NCP: Neural Correspondence Prior for Effective Unsupervised Shape Matching
Authors: Souhaib Attaiki
Maks Ovsjanikov
Venue: NeurIPS 2022 Accept (2022)
Keywords: 3D Shape matching
Deep Learning
Neural Prior
Geometry Processing

Abstract: We present Neural Correspondence Prior (NCP), a new paradigm for computing correspondences between 3D shapes. Our approach is fully unsupervised and can lead to high quality correspondences even in challenging cases such as sparse point clouds or non-isometric meshes, where current methods fail. Our first key observation is that, in line with neural priors observed in other domains, recent network architectures on 3D data, even without training, tend to produce pointwise features that induce plausible maps between rigid or non-rigid shapes. Secondly, we show that given a noisy map as input, training a feature extraction network with the input map as supervision, tends to remove artifacts from the input and can act as a powerful correspondence denoising mechanism, both between individual pairs and within a collection. With  these observations in hand, we propose a two-stage unsupervised paradigm for shape matching, by (i) performing unsupervised training by adapting an existing approach to obtain an initial set of noisy matches, (ii) using these matches to train a network in a supervised manner. We demonstrate that this approach significantly improves the accuracy of the maps, especially when trained within a collection. We show that NCP is data-efficient, fast, and achieves state-of-the-art results on many tasks. Our code will be released after publication.

NCP: Neural Correspondence Prior for Effective Unsupervised Shape Matching
Souhaib Attaiki
Maks Ovsjanikov LIX, École Polytechnique, IP Paris maks@lix, wol.iiic..tyttcniuuu......
LIX, École Polytechnique, IP Paris attaiki@lix. potttchiii.....ttechii.i.... frr
Abstract
We present Neural Correspondence Prior (NCP), new paradigm for computing correspondences between 3D shapes. Our approach is fully unsupervised and can lead to high-quality correspondences even in challenging cases such as sparse point clouds or non-isometric meshes, where current methods fail. Our first key observation is that, in line with neural priors observed in other domains, recent network architectures on 3D data, even without training. tend to produce pointwise features that induce plausible maps between rigid non-rigid shapes. Secondly, we show that given noisy map as imput, training a feature extraction network with the input map supervision tends to remove artifacts from the input and can act as a powerful correspondence denoising mechanism, both between individual pairs and within collection. With these observations in hand, propose two-stage unsupervised paradigm for shape matching by (i) performing unsupervised training by adapting existing approach obtain an initial of noisy matches, and (ii) using these matches to train a network in supervised manner. We demonstrate that this approach significantly improves the accuracy of the maps, especially when trained within a collection. We show that NCP is data-efficient, fast, and achieves state-of-the-art results on many tasks. Our code can be found online: attps://gitann thub. com/pvnieo/NCPP
Introduction
Establishing dense correspondences between 3D shapes is fundamental problem in computer vision and computer graphics, as enables many downstream applications such as statistical shape analysis [1, 2]. texture [3] and deformation transfer [4, 5], and registration [6], to name few.
A particularly challenging setting for this task is the computation of correspondences between 3D shapes that undergo non-rigid, non-isometric deformations, and that may exhibit some partiality, such missing semantic parts, and can be represented sparse point clouds.
The standard approach is to formulate shape correspondence supervised learning problem, by relying on ground truth maps within large datasets of shape pairs. Given such ground truth, is possible to train neural networks to either produce deformation fields [7], segmentation maps [8, 9, 10] or functional maps [11, 12, 13] between the input shapes. However, all such methods rely on the presence of labeled point-to-poine correspondences, which are expensive to obtain, and are only available a handful of cases.
At the same time, unsupervised techniques try to solve the matching problem by imposing structural properties the maps, either in the intrinsic [14, 15, 16] extrinsic domains [17, 18]. However, the priors used by these methods tend to be purely geometric (e.g., promoting near isometry or divergence- free deformation fields), and, challenging non-isometric settings.
demonstrate below, not always applicable, especially in
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

Shape Collection
Approximate
Unsuservised Methodd
Stage
Refined Maps
rearai reigh por
Figure I: We present NCP-UN, an unsupervised shape matching method applicable even in challeng- ing cases non-isometric non-rigid shapes or sparse point clouds. Our method composed two stages (detailed in Algorithm 1). Namely, given a collection of shapes, vised method to obtain an initial set potentially approximate, highly noisy maps. These maps are then used supervision for learning- based shape-matching method in Stage 2. Remarkably, show that the resulting maps are of better quality than the maps used for training.
existing unsuper-
In this work, we demonstrate that the neural network itself can act as powerful prior for shape correspondence problems. For this, we first observe that even without training, recent architectures for 3D shape analysis [19, 20, 21] produce pointwise features that capture local geometry and lead to well-structured maps between shapes. Secondly and perhaps more importantly, we demonstrate an effect that call neural correspondence prior. Specifically, given noisy input maps, formulate supervised learning problem, where we aim to learn pointwise features that, when compared, would recover the input maps. Remarkably, we show that the correspondences computed by the trained networks are typically of higher quality than the input. Our results are consistent with the notion of noise impedance observed in other works neural priors pioneered in [22], and neural bias [23] which states that neural networks tend to optimize lower frequency signals before higher-frequee frequency ones. We show that in the context of shape matching, these effects imply that neural networks can learn powerful features by training to overfit to given noisy or incoherent correspondences.
Based on these observations, we develop a two-stage algorithm for unsupervised shape matching (see Fig. 1). In the first stage, adapt existing unsupervised neural network, using variants of standard methods from the literature. We then train a new network from scratch a supervised manner using the noisy maps as ground truth. We demonstrate that this second stage not only helps to denoise the maps but also improves the matching result overall and achieves state-of-the-a results on multiple tasks. We show that neural correspondence prior applicable within both collections and on individual shape pairs, to provide test time refinement of dense input maps. Since this method makes no assumptions about the geometry of the shapes, can be used for complex cases such as non-isometric matching.
Overall, our contributions can be summarized as follows:
We demonstrate that untrained neural networks for 3D shapes can produce pointwise features that are on par with complex axiomatic local descriptors.
We show that when trained using artifact-laden maps for supervision, the features learned by neural networks lead to correspondences that are more coherent and of higher quality than the input. We call this effect Neural Correspondence Prior.
Based these insights, we develop two-stage algorithm for unsupervised 3D non-rigid shape matching that achieves SOTA results for difficult matching scenarios such as non- isometric and point cloud matching.
Using our unsupervised point-to-point correspondence method, we propose an approach for few-shot keypoint detection.

---

2
Related work
Shape matching is well-studied area in computer vision and graphics, and full overview is beyond the scope of this paper. We refer the interested readers to recent surveys [24, 25, 26, 27] for a more in-depth treatment of this field. Below, we review methods that are most related to our work, with a focus on learning-based techniques, both supervised and unsupervised.
3D non-rigid shape correspondence There is a vast literature on learning non-rigid shape cor- respondences. In terms of supervised methods, an important direction is to consider the matching problem as dense semantic segmentation problem where the labels are vertices on some template shape [8, 9, 10], or by mapping via template deformation [7]. However, these techniques are known to require a considerable amount of data, and may fail when discretization changes [19, 11]. Another approach is to use the functional map framework that was introduced in [28] and extended in many follow-up works [29, 30, 31, 32, 33, 34, 35, 36, 37] to name a few (see [38] for an overview). The functional map (fmap) framework is based on encoding and optimizing maps as small matrices in a reduced basis. This formulation has been successfully used in the supervised setting to establish correspondences between complete and partial non-rigid shapes [11, 12, 13]. However, it is typically restricted to near- -sommiisoettic matching between relatively clean discretizations. The fmap framework has also been adapted to the unsupervised setting. either by imposing structural properties the functional maps [15, 39]. by enforcing cycle consistency [40], or by combining intrinsic and extrinsic properties [17, 18]. Another line of work proposes to perform unsupervised shape matching by reconstructing the input shapes in canonical order, based on an autoencoder [41, 42, 43]. However, these methods require lot of shapes and learning time, and have only successfully been applied to man-made shapes that do not undergo significant non-rigid deformations.
Neural prior The neural prior was first introduced in the seminal work of Ulyanov et al. [22], where the authors introduced the Deep Image Prior (DIP), and showed that randomly initialized convolutional neural network can serve as good prior for many 2D inverse problems, such as inpainting and denoising. Since then, several modifications and improvements have been made to DIP [44, 45, 46, 47, 48], in order to improve its performance or adapt it to new tasks. Using untrained features for matching was recently used in the context of 2D keypoint matching in [49]. Another line of work [50, 51] has theoretically studied DIP either through the lens of regularization theory or through its connection to Gaussian processes. Despite its success in 2D computer vision, neural prior has been little exploited in the 3D domain, the only application being shape reconstruction. Indeed, [52, 53] attempt to reconstruct a water-tight mesh from point cloud, either by overfitting several local patches to the point cloud or by shrinking an input mesh by a neural network. The objective of our work is to fill this gap and to propose a neural prior technique for 3D matching.
Unsupervised keypoint detection Unlike the 2D case [54, 55, 56, 57], unsupervised 3D keypoint detection is relatively under-explored in the literature. The recently introduced KeyPointNet [58] benchmark provides a good testing bed. Prominent unsupervised keypoint detection methods are based on self-supervised paradigm using an encoder-decoder network, with reconstruction error. The encoder of Skeleton Merger [59] predicts a set of salient keypoints, that the decoder uses to produce skeleton of the shape. UKPGAN [60] harnesses the power of GANs to detect keypoints, by forcing the decoder to reconstruct the input shape using solely the set of keypoints produced by the encoder. Finally, KeypoinlDeformer [61] predicts a set of keypoints that allow for efficient shape manipulation. In particular, given a source and a target shape, the keypoints predicted on the source shape are used to deform to the target shape using a cage skinning deformer [62].
3 Motivation & Method overview
Our main goal is to develop an unsupervised Iearning- bassd shape correspondence method that would take as input a pair of shapes M and A and produce dense correspondence, also called map, T : M + +. For this, we explore the direction of neural priors and show how they can be used for 3D shape matching. The motivation behind this direction twofold. First, as in the 2D case [63, 46), we hope that the structure of an untrained 3D neural network captures the low-level statistics of single 3D shape, and thus can enable fine-tunning using a single pair. Second, as demonstrated in DIP [22], deep neural networks have high noise impedance, which allows them to learn good features from noisy inputs. This property can be used to guide the network to good local minima while trying to use, in our case, noisy point-to-point (p2p) matches for supervision.
3

---

Methods
Method type
FAUST
SCAPE
10'
JKS [69] NN
Axicm
26.8
28.6
WKS [70] + NN
Aciim
24.8
26.0
SHOT [7]]+ NN
Axiom
37.7
38.4
NCE loss (
10'
10
Geodesic error
DiffusionNer [19] + NN
Rond.inir
17.8 = 0.1
20.3 + 0.3
BCICP [35]
Axiom
16.
FMNel [13]
Sup
11.
17.
SURI-MNet [15]
Unsup
12.
10
Unsup FMNet [14]
Unsup
10.
16.
map (6)-
10
map (b)- 14,8
HKS FMAP
Astom
20.2
26.9
map 28.5
WKS + FMAP
Assiom
20.1
27.1
map (d)
SHOTT FMAP
Asiom
10
10"
10'
10"
10"
DiffusionNer FMAP
Rnid
8.9 +
0.02
13.9 0.3
Optimization iterations
Figure 2: Motivation for the NCP effect. Left: Matching accuracy of DiffusionNet with random
weights (mean and standard deviation three random runs), compared to baseline shape matching methods, on the test sets of FAUST and SCAPE. Right: The evolution of the training NCE loss and geodesic error during optimization when using the ground truth map (map (a)), and noisy maps with different levels of noise (maps (b). (c) and (d)) as supervision. The legend reports the geodesic error of the input maps. Observe the noise impedance and geodesic error decrease of intermediate maps.
Feature embeddings. In this work, we formulate the shape-matchine problem for both man-made and organic shapes, by learning pointwise features. Such features can be used to compute correspondences either through nearest neighbor search or with minimal post proccessoss.ng. Specifically, the network N takes as input 3D shape M and produces, output, feature vector N(M), Rd for every point of the shape x E .M. When considering the entire object as a whole, we call the set N(M),, for all M feature embedding of shape M.
Our method is based we call neural correspondence prior. Below we describe each of these effects and then describe our unsupervised shape-matching method in Sec. 4.1.
main observations: neural bias for untrained networks, and effect that
3.1 Neural bias for pointwise features
Our first observation is that feature embeddings produced by neural networks have a particular structure, which can be exploited in the context of computing features for shape correspondence. To highlight this, we considered the test sets of the FAUST-Remm Remeshed (FAUST) and SCAPE- Remeshed SCAPE-Remeshed
(SCAPE) datasets introduced in [33], and used in many recent works [11, 17, 39], and computed the correspondences using the pointwise features extracted by DiffusionNet network [19] with randomly set weights. We use the default variant of DiffusionNet, which takes as input the XYZ coordinates of the shapes and produces the weights of the network are set randomly and without any training. We then used the extracted features produce p2p maps either via a simple nearest neighbor (NN) search in the feature space or with the functional map framework [64] (FMAP). We compare the results to maps produced using the same procedure with classical axiomatic features such as SHOT [65], or using other axiomatic and training- based non-rigid correspondence methods.
128-dimensional feature vector for every point. We emphasize that
As shown in Fig. 2-Left, remarkably, the features produced by an untrained DiffusionNet network perform on par or better than axiomatic features and even outperform the supervised learning method, FMNet [66]. based on training MLPs to refine pre-defined SHOT features. We attribute the fact that a randomly-initializer
NiffusionNet performs better than trained point-wise MI.Ps to the spatially- aware nature of the architecture of the network, which uses diffusion to simulate intrinsic convolution, thereby providing a strong neural prior [67, 68, 53].
3.2 NCP: Neural Correspondence Prior
While the previous effect relates properties of untrained networks, in this work, we also observe another, complementary phenomenon that we call Neural Correspondence Prior, and which we exploit in our approach below. This effect can formulated follows. Suppose we are given fixed map TMN, between some shapes M and N. We can formulate an optimization problem, where we aim to learn the feature embedding of M and N so that the induced map, e.g., computed via nearest neighbors in the feature space: i.e. + arg min, [IM()), N(V), is as close as possible to TMN- Note that unlike the neural bias mentioned the previous section, here we formulate an optimization problem, where the parameters of the network N are optimized to fit the given input

---

map. Our key observation is that when the target map Tuw is noisy approximate, networks tend to produce well-structured feature embeddings until the very late stages of the optimization. Le.., throughout the early stages of training, the maps computed by the optimized features tend to be of higher accuracy than the input noisy map used for supervision.
To demonstrate the effect of NCP quantitatively, we performed an experiment in which we corrupted the ground truth map between a pair of shapes from the FAUST dataset, and then trained the Diffu- sionNet network, with a gradient descent optimizer using the standard NCE loss [72] to overfit to those corrupted maps. Specifically, to supervise this training. we use: (a) the ground truth (GT) map between the shapes. (b) the GT map where 25% of its entries have been assigned to random vertices, (c) the GT map with 50% noise, (d) the GT map with 75% noise. The results of this experiment are shown in Fig. 2-Right. We plot both the NCE loss being optimized as well as the geodesic error of the intermediate maps with respect to the ground truth.
As shown in Fig. 2-Right, while the network can easily adapt to the correct map, it has difficulty minimizing the loss in the case of noisy maps. Moreover, and perhaps more remarkably, observe that the geodesic error starts to decrease towards lower values than the input map, in the first iterations of optimization and diverges only in the later stages, when the network starts to overfit to the noise in the training map. This implies that such optimization with neural network and early stopping can be used as an effective map denoising mechanism (Sec. 4.2). It should be mentioned that this effect does not depend on the shape pair, and other examples of different pairs are provided in Sec. 5.1.2 and the supplementary.
We observe this effect to hold very broadly in shape correspondence problems. For a single shape pair with noisy input map, eventually, any map can be learned by the network if the parametrization and the number of training iterations are sufficient. However, the network architecture strongly regularizes the map search space, providing low impedance to 'signal' and high impedance to 'noise", resulting in an optimization trajectory that either converges to a good local minimum or passes near one [22]. Furthermore, in the presence of shape collections, this effect is even stronger and, as we demonstrate below, significantly helps to regularize computed features so that no early stopping or post-pun post-processing of the results becomes necessary (see our Algorithm that we explain in detail in Sec. 4.1).
In addition to the network architecture itself, we attribute the Neural Correspondence Prior to the spectral bias principle [23, 73], which states that neural networks tend to learn low frequencies in the early stages of training and that low frequencies are more robust to random perturbations of network parameters. Since we formulate shape matching as the problem of computing optimal features, spectral bias, in our context, implies that even given a noisy or artifact-laden map as input, the feature embeddings produced by the optimized neural network tend to be smooth, especially in the early stages of the optimization. In other words, is significantly harder for the network to produce a noisy, gighffreueency feature embedding than smooth one. Furthermore, smooth feature embedding will tend to produce smooth correspondences between shapes, as suggested by the following theorem, which we state here and prove in the supplementary.
Theorem 1. Let M and N be two compact smooth surfaces (smooth manifolds of dimension 2). Let M and N be their feature embeddings in Rdd, given by some functions: 1/2 : MM- + Rd and : N > Rdd, so that M = viM) and N =(NN.. For example, va and & can be given by some neural network. Suppose that t/ and & are both smooth and injective. Then up to arbitrarily small perturbations of o, v, the map Twn .M- + NN given by (xn(z) = arg minyeN [1((())) - +(y))l must be smooth up to sets of measure 0 on M.
This result highlights the fact that smooth feature embeddings, generically, translate into smooth maps, which desirable property in most typical correspondence scenarios.
4 Method description
Based on the observations above, in this section, we introduce contexts in which the NCP can be exploited, resulting in two algorithms: one for unsupervised shape matching, and the second for p2p map denoising.
4.1 NCP within shape collection
Given collection of shapes [NG]. we propose to exploit the NCP in two ways. The first configuration assumes that we are also given a collection of artifact-laden maps } between some shape pairs in
5

---

Algorithm 1: NCP-UN:
based UNsupervised Shape Matching
Input: Collection of shapes (Ma)
Output: p2p maps between shapes of the test set of (Ni
1: Train variant of off-the-shelf unsupervised matching method UM on the train set of (Ni)
Stage
2: Predict p2p maps } on the train set of {Ni} using trained UM
3: Use {T; to supervise the training of randomly-initialize neural network RN with a p2p loss
Stage? 2
4: Predict Feature Embedding (RN(N)) on the test set of }Na using the trained RN
55: Use (R)( to establish correspondences either using the nearest neighbor or functional
map pipeline.
[Mi as input. In that case, we propose to train neural network to produce feature embedding for each shape, s.t., the induced maps are as close as possible to the target maps {T₂}. We provide the exact choice of the loss function depending on the nature of the underlying shapes in Sec. 5 below.
In the second configuration, no maps are given as input. In this case, we propose a two-stage pipeline where in the first stage, we adapt an existing unsupervised correspondence method to obtain possibly approximate very noisy initial correspondences. We then formulate supervised learning problem, where we learn feature embeddings that would induce these computed correspondences as a target. Once the network is trained, we establish final correspondences via simple nearest neighbor search in the feature space. Note that the trained network can be used to establish correspondences both between shapes within the given shape collection, but also across new, never seen shape pairs. In our evaluation below, and unless otherwise noted, we always evaluate on a test set, never seen during training. We dubbed the resulting algorithm NCP-UN, and we summarize it in Algorithm I and Fig. 1. This is the algorithm that is used throughout most of our experiments.
We argue that it is advantageous to perform Stage 2 simultancously on an entire shape collection, instead of doing it for each pair individually for two main reasons: first, it is faster and avoids the need for test-time optimization (see Sec. 4.2), and second, the collection provides an additional layer of regularization. In fact, since optimize for features features that would overfit to all O(N²) noisy maps in a collection of N shapes, as the artifacts across pairs are typically not consistent. We observe that the noise impedance property is even stronger in the case of a collection, and the produced maps are better than if the refinement is done per pair.
each shape, it is difficult to compute
Intuition behind Stage 2 of Algorithm 1 Our intuition for Stage 2 is threefold. First, and motivated by the results of Sec. 3.2 (the NCP effect), we showed that neural networks tend to have difficulty overfitting to corrupted inputs, and tend to favor smooth outputs. Thus, in the process of over- ever-fitting to approximate maps by a neural network, the output maps tend to be of better quality than the input. This motivates the use of artifact-laden maps as supervision for network thus capitalizing on the NCP effect to recover better correspondences. Second, as we mentioned above, given a collection of approximate maps, their errors tend to be inconsistent, and training network using such maps for supervision adds an extra layer of regularization, as it is difficult for the network to adapt to all inconsistent errors on all pairs at once. Finally, in our formulation of Stage 2, we learn one feature embedding per shape and compute correspondences via nearest neighbor search between feature embeddings. This also adds strong regularization, since makes it difficult for the network to learn a feature embedding, which can reproduce errors in the maps between all shape pairs.
For our applications (see Sec. 5), we used the following design choices. For the UM method, we used the unsupervised functional map method [15, 39], with a DiffusionNee [19] backbone for 3D meshes, and Point-MLP [74] backbone for point clouds. The same backbone was used as RN for step 3. in Algorithm 1. Concerning the p2p loss in step 3, we used the LIE loss [75] for 3D meshes and point clouds. We also used a variant of our approach, dubbed NCP-UN that uses the FMAP framework and loss from [1]] in steps and 5 of Algorithm 1, especially on organic shapes, where the Laplacian basis tends to be of high quality.
We want to emphasize that our observations regarding NCP are independent of the choice of loss and network architecture. A demonstration of the generality of NCP, as well as more details about the implementation, are provided in the supplementary. Across all cases, we observe significant improvement in results when using the second stage of our NCP-UN pipeline (supervised learning) compared to the initial input maps.

---

Finally, we want to emphasize that it is possible to repeat Stage 2 several times, and we observed that there slight improvement in some cases, but most often it stagnates after the first iteration. We experimented with this and chose to keep the method simple and use only one iteration, thus, avoiding an additional tunable hyperparameter.
4.2 Test time denoising
In addition to the previous setting, we also observe that NCP can be used to denoise correspondence when given a single shape pair. Specifically, given a noisy initial map, we train a neural network to produce feature embeddings that would induce the given input map, used supervision. Again, tailor the exact choice of loss to the nature of the shapes and describe one option in Sec. 5.1.2 below.
Test time denoising can be advantageous in the absence of a shape collection. At the same time, as mentioned in the original DIP work [22], in the case of single shape pair, early stopping must be used in order to stop training at the best local minimum. For the shape-matching task, we use a cycle loss the criterion for stopping. Given a pair of shapes M, and two maps between them 1. and M computed via nearest neighbors between optimized features, and represented as soft binary matrices (see Appendix A for computation details), we compute Leyele = - M Xxll I where XNv is the matrix of XYZ coordinates of shape N We overfit the network for a number of iterations and stop when Leyele stops improving after predetermined patience period. The weights that produce the minimum value of Leyele are used to compute feature embeddings that we then use to compute the p2p map, via nearest neighbor search.
4.3
Correspondences as a tool for downstream tasks
Given computed correspondences, we also use them to solve multiple downstream tasks. Indeed, since our proposed algorithm is unsupervised, we can use its output, i.c., p2p maps, to transfer multiple quantities between shapes, such as keypoints. For this, we only need few labeled examples with the quantities we want to transfer, resulting in a few-shot algorithm.
In the following, we present a method for few-shot keypoint detection, dubbed FSKD hereafter, and based on the p2p maps obtained by unsupervised algorithm NCP-UN. A particular challenge in this setup that some keypoints and parts appear exclusively on some shapes and not in others, and so account for the keypoint appearing there, multiple labeled source shapes must be used.
We begin FSKD by randomly selecting N shapes that cover the largest set of keypoints. These N shapes will be labeled and we transfer their keypoints to the shapes in the test set, in order to "discover" their keypoints.
Given target shape, FSKD is composed of three steps: 1. Detection of potential keypoints by transferring keypoints from labeled shapes, 2. Filtering to remove keypoints that are likely not to exist the target shape, 3. Combination: merge transferred keypoints if multiple points the target shape are assigned to the same keypoint ID. We provide the implementation details for these three steps in the supplementary (see Appendix C).
Results & Applications
In this section, we provide results in a wide range of challenging tasks, showing the efficiency and robustness of our approach to different types of data and tasks. In particular, we consider the tasks of shape correspondence between man-made shapes in Sec. 5.1.1, non-rigid shape correspondence in Sec. 5.1.2, part segmentation in Sec. 5.2, and few-shot keypoint detection in Sec. 5.3. The fact that our model does not rely on a geometrie prior allows provide good results on different types of data, including isometric and non isometric,, organic, and man man-made data.
We conducted our experiments on four different datasets. To the best of our knowledge, there is no point cloud dataset that provides dense point-to-poo to-nt ground-trun correspondences. For this purpose, we follow the setup of [41], and use KEYPOINTNET dataset [58] for the man-made shape correspondence experiment in Sec. 1. We also use this dataset for our few-shot keypoint detection experiment in Sec. 5.3. KEYPOINTNET is composed of over 8K models and 100K semantic keypoints labeled by professional humans. We use the same train/validation/ess splits as the original paper [58]. For the part segmentation experiment in Sec. 5.2, we use the PARTNET dataset [76],
Datasets
7

---

Airpinne
Bathhub
Bied
Table
100
125
EXC T40
DDC. R.4
Aduches
%
INc 421
3
Dan 11:454
(inge 21-23
singe 21
Geodess (020)))60)
Geodesic (uar0
Goodesoo (x100)
Geodsti (X100)
Figure 3: Correspondence accuracy on 4 categories in the KEYPOINTNET dataset.
which is composed of 16881 shapes of 16 categories, where each category has a different number of parts ranging from 2 to 6.
For 3D meshes, we use the organic, non-isometric non-rigid SMAL [77, 36] dataset and the SHREC' 20 Non-Isometric benchmark [78] for our non-rigid correspondence experiment in Sec. 5.1.2, these challenging non-isometric datasets. The former dataset is composed of 50 animal shapes represented as 3D meshes. 25 shapes are used for training. and the remaining 25 used for testing. It should be noted that the set of animals and poses seen in training is different from the one used in testing. The SHREC' 20 benchmark consists of 14 scans of different animals, some of which contain holes and topological deformations. This, in addition to the fact that the ground truth maps are given as a set of sparsely annotated keypoints, means that most supervised methods cannot be applied.
5.1 3D shape correspondences
5.1.1
Correspondences on point clouds
We test the quality of the matches produced by NCP-UN on sparse point clouds using the KEYPOINT- NET dataset. We train the first stage of our method using the unsupervised functional map method [15, 39], with PointMLP backbone [79] and XYZ coordinates as input, and impose the bijectivity loss on the functional map. The second stage is trained using LIE loss, and maps are extracted using the nearest neighbor in the feature space. For testing, we follow the same setup as [41], by generating pairs of shapes between all point clouds in a given category and removing the pairs that do not share the same set of semantic keypoints. We compute the L2 distance between the transferred keypoint and the ground truth one. Our method is evaluated against sfate-of-the-an learning-base correspondence prediction approaches, including AtlasNetV2 [80]. FoldingNet [8]]. IDC [43], CPAE [41], in addition to the nearest neighbor baseline between the XYZ coordinates of the points, which turns out to be competitive baseline, not considered by previous methods.
3D dense
In Fig. 3, we report the percentage of testing pairs where the distances between predicted and ground truth maps are below a given threshold for 4 categories and report the remaining categories in the supplementary. Our method outperforms all methods in 13 out of 16 categories. Note that our method obtains SOTA results although the method used in stage is not fully designed to work on point clouds (the quality of the Laplacian is not good, and the used bijectivity loss is weak). We believe that using a more adapted method for the first stage will further improve the results.
5.1.2 Correspondences between non-rigid meshes
For the non-rigid non-isometric application, we trained the unsupervised functional map network from [19] and used LIE loss for the second stage (step 3. in Algorithm 1). P2P maps were obtained using nearest neighbors between feature embeddings. Because the SHREC' 20 dataset is limited (only 14 shapes), we follow previous work [82] by training and testing on the same set, whereas for SMAL, the training and testing sets are different. For fair evaluation, we compared our method to many state-of-the-aa supervised and unsupervised methods. For the supervised methods, we compared to FMNet [13], GeomFMaps [11], GeomFMaps + DiffusionNet [19] and LIE [75]. For the unsupervised methods, we compared our method to WSupFMNet [39], Smooth Shells [18], Deep Shells [17], NeuroMorph [82] and WSupFMNet + DiffusionNet [19].
As shown in Fig. 4, our method achieves of-te-of results among unsupervised methods and supervised methods (we follow the standard Princeton protocol [83] for evaluation). The majority of unsupervised methods and supervised methods fail because of the near-isometry assumption they make. Since our method makes no assumptions about the topology of the shape, it can perform very well in difficult scenarios such as non-isometric animals. should be noted that these results would not be possible without the second stage of our algorithm, which is enabled by NCP. For example, for
8

---

Methods
SMAL
Shree'20
WSupFMNet [39]
13.3
100
Smooth Shells [18]
16.3
Unsup
DeepShells [17]
15.2
4/ Correspondeness
80
Morph [82]
23.1
Ours (Stage
7.8
60
SURKMNer
UnsupPMNett 40.3
FMNet [13]
47.1
ZoomOut -44.2
40
Smooth Shells
Sup
GeomFMaps [11]
8.8
Deep Shells- 35.5
GeomFMaps + DiffusionNer [19]
6.6
20
NeuroMorph- 10.5
LIE [75]
20.0
Our (stage 1)- 10.6
(stage
Ours (Stage2)
6.0
0
Unsup
Ours (Stage2 fmap)
5.8
0
20
40
60
80
100
Ours (Test-Time Optimization)
6.4
Geodesic error (x100)
Figure 4: Non-isometric matching results. Left: Matching accuracy on the SMAL test set. Values mean geodesic methods on the test set of SHREC' 20 Non-Isometric dataset.
x x100 on unit-area shapes. Right: Correspondence quality of different
Stage 2 (Test Time
Stage 2 (NCP-UN)
Source
Target
NeuroMorph
Stage 1
Optimization)
Figure 5: Qualitative results on SMAL dataset, showing the maps produced by Stage 1, and refined by Stage 2, either with test time optimization (see Sec. 4.2) or using NCP-UN (see Algorithm 1)
the SMAL dataset, the result of the first stage is 7.8, and the second stage improves this result to 5.8, which is 25% improvement, achieving state-of-the-arr results.
We provide in Fig. 5 some qualitative results showing the performance of our algorithm against the state-of-the-ard NeuroMorph method on the SMAL dataset. It can seen that our method produces visually plausible maps, especially after applying the second stage.
Test time optimization To show the utility of test time optimization, well as the early stopping criteria we introduced, we performed
following experiment. After training the first stage of our algorithm, the trained model was used to predict p2p on the test set of SMAL dataset. Pairs of shapes are then created, and we use our test time optimization method (see Sec. 4.2) to improve upon the maps produced by the first stage. Results are reported in Fig. 4-Left, see row "Ours (Test-Time Optimization)". It can be seen that similar to the previous case, training a random network on top of artifact-laden maps helps to denoise them. We also show in Fig. 5 some qualitative results that illustrate the effect of test time optimization and compare it to NCP-UN.
The results of this experiment confirm our hypothesis that doing the second stage on a collection, the results are better, in
instead of individual pairs, provides a second layer of regularization, addition to improving computational complexity. In fact, the second stage using the whole collection takes around 15 minutes, meanwhile, doing it each pair individually takes around 160 minutes, which represents more than 10x improvement in running time.
5.2 Part segmentation transfer
We further validate our approach on the part label transfer task on the PARTNET dataset following the setup of [41]. Results are summarized in Tab. 1, where the average reported. Our method outperforms the baselines on 14 out of 16 categories, achieving of the-art results by more than 3% IOU on average. In classes with large intra-class variations such as Lamps, our method outperforms the baselines by up to average IOU points. addition to being performant, our method is extremely fast. In fact, training both stages of our method takes average
IOU

---

Table 1: Part label transfer (in the average IOU(%) for 16 categories in the PARTNET dataset.
pla.
bag
cap
car
cha.
ear.
gui.
kni.
lap.
bik.
mug.
pis.
ska.
tab
arg.
TDC [43]
60.1
56.2
59.7
72.2
45.3
81.5
66.4
42.6
88.5
40.5
87.5
66.4
37.2
50.7
70.4
61.7
CPAE [41]
61.3
59.3
61.6
72.6
55.5
78.9
71.3
53.2
89.9
55.4
86.5
66.2
40.2
61.8
72.5
65.8
Ours
63.7
66.7
68.7
57.4
80.2
59
78.8
72.5
61.9
91.4
57.2
89.5
61.4
44.2
63.6
79.2
69.2
Training shapes (N=3)
Ground truth (human labeled)
Our predictions
0011
FRENGAS
Strohhod
(a)
(b)
Figure 6: Few-shot keypoint detection results.(a) mIoU results of our simple few-shot keypoint detection algorithm on KEYPOINTNET using only labeled examples. Our simple method achieves good results and is on par with or better than recent keypoint detection methods. (b) Qualitative results on the table category, showing the shapes used for training (N=3, left), human annotations on the testset (center) as well as our predictions (right) on four examples.
1 hour (see the computational specifications in Appendix B ), which is more efficient in comparison to the baselines, where CPAE for example takes around 24 hours to be trained for one category.
5.3 Few-shot 3D keypoint detection
We evaluated our few-shot keypoint detection algorithm FSKD on the KEYPOINTNET dataset, following the setup of [60], and using only 3 labeled source shapes (the resulting algorithm is thus three-shot keypoint detection algorithm). The goal of this task is to predict keypoints that correlate with human annotation. The performance is evaluated by the mean Intersection over Union (mloU). The intersection is counted if the geodesic distance between a predicted keypoint and its nearest ground truth keypoint is less than some geodesic threshold. The union is simply the union of the detected keypoints and ground truth keypoints.
We compare FSKD against multiple baselines, including SIFT-3D [84], HARRIS-31 33D [85], ISS [86], D3Feat [87], USIP [88] and UKPGAN [60]. Results summarized in Fig. 6 left. It can be that our simple, correspondence-basee based method outperforms the baselines tailored specifically for this task. In particular, our method achieves state-of-the-art results for threshold superior to 0.035, using only three labeled shapes. We provide some qualitative results in Fig. 6 right.
Conclusion & Limitations
In this work, we showed that using noisy maps as a supervisory neural network training signal for the 3D shape-matching task can lead to significantly higher quality correspondences. We named this effect Neural Correspondence Prior (NCP). Through extensive experiments, we shed light on the properties of NCP and developed two-stage algorithm for unsupervised 3D shape matching. We demonstrate the effectiveness and generality of our algorithm on a range of shape-matching problems, including unsupervised shape matching on both man-made and organic non-rigid shapes, achieving tate-of-thhe-ar results on wide range of benchmarks.
One limitation of our approach is that it assumes that the p2p maps used to train the second stage of the algorithm still have some structure. In fact, if the unsupervised method used in the first stage fails completely and provides random maps, our method will not converge. This problem can be mitigated by using a good application-spectn -speciffc unsupervised method for the first stage, especially with the recent interest and increase in unsupervised 3D learning methods.
The authors would like to thank Frédéric Chazal and Mathijs Wintraecken for many useful discussions related to Theorem 1. We also acknowledge the anonymous reviewers for their valuable suggestions. Parts of this work were supported by the ERC Starting Grant No. 758800 (EXPROTEA) and the ANR AI Chair AIGRETTE.
Acknowledgements
10

---

References
[1] Federica Bogo, Javier Romero, Matthew Loper, and Michael Black. FAUST: Dataset and evaluation for 3D mesh registration. In 2014 IEEE Conference Computer Vision and Pattern
Recognition. IEEE, June 2014. 1
[2] Leonid Pishchulin, Stefanie Wuhrer, Thomas Helten, Christian Theobalt, and Bernt Schiele. Building statistical shape spaces for 3D human modeling. Pattern Recogn., 67:276-286, July 2017.
[3] Huong Quynh Dinh, Anthony Yezzi, and Greg Turk. Texture transfer during shape transforma- tion. ACM Trans. Gruphics, 24(2):289-310, April 2005. 1
[4] Robert W. Sumner and Jovan Popović. Deformation transfer for triangle meshes. ACM Trans. Graphics, 23(3):399 405, August 2004.
[5] Ilya Baran, Daniel Vlasic, Eitan Grinspun, and Jovan Popović. Semantic deformation transfer. ACM Trans. Graphics, 28(3):1-6) July 2009. 1
[6] Qian- Yi Zhou, Jaesik Park, and Vladlen Koltun. Fast global registration. In Computer Vision ECCV 2016, pages 766-782. Springer International Publishing, 2016. 1
[7] Thibault Groueix, Matthew Fisher, Vladimir G. Kim, Bryan C. Russell, and Mathicu Aubry, A papier-mache approach to learning 3D surface generation. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 230-246. IEEE, June 2018. 1,3
[8] Jonathan Masci, Davide Boscaini, Michael M. Bronstein, and Pierre Vandergheynst. Geodesic convolutional neural networks on Riemannian manifolds. In 2015 IEEE International Con- ference on Computer Vision Workshop (ICCVW), pages 37-45, IEEE, December 2015. 1,
3
[9] Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M. Bronstein. Geometric deep learning graphs and manifolds using mixture model CNNs. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 5115-5124. IEEE, July 2017. 1,3
[10] Adrien Poulenard and Maks Ovsjanikov. Multi-directional geodesic neural networks via equivariant convolution. ACM Trans. Graphics, 37(6))1-14, December 2018. 1,
[11] Nicolas Donati, Abhishek Sharma, and Maks Ovsjanikov. Deep geometric functional maps: Robust feature learning for shape correspondence. In 2020 IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), pages 8592-8601. IEEE, June 2020. 1, 3, 4, 6, 8, 9
[12] Souhaib Attaiki, Gautam Pai, and Maks Ovsjanikov. DPFM: Deep partial functional maps. In 2021 International Conference 3D Vision (3DV). IEEE, December 2021. 1,
[13] Or Litany, Tal Remez, Emanuele Rodola, Alex Bronstein, and Michael Bronstein. Deep func- tional maps: Structured prediction for dense shape correspondence. In 2017 IEEE International Conference Computer Vision (ICCV), pages 5659-5667. IEEE, October 2017. 1,3,4,
[14] Oshri Halimi, Or Litany, Emanuele Rodola Rodola, Alex M. Bronstein, and Ron Kimmel. Unsupervised learning of dense shape correspondence. In 2019 IEEE/CVF Conference on IEEE, June 2019. 1,
Computer Vision and Pattern Recognition (CVPR), pages
[15] Jean-Michel Roufosse, Abhishek Sharma, and Maks Ovsjanikov. Unsupervised deep learning for structured shape matching. In 2019 JEEE/CVF International Conference on Computer Vision (ICCV), pages 1617-1627. IEEE, October 2019. 1, 4, 8,
[16] F. Rodolà, L. Cosmo, M. M. Bronstein, A. Torsello, and D. Cremers. Partial functional correspondence. Comput. Graph. Forum, 36(1):222-236, February 2016.
[17] Marvin Eisenberger, Aysim Toker, Laura Leal-Taixé. and Daniel Cremers. Deep shells: Unsu- pervised shape correspondence with optimal transport. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances Neural Information Processing Systems, vol-
ume 33, pages 10491-10502. Curran Associates, Inc., 2020. 1, 3, 4, 8,

---

[18] Marvin Eisenberger, Zorah Lähner, and Daniel Cremers. Smooth shells: Multi-scale shape registration with functional maps. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 12262-12271, 2020. 1, 3, 8,9
[19] Nicholas Sharp, Souhaib Attaiki, Keenan Crane, and Maks Ovsjanikov. DiffusionNet: Dis- cretization agnostic learning on surfaces. ACM Trans. Graphics, 41(3):1-16, June 2022. 2, 3, 4,
6, 8, 9
[20] Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph CNN for learning on point clouds. ACM Trans. Graphics, 38(5):1-12. October 2019.
[21] R. Qi Charles, Hao Su, Mo Kaichun, and Leonidas J. Guibas. PointNet: Deep learning on point sets for 3D classification and segmentation. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 5105-5114, IEEE, July 2017. 2
[22] Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Deep image prior. Int. J. Comput. Vision, (88(7)1867-188 March 2020. 2, 3, 7
[23] Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. In Proceedings of the 36th International Conference on Machine Learning, Proceedings of Machine Learning Research, pages 5301-5310. PMLR, 2019. 2,5
[24] Yusuf Sahillioglu. Recent advances in shape correspondence. The Visual Computer, 36(8):1705
1721, September 2019. 3
[25] S. Biasotti, A. Cerri, A. Bronstein, and M. Bronstein. Recent trends, applications, and perspec- tives in 3D shape similarity assessment. Comput. Graph. Forum, 35(6):87-119, October 2015.
3
[26] Bailin Deng, Yuxin Yao, Roberto M Dyke, and Juyong Zhang. A survey of non-rigid 3D registration. arXiv preprint arXiv:2203.07858, 2022.
[27] G. K. L. Tam, Zhi- QhanQuan Cheng, Yu- Lai, F. C. Langhein, Yonghuai Liu, D. Marshall, R. Martin, Xian-Fang Fang Sun, and P. L. Rosin. Registration of 3D point clouds and meshes: A survey from rigid to nonrigid. IEEE Truns. Visual Comput. Graphics, 19(7): (177):11994:) 199-1127, July 2013.
[28] Maks Ovsjanikov, Mirela Ben-Chen, Justin Solomon, Adrian Butscher, and Leonidas Guibas. Functional maps. ACM Trans. Graphics, 31 11): August 2012. 3
[29] Emanuele Rodolà, Luca Cosmo, Michael M Bronstein, Andrea Torsello, and Daniel Cremers. Partial functional correspondence. In Computer Graphics Forum, volume 36, pages 222-236.
Wiley Online Library, 2017. 3
[30] Oliver Burghard, Alexander Dieckmann, and Reinhard Klein. Embedding shapes with green's s functions for global shape matching. Computers &amp; Graphics, 18:1-10, November 2017.
[31] Davide Eynard, Emanuele Rodola, Klaus Glashoff, and Michael M. Bronstein. Coupled functional maps. In 2016 Fourth International Conference on 31) Vision (3DV), pages 409. IEEE, IEEE, October 2016. 3
[32] Meged Shoham, Amir Vaxman, and Mirela Ben-Chen. Hierarchical functional maps between subdivision surfaces. Comput. Graph. Forum, 38(5)555-73, August 2019. 3
[33] Jing Ren, Adrien Poulenard, Peter Wonka, and Maks Ovsjanikov. Continuous and orientation- preserving correspondences via functional maps. ACM Trans. Graphics 37 December 2018. 3,
[34] Jing Ren, Adrien Poulenard, Peter Wonka, and Maks Ovsjanikov. Continuous and orientation- preserving correspondences via functional maps. ACM Trans. Graphics, 37(6):1-16, December 2018.7
[35] Dorian Nogneng and Maks Ovsjanikov. Informative descriptor preservation via commutativity for shape matching. Comput. Graph. Forum, 36(2):259-267, May 2017. 3
12

---

[36] Riccardo Marin, Souhaib Attaiki, Simone Melzi, Emanuele Rodolå, and Maks Ovsjanikov. Why you should learn functional basis, 2021. 3.8 8
[37] Lei Li, Souhaib Attaiki, and Maks Ovsjanikov. SRFeat: Learning locally accurate and globally consistent non-rigid shape correspondence. In 2022 International Conference on 3D Vision (3DV). IEEE, September 2022. 3
[38] Maks Ovsjanikov, Etienne Corman, Michael Bronstein, Emanuele Rodolå, Mirela Ben-Cher Leonidas Guibas, Frederic Chazal, and Alex Bronstein. Computing and processing correspon- dences with functional maps. In ACM SIGGRAPH 2017 Courses. ACM, July 2017. 3
[39] Abhishek Sharma and Maks Ovsjanikov. Weakly supervised deep functional maps for shape matching. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Prucessing Systems, volume 33, pages 19264-19276 Curran Associates, Inc., 2020. 3, 4, 6, 8,9
[40] Dvir Ginzburg and Dan Raviv. Cyclic functional mapping: Self-supervised correspondence between non-isometric deformable shapes. In Computer Vision ECCV 2020, pages 36-52.
Springer International Publishing. 2020. 3
[41] An-Chieh Cheng, Xueting Li, Min Sun, Ming-Hsuan Yang, and Sifei Liu. Learning 3D dense correspondence via canonical point autoencoder, 2021. 3, 7, 8, 9, 10
[42] Zerong Zheng, Tao Yu, Qionghai Dai, and Yebin Liu. Deep implicit templates for 3D shape representation, 2020.
[43] Feng Liu and Xiaoming Liu. Learning implicit functions for toplogy-varying dense 3D shape correspondence, 2020. 3, 8, 10
[44] Reinhard Heckel and Paul Hand. Deep decoder: Concise image representations from untrained non-convolutional networks. International Conference on Learming Representations, 2019. 3
[45] Reinhard Heckel and Mahdi Soltanolkotabi. Compressive sensing with un-trained neural networks: Gradient descent finds the smoothest approximation. ArXiv, abs/2005.03991, 2020.
[46] Gary Mataev, Michael Elad, and Peyman Milanfar. DeepRED: Deep image prior powered by RED. ArXiv, abs/ 90ts/190.1110 10176 2019. 3
[47] Jiaming Liu, Yu Sun, Xiaojian Xu, and Ulugbek S. Kamilov. Image restoration using total variation regularized deep image prior. In ICASSP 2019 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, May 2019. 3
[48] Daniel Otero Baguer, Johannes Leuschner, and Maximilian Schmidt. Computed tomography reconstruction using deep image prior and learned reconstruction methods. Inverse Probl., 36(9):094004, September 2020. 3
[49] Sunghwan Hong and Seungryong Kim. Deep matching prior: Test-time optimization for dense correspondence. In 2021 IEEE/CVF International Conference Computer Vision (ICCV), pages 9907-9917. IEEE, October 2021.
[50] Sören Dittmer, Tobias Kluth, Peter Maass, and Daniel Otero Baguer. Regularization by architecture: deep prior approach for inverse problems. Math. Imaging Vis., 62(3))456-470, October 2019. 3
[51] Zezhou Cheng, Matheus Gadelha, Subhransu Maji, and Daniel Sheldon. A Bayesian perspective on the deep image prior, 2019. 3
[52] Francis Williams, Teseo Schneider, Claudio Silva, Denis Zorin, Joan Bruna, and Daniele Panozzo. Deep geometric prior for surface reconstruction. In 2019 IEEE/CV Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, June 2019. 3
[53] Rana Hanocka, Gal Metzer, Raja Giryes, and Daniel Cohen-Or. Point2Mesh. ACM Trans. Graphics, 39(4), August 2020. 3,
13

---

[54] Tomas Jakab, Ankush Gupta, Hakan Bilen, and Andrea Vedaldi. Unsupervised learning of object landmarks through conditional image generation. In NeurlPS, 2018. 3
[55] Tomas Jakab, Ankush Gupta, Hakan Bilen, and Andrea Vedaldi. Self-supervised learning of interpretable keypointss from unlabelled videos. In 2020 IEEE/CVF Conference on Computer Vision and Puttern Recognition (CVPR). IEEE, June 2020.
[56] James Thewlis, Hakan Bilen, and Andrea Vedaldi. Unsupervised learning of object landmarks by factorized spatial embeddings. In 2017 IEEE International Conference Computer Vision (ICCV). IEEE, October 2017. 3
[57] Olivia Wiles, A. Sophia Koepke, and Andrew Zisserman. Self-supervised learning of class embeddings from video. In 20/9 IEEE/CVF International Conference Computer Vision Workshop (ICCVW). IEEE, October 2019.
[58] Yang You. Yujing Lou, Chengkun Li. Zhoujun Cheng, Liangwei Li, Lizhuang Ma, Cewu Lu, and Weiming Wang. KeypointNet: A large-scale 3D keypoint dataset aggregated from numerous human annotations. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. 3.7
[59] Ruoxi Shi. Zhengrong Xue, Yang You, and Cewu Lu. Skeleton merger: An unsupervised aligned keypoint detector. 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2021. 3
[60] Yang You, Wenhai Liu, Yanjie Ze, Yong-Lu Li, Weiming Wang, and Cewu Lu. UKPGAN: general self-supervised keypoint detector, 2020. 3, 10
[61] Tomas Jakab, Richard Tucker, Ameesh Makadia, Jiajun Wu, Noah Snavely, and Angjoo Kanazawa. KeypointDeformer: Unsupervised 3D keypoint discovery for shape control. 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2021. 3
[62] Wang Yifan, Noam Aigerman, Vladimir G. Kim, Siddhartha Chaudhuri, and Olga Sorkine- Hornung, Neural cages for detail-preserving 3D deformations. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 75-83. IEEE, June 2020. 3
[63] Yconsik Jo, Se Young Chun, and Jonghyun Choi. Rethinking deep image prior for denoising. In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 5087-5096. IEEE, October 2021.
[64] Maks Ovsjanikov, Mirela Ben-Chen, Justin Solomon, Adrian Butscher, and Leonidas Guibas. Functional maps. ACM Trans. Graphics, 31(4) August 2012. 4
[65] Federico Tombari, Samuele Salti, and Luigi Di Stefano. Unique signatures of histograms for local surface description. European conference on computer vision, pages 356-369. Springer, 2010. 4
[66] O. Litany, E. Rodolå, A. M. Bronstein, and M. M. Bronstein. Fully spectral partial shape matching. Comput. Graph. Forum, 2688):247-258, May 2017. 4
[67] Adam Gaier and David Ha. Weight agnostic neural networks. In H. Wallach, H. Larochelle,
A. Beygelzimer, F. d' Alché-Buc, E. Fox, and R. Gareett, editors, Advances in Neural Informa- tion Processing Systems, volume 32. Curran Associates, Inc., 2019. 4
[68] Andrew M. Saxe, Pang Wei Koh, Zhenghao Chen, Maneesh Bhand, Bipin Suresh, and Andrew Y. Ng. On random weights and unsupervised feature learning. In In ICML, 2011.
[69] Jian Sun. Maks Ovsjanikov, and Leonidas Guibas. A concise and provably informative multi- scale signature based on heat diffusion. In Computer graphics forum, pages 1383-1392. Wiley Online Library, 2009.
[70] Mathieu Aubry, UIrich Schlickewei, and Daniel Cremers. The wave kernel signature: A quantum mechanical approach to shape analysis. In 2011 / IEEE International Conference on Computer Vision Workshops (ICCV Workshops), pages 1626-1633. IEEE, IEEE, November 2011. 4
14

---

[71] Samuele Salti, Federico Tombari, and Luigi Di Stefano. SHOT: Unique signatures of histograms for surface and texture description. Comput. Vis. Image Und., 125:251-264. August 2014. 4
[72] Saining Xie, Jiatao Gu, Demi Guo, Charles R Qi, Leonidas Guibas, and Or Litany. PointContrast: Unsupervised pre-training for 3D point cloud understanding. In European Conference on Computer Vision, pages 574-591. Springer, 2020. 5
[73] Yuan Cao, Zhiying Fang, Yue Wu, Ding-Xuan Zhou, and Quanquan Gu. Towards understanding the spectral bias of deep learning. In Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence. International Joint Conferences Artificial Intelligence Organization, August 2021.
[74] Xu Ma, Can Qin, Haoxuan You, Haoxi Ran, and Yun Fu. Rethinking network design and local geometry in point cloud: A simple residual MLP framework, 2022. 6
[75] Riccardo Marin, Marie-Julie Rakotosaona, Simone Melzi, and Maks Ovsjanikov, Correspon- dence learning via mmeaaly-invvrian embedding. ArXiv, abs/2010. 131 1366, 2020. 6, 8,
[76] Li Yi, Vladimir G. Kim, Duygu Ceylan, I-Chao Shen, Mengyan Yan, Hao Su, Cewu Lu, Qixing Huang, Alla Sheffer, and Leonidas Guibas. A scalable active framework for region annotation in 3D shape collections. ACM Trans. Graphics, 35(6): 1-2 November 2016. 7
[77] Silvia Zuffi, Angjoo Kanazawa, David W. Jacobs, and Michael J. Black. 3D menagerie: Modeling the 3D shape and pose of animals. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, July 2017. 8
[78] Roberto M. Dyke, Yu-Kun Lai, Paul L. Rosin, Stefano Zappalà, Scana Dykes, Daoliang Guo, Kun Li, Riccardo Marin, Simone Melzi, and Jingyu Yang. SHREC' 20: Shape correspondence with non-isometric deformations. Computers &amp; Graphics, 92:28 43, November 2020. 8
[79] Xu Ma, Can Qin, Haoxuan You, Haoxi Ran, and Yun Fu. Rethinking network design and local geometry in point cloud: A simple residual MLP framework. In International Conference on Learning Representations, 2022.
[80] Thibault Groueix, Matthew Fisher, Vladimir G. Kim, Bryan C. Russell, and Mathicu Aubry, A papier-mache approach learning 3D surface generation. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, volume 32. IEEE, June 2018.
[81] Yaoqing Yang, Chen Feng, Yiru Shen, and Dong Tian. FoldingNet: Point cloud auto-encoder via deep grid deformation. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. IEEE, June 2018. 8
[82] Marvin Eisenberger, David Novotný, Gael Kerchenbaum, Patrick Labatut, Natalia Neverova, Daniel Cremers, and Andrea Vedaldi. NeuroMorph: Unsupervised shape interpolation and correspondence in one go. 202/ IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 7469-7479, 2021. 8,9
[83] Vladimir G. Kim, Yaron Lipman, and Thomas Funkhouser. Blended intrinsic maps. ACM Trans. Graphics, 30(4): July 2011. 8
[84] Blaine Rister, Mark A. Horowitz, and Daniel L. Rubin. Volumetric image registration from invariant keypoints. IEEE Trans. Image Process., 26(10):4900-4910, October 2017. 10
[85] Ivan Sipiran and Benjamin Bustos. Harris 3D: A robust extension of the Harris operator for interest point detection on 3D meshes. The Visual Computer, 27(11):933-97,, July 2011. 10
[86] Yu Zhong. Intrinsic shape signatures: A shape descriptor for 3D object recognition. In 2009 IEEE 12th International Conference Computer Vision Workshops, ICCV Workshops. IEEE, September 2009. 10
[87] Xuyang Bai, Zixin Luo, Lei Zhou, Hongbo Fu, Long Quan, and Chiew-Lan Tai. D3Feat: Joint learning of dense detection and description of 3D local features. 2020 IEEE/CVE Conference on Computer Vision and Pattern Recognition (CVPR), pages 6359-6367. IEEE, June 2020. 10
15

---

[88] Jiaxin Li and Gim Hee Lee. USIP: Unsupervised stable interest point detection from 3D point clouds. In 2019 IEEE/CVF International Conference on Computer Vision (ICCV). IEEE, October 2019. 10
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Sec. 6
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See supplementary
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See supple- mentary
(b) Did you include complete proofs of all theoretical results? [Yes] See supplementary
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [Yes] Our code can be found online: https: / github..... com/ http///niiio/////
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they chosen)? [Yes] See Sec. 5 and Supp Mat
(c) Did you report error bars (e.g., with respect the random seed after running ex- periments multiple times)? [Yes] In some experiments when it's not prohibitively expensive
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] See Supplementary
4. If you using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any assets either in the supplemental material URL? [No] using/curating? [No] The used datasets are publicly available.
(d) Did you discuss whether and how consent was obtained from people whose data you't 're
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
16


Reviews:
[{"review_id": "roJ5XXilQ3", "reviewer": "Reviewer_41ct", "summary": "The authors analyze the power of random features from 3D encoders and use it to provide methods for point cloud matching, non-isometric shape matching on meshes, part segmentation transfer and keypoint transfer. It is shown that the random features of several architectures provide a strong prior that automatically resolves matching errors in the early iterations of training. Together with a novel stopping criteria, this leads to a method for refining dense correspondences between shapes.", "questions": "- Are many-to-one mappings theoretically allowed / is it just assumed that there is an injective mapping from shape to feature space, as in the requirement of Theorem 1?\n- Related to the last question: DiffusionNet is a method that incorporates global information through the global diffusion mechanism. However, other networks, such as PointNet and variants are inherently local, meaning that many-to-one feature mappings will be much more likely, even for isometric shapes. Does the method require a global backend, such as as DiffusionNet (or seemingly PointMLP, not so sure on this one) to work, do the insights in Section 3 only hold for those backends, and in case of a local operator, would an additional consensus method (like [1]) help?\n- Would the following statement be correct: The first part of the method finds dense correspondences based on a global feature extractor and fmaps and relies on the near-isometry assumption. Refining the random features during test time is able to resolve errors made due to non-isometric correspondences (in this case, \"noise\" would not be a reasonable description of such errors). Is this due to the local nature of the random features? \n- Many correspondence methods first do local feature matching and then resolve mistakes by introducing global information in a second stage. Here, it seems to be the other way around. Could the authors discuss the implications of this and which advantages/downsides this variant has?\n- I would like to know where the proposed method breaks - how mismatching do the shapes have to be? Since the method seem to rely on a global feature extractor backend, how well does it work on partial meshes? Could a similar technique as in the keypoint extractor be applied?\n- I think the paper would benefit from an analysis of different initialization schemes, when using random features. In the end, this seems to be the main part of the paper.\n\n[1] Fey et al.: Deep Graph Matching Consensus, ICLR 2020\n\nMinor:\n- l. 61: Correspondnece\n- l 281: does not relies\n- l 285: \"point cloud dataset that provide\"\n- Figure 2 labels are way too small\n- Content of the supplementary materials should be explicitly referenced in the main paper (most important: keypoint application).", "limitations": "The authors shortly mention some limitations of the method, as it depends on the results of the applied step 1. Discussion of limitations could be improved.\nPotential negative societal impact is not discussed but I also see no immediate reason to do so. ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper presents a lot of insight into features extracted from 3D operators. The strength of random features was known before (in CNNs) but this is the first time, it was analyzed for 3D architectures on point clouds/meshes\n- The described neural correspondence prior in combination with the cycle loss as early-stopping criteria is an interesting and strong mechanism\n- There are a lot of state-of-the-art results beaten here (or on par): point cloud matching, shape matching on non-isometric shapes, part segementation transfer, 3D keypoint detection on point clouds\n- The paper is well written\n\nWeaknesses:\n- The paper raises a lot of questions (see below), I am not sure if I fully understood the effect of the second stage of the network. It would be good if the authors made an additional effort describing the main intuitions.\n- Theorem 1 seems to be not that important to me. The crucial parts are the assumption of smooth encoders and injectivity, which are often not given for existing methods (especially local encoders like PointNet). The consequence of smooth encoders and injectivity is pretty straight-forward. It would be more interesting if the authors would additionally show that the used encoders are indeed injective and smooth.\n- It seems to me that the term \"noise\" is used in a hand-wavy fashion in this work. I would appreciate a more clear definition of \"noise\" and comparisons between different types of noise if they are applied artificially. I am not convinced that the output of stage 1 of the approach should be called \"noisy\". The errors seem to be more systematic.\n- This paper seems to be divided into to main messages, (1) the analysis of random features/neural correspondence prior, and (2) Theorem 1 and its application using the respective networks (DiffusionNet and PointMLP + fmaps). This makes the paper hard to follow in some parts.\n\nEven if the weaknesses list is longer, I think this is a good paper. I liked reading it and it gave me several insights and a few ideas, which is a good sign. In order to make it really great I would encourage the authors to work a bit more on the structure, discussions of main implications and several details that need to be answered, see below.", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "n0L6VJYeTZS", "reviewer": "Reviewer_2gYm", "summary": "An unsupervised shape correspondence method is proposed :\n\n1. Inputs : training set of shapes (without ground truth)\n2. Train an off-the-shelf unsupervised matching \"teacher\" network on the training set\n3. Train a \"student\" network from scratch on the training set in a supervised manner using the \"teacher\" predictions as ground truth\n4. Use the \"student\" network to establish correspondences between unseen shape pairs\n\nThis method is motived by two empirical observations :\n\n1. \"Neural bias\" : It experimentally shows that features extracted with a randomly initialized DiffusionNet produce good matching performances on FAUST remeshed and SCAPE-remeshed datasets .\n2. \"Neural Correspondence Prior\" : It experimentally shows that the \"Deep Image Prior\" holds for a shape correspondence problem.\n\nThe proposed method either outperforms or is on par with the sota methods on the KeyPointNet dataset, the SHREC20 dataset,  the SMAL dataset and the PartNet dataset.", "questions": "Please see \"Weaknesses\" above.", "limitations": "-", "rating": 3, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n1. Extending the \"Deep Image Prior\" to shape matching is an interesting idea.\n2. The paper is well written.\n3. The overall approach is simple.\n\n### Weaknesses\n\n#### 3.1 Neural bias for pointwise features\n\n1. This part of the paper experimentally shows that features extracted with a randomly initialized diffusionNet produce good matching performances on FAUST remeshed and SCAPE-remeshed datasets. I believe the results of a randomly initialized pointwise MLP should be added, otherwise the section title should be changed to \"diffusionNet bias for pointwise features\".\n2. The link between this neural bias (i.e the fact that an untrained network produces good matches) and the proposed method is unclear to me, since both the teacher and the student networks are trained.\n\n#### 4.1 NCP within a shape collection\n\n1. Why not using the standard terminology teacher network / student network ? I think it would make the whole paper easier to read.\n2. I was expecting the teacher network to be an untrained diffusionNet, especially after reading section 3.1. It seems to me that section 3.1 is not tightly connected to the rest of the paper.\n\n#### 5.1.1 Correspondences on Point Clouds\n\n1. The proposed method outperforms the sota methods but I could not find the performances of the teacher network (i.e unsupervised functional map method + PointMLP backbone).\n\n#### 5.1.2 Correspondences between non-rigid meshes\n\n1. Table 2 shows that the proposed method only slightly outperforms the DiffusionNet-based methods, for instance \"WSupFMNeT + DiffusionNet\" is an unsupervised method that is only slighly outperformed by the proposed method.\n2. Figure 4 shows the results of NeuroMorph which is the worst unsupervised method in Table 2. Why not showing the results of \"WSupFMNeT + DiffusionNet\" instead ?\n\nOverall the proposed method is probably interesting for the shape matching community but the novelty is limited. Moreover, it seems the proposed method only slightly outperforms Diffusion-based methods.", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Sl_aXdWGuIL", "reviewer": "Reviewer_ZBSs", "summary": "This paper applies the idea of deep prior to predict 3D correspondences. It does so using two ideas, which are demonstrated experimentally in the first part of the paper: first, it shows that randomly intialized diffusionNet features actually provide better results than spectral and SHOT   descriptors, and combined with fonctional maps even improve over learned metods; second, it shows that when training a network to predict features to fit noisy correspondences, the quality of correspondences/features obtained at the beginning of training are actually better than the noisy target. This naturally leads to a method to predict correspondences, obtaining features from a randomly initialized network, then training a second one to predict them and stoping early (using a cycle consistency loss to decide when)", "questions": "See weaknesses (in particular FAUST evaluation)", "limitations": "yes", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I liked this paper, the idea is simple, well explained and well demonstrated. To the best of my knowledge using neural prior for shape correspondence is novel, if this is indeed the case I think the paper should clearly be accepted.\n\nAs for weaknesses:\n- Why aren't correspondences evaluated on FAUST? I would really want to see this evaluation, which would also ease comparison with other methods.\n- smaller weaknesses:\n* I liked the separation of section 3 and 4, but then section 4 points several times to section 5 for important details about the method, I think this makes the reading more complicated than needed, please move all details about the method to section 4.\n* I am not a fan of theorem 1. It is actually not related directly to the fact that the mapping are neural nets, it would need a demonstration that this is the general case (i.e. with random weights) for the neural net architecture considered, and both formulation and demonstration in the supmat are very handwavy (for example the theorem is clearly false for d<3, which doesn't appear anywhere). I would thus suggest to completely remove it, or at least not call it a theorem for which I would like a much more rigorous demonstration\n* too many unnecessary abbreviations make the text harder to read (FR, SR, NCP, NCP-UN, UM, RN not counting the ones for previous methods ), I would remove at least 50% of them and be sure to regularly remind the reader of the meaning of the ones kept (none, or maybe only NCP would be great)\n* fig 2 is not readable in the printed version", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "HA20apUD-XF", "reviewer": "Reviewer_4Seq", "summary": "This paper proposed an unsupervised shape matching method, where the core idea is to first generate noisy maps from existing unsupervised shape matching methods, and refine them by learning pointwise features from the noisy maps, and build a new map by matching those features. The new map is empirically more accurate than the input noisy maps. Experiments are conducted to support the claim.\n", "questions": "1. Is it possible if one repeats the process of learning features from a map and matching the features to produce a map, i.e., steps 2-5, would one get further improvement? \n2. What is the main difference between 4.1 NCP and 4.2 test time denoising? In 4.2 do you train an extra network beyond that already trained in 4.1 NCP? Is the purpose of 4.2 to propose an early stopping criterion for NCP?\n", "limitations": "N/A", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The reviewer finds the work novel and interesting. The main concerns are as follows:\n\n1. How is the loss in line 260 differentiable with respect to the network parameters? Since it appears that Pi_MN lies in the set of permutations, which is a discrete set.\n2. In section 3.2 it says early stopping is needed for training the random-initialized network to learn features. However, it appears that early stopping criterion is only proposed in 4.2 but not 4.1. Is it true that 4.2 is not an optional step of denoising, but actually an important step for early stopping?\n\n\nMinor issues on presentation: \n3. The notations of Theorem 1 (lines 201-204) look a bit confusing. Does N(M)_x mean the mapping N restricted on M applied on x \\in M? \nIf so, I wonder if N(M)_x a standard notation, or if N|_M(x) is more standard? Since in the former, N(M) could mean the image of M under N.\nIf not, what does this notation mean? \n4. The fonts in the figures could be improved, in particular, they are way too small compared to the body text, and it seems that there is space.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept