some extent resembles the 2D-GEM method, which limits the novelty. As presented in the limitation, since it uses the vertex neighborhood it is vulnerable to remeshing, and it counters the statements in the paper about the method usefulness in real data scenarios (overclaiming). \nSmall issues: I think [1] should be referred to because of its similarly to the ideas presented in the paper, they also use alignment processing of the initial descriptors by reducing geodesic distance error.\n\n[1] Bracha, A., et al., 2020. Shape correspondence by aligning scale-invariant LBO eigenfunctions. 3DOR: Eurographics Workshop on 3D Object Retrieval", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the Reviewer for the insightful and constructive comments. We address some of the Reviewer’s comments and suggestions below:\n\n# Weaknesses:\n- W1: The new method to some extent resembles the 2D-GEM method, which limits the novelty. As presented in the limitation, since it uses the vertex neighborhood it is vulnerable to remeshing, and it counters the statements in the paper about the method usefulness in real data scenarios (overclaiming). Small issues: I think [1] should be referred to because of its similarly to the ideas presented in the paper, they also use alignment processing of the initial descriptors by reducing geodesic distance error.\n[1] Bracha, A., et al., 2020. Shape correspondence by aligning scale-invariant LBO eigenfunctions. 3DOR: Eurographics Workshop on 3D Object Retrieval We thank the reviewer for pointing us to this insightful work.\n- RW1:  We will add this work to our related work section, since indeed [1] also proposed to address some concerns with the LBO basis for shape matching. Specifically, [1] proposed to solve the problem caused by rotation, reflections, sign flips and other symmetries in the LBO basis by first aligning the eigenfunctions (vectors) of the LBO of two nearly isometric shapes and then using these aligned eigen-vectors as the new basis for the shape matching problem.  We also agree with the Reviewer that the limitation incurred by HOPE from using the mesh of the shape is a real one since in the real world most data is meshed independently of each other and so the meshes may not be correlated. We are trying at the moment to study this hard problem and considering how to lift this limitation in a near future work, and will welcome any suggestions from the Reviewer. However, to appease any other concerns from the Reviewer with regards to this, we recall with the Reviewer that though HOPE suffers from this limitation, our experiments in Section 5.6 specifically lines 305-310 as well as figure~5 both help to show that though HOPE does not drastically outperform other baselines on re-meshed shapes, it is still competitive by either performing comparatively or better. Moreover, we attached some more experiments on the SMAL_r (challenging dataset, see PDF attached) as well as some qualitative results on different datasets, which both show again the benefit of choosing HOPE over competitors such as 2D-GEM, DIR, or ZoomOut.", "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:
# Comment To All Reviewers
- First, we thank the Reviewer for their comments and please direct each Reviewer to specific responses to each as well as to the general responses here and the attached PDF.
- We added experiments on intra-class matching on the SMAL_r dataset as suggested by Reviewer NAeg. We hope it convinces the Reviewer about the quality of our work, as this experiment further demonstrates the choice of HOPE over existing baselines. We used the same parameters as on other datasets in the main paper since one Key benefit of HOPE is that we do not need any parameter tweaking. For this experiment, we match 298 intr-class shape pairs and report he average geodesic curves of HOPE, 2D-GEM, DIR, and ZoomOut.
- We added some Qualitative visuals on the maps from different baselines as suggested by Reviewer dry5. However, due to space we could not add on other datasets besides the three chosen. We will add more to the main paper in the appendix in case of acceptance
- We will equally add all related works suggested by all Reviewers to the main paper in case of acceptance. 
- We apologize for any miss understanding with Reviewer  E6Vo and plead for a look at our rebuttal as well as ask us any further questions as we really hope to address all concerns and convince the Reviewer about the quality of our work.

Meta-review: The paper obtained mixed reviews, with two reviewers voting for reject and two voting for accept. On the positive side, good presentation, theoretical contribution (simple and sound methodology) and good performance are noted. On the downside, limited novelty, limited evaluation, and a negative trade-off with smoothness of the resulting correspondence. The extensive rebuttal is acknowledged. In particular, we weighted down one very negative review in light of its divergent complaints regarding the presentation and lack of understandability that could not be corroborated. We urge the authors to still improve the paper regarding the raised concerns, though. Overall, we follow the positive reviews and the paper should be accepted in light of its theoretical contribution and practical advantages.
Decision: Accept (poster)