Original: Correspondence-Free Point Cloud Registration with SO(3)-Equivariant Implicit Shape RepresentationsAnonymous Author(s)
Revised: Correspondence-Free Point Cloud Registration with SO(3)-Equivariant Implicit Shape Representations

Original: AfﬁliationAddress

Original: Abstract: This paper proposes a correspondence-free method for point cloud rotational registration.
Revised: Abstract: This paper proposes a correspondence-free method for point cloud rotational registration.

Original: We learn an embedding for each point cloud in a featurespace that preserves the SO(3)-equivariance property, enabled by recent developments in equivariant neural networks.
Revised: We learn an embedding for each point cloud in a feature space that preserves the SO(3)-equivariance property, enabled by recent developments in equivariant neural networks.

Original: The proposed shape registration methodachieves three major advantages through combining equivariant feature learningwith implicit shape models.
Revised: The proposed shape registration method achieves three major advantages through combining equivariant feature learning with implicit shape models.

Original: First, the necessity of data association is removedbecause of the permutation-invariant property in network architectures similar to
Revised: First, the necessity of data association is removed because of the permutation-invariant property in network architectures similar to

Original: PointNet.
Revised: PointNet.

Original: Second, the registration in feature space can be solved in closed-formusing Horn’s method due to the SO(3)-equivariance property.
Revised: Second, the registration in feature space can be solved in closed-form using Horn’s method due to the SO(3)-equivariance property.

Original: Third, the registration is robust to noise in the point cloud because of the joint training of registrationand implicit shape reconstruction.
Revised: Third, the registration is robust to noise in the point cloud because of the joint training of registration and implicit shape reconstruction.

Original: The experimental results show superior performance compared with existing correspondence-free deep registration methods.
Revised: The experimental results show superior performance compared with existing correspondence-free deep registration methods.

Original:  Introduction
Revised:  Introduction

Original: Point clouds are a major form of 3D information representation in perception with numerous applications today, including intelligent robots and self-driving cars.
Revised: Point clouds are a major form of 3D information representation in perception with numerous applications today, including intelligent robots and self-driving cars.

Original: The registration of two differentpoint clouds capturing the same underlying object or scene refers to estimating the relative transformation that aligns them.
Revised: The registration of two different point clouds capturing the same underlying object or scene refers to estimating the relative transformation that aligns them.

Original: Point cloud registration enables information aggregation across severalobservations, which is essential for many tasks such as mapping, localization, shape reconstruction,and object tracking.
Revised: Point cloud registration enables information aggregation across several observations, which is essential for many tasks such as mapping, localization, shape reconstruction, and object tracking.

Original: Given perfect point-to-point pairings between the point-cloud-pair, the relativetransformation can be solved in closed-form using Horn’s method [1].
Revised: Given perfect point-to-point pairings between the point-cloud-pair, the relative transformation can be solved in closed-form using Horn’s method [1].

Original: However, such a pairing is not available or  corrupted in real-world problems, posing a signiﬁcantchallenge for the point cloud registration task, namely data association.
Revised: However, such a pairing is not available or is corrupted in real-world problems, posing a signiﬁcant challenge for the point cloud registration task, namely data association.

Revised: ICP matches the closest points in Euclidean space and iteratively updates the matching using the current pose estimation [2].

Original: In the deep learning era,the strong feature learning capacity of neural networks enables better description and discriminationof points, improving their matching.
Revised: In the deep learning era, the strong feature learning capacity of neural networks enables better description and discrimination of points, improving their matching.

Original: Overall, these methods can be categorized as correspondencebased registration.
Revised: Overall, these methods can be categorized as correspondence-based registration.

Original: An alternative approach, called correspondence-free registration, is to completely circumvent thematching of points  by learning a global representation for the entire point cloud and estimatingthe transformation by aligning their representations.
Revised: An alternative approach, called correspondence-free registration, is to completely circumvent the matching of points [3] by learning a global representation for the entire point cloud and estimating the transformation by aligning their representations.

Original: Similar to correspondence-based registration,correspondence-free registration can also be with or without the involvement of deep learning.
Revised: Similar to correspondence-based registration, correspondence-free registration can also be with or without the involvement of deep learning.

Original: Themain obstacle in thistype of approach is that the mapping between the input Euclidean space andthe output feature space realized by neural networks is nonlinear and obscure.
Revised: The main obstacle in this approach is that the mapping between the input Euclidean space and the output feature space realized by neural networks is nonlinear and obscure.

Original: Thus it is commonto rely on iterative optimization  with local linearization like Gauss-Newton [2].
Revised: Thus it is common to rely on iterative optimization methods with local linearization such as Gauss-Newton [4].

Original: The performance ofthese approaches deteriorates when the initial pose difference gets larger.
Revised: The performance of these approaches deteriorates when the initial pose difference gets larger.

Revised: A method is called global if the registration result is independent of initial pose, or local otherwise [5].

Original:  In this paper, weinnovatively employ equivariant neural networks and implicit shape learning intocorrespondence-free registration by constructing a feature space with two major advantages.
Revised: Most existing correspondence-free methods are local. In this paper, we employ equivariant neural networks and implicit shape learning into correspondence-free registration, achieving two major advantages.

Original: First,thanks to the equivariance property, the feature space preserves the same rotation operation as the Euclidean input space, allowing us to solve the feature-space registration in  closed form using Horn’smethod.
Revised: First, because of the equivari- ance property, the feature space preserves the same rotation operation as the Euclidean input space, allowing us to solve the feature-space registration in a closed form using Horn’s method.

Original: As a result, the accuracy is independent of the initial pose difference.
Revised: As a result, our method is global , i.e., invariant to the initial pose difference.

Original: Second, we improve the registration robustness against noise in point clouds by training the features for both the implicitshape reconstructiontask andthe registration task.
Revised: Second, we improve the registration robustness against noise in point clouds to better handle challenges in real-world applications through training the features for both the implicit shape reconstruction and registration tasks.

Original: The implicit shape learning encourages thenetwork to learn features describing the underlying geometry rather than speciﬁc points sampled onthe geometry, and the features are better prepared for registration by including the registration in thetraining loop.
Revised: The implicit shape learning encourages the network to learn features describing the underlying geometry rather than speciﬁc points sampled on the geometry; thus, the features are better prepared for registration by including the registration in the training loop.

Original: In this work, we focus on the rotational registration task.
Revised: In this work, we focus on the rotational registration task.

Original: In particular,this work has the following contributions.
Revised: In particular, the main contribution of this paper are:

Original: We propose a correspondence-free point cloud registration method via SO(3) -equivariance representation learning and closed-form pose estimation.
Revised: We propose a correspondence-free point cloud registration method via SO(3) -equivariance representation learning and closed-form pose estimation 1 . 2.

Original: The proposed method achieves accurate registration results independent of initial pose dif- ferences with robustness against noisy points. Open source software will be available (after the ﬁnal decision).
Revised: The proposed method is global, delivering consistent registration regardless of initial rotation error. The proposed method is robust against imperfections in input point clouds, including noise, sampling, and density differences.

Original:  Preliminaries on Equivariance and Vector Neurons
Revised:  Preliminaries on Equivariance and Vector Neurons

Original: Symmetry (or equivariance) in a neural network is an important ingredient for efﬁcient learning andgeneralization.
Revised: Symmetry (or equivariance) in a neural network is an important ingredient for efﬁcient learning and generalization.

Original: It also allows us to build a clear connection between the changes in the input spaceand the output space.
Revised: It also allows us to build a clear connection between the changes in the input space and the output space.

Original: For example, convolutional neural networks are translation-equivariant.
Revised: For example, convolutional neural networks are translation-equivariant.

Original: Itmakes a network generalize to image shifting and enables huge parameter saving.
Revised: It makes a network generalize to image shifting and enables huge parameter saving.

Original: However, othersymmetries are less explored.
Revised: However, other symmetries are less explored.

Original: For example, most point cloud processing networks (e.g., PointNet [3], DGCNN [4]) are not equivariant to rotations.
Revised: For example, most point cloud processing networks (e.g., PointNet [6], DGCNN [7]) are not equivariant to rotations.

Original: We leverage the recently proposed Vector Neuron layers [5] to enable rotation-equivariant featurelearning for point clouds.
Revised: We leverage the recently proposed Vector Neuron layers [8] to enable rotation-equivariant feature learning for point clouds.

Original: For the sake of completeness, we brieﬂy introduce the preliminary onequivariance and the design of layers.
Revised: For the sake of completeness, we brieﬂy introduce the preliminary on equivariance and the design of layers.

Original: A function f : X → X is equivariant to a set of transformations G , if for any g ∈ G , f and gcommutes, i.e., g · f ( x ) =
Revised: A function f : X → X is equivariant to a set of transformations G , if for any g ∈ G , f and g commutes, i.e., g · f ( x ) =

Original: f ( g · x ) , ∀ x ∈ X .
Revised: f ( g · x ) , ∀ x ∈ X .

Original: For example, applying a translation on a 2D image andthen going through a convolution layer is identical to processing the original image with a convolution layer and then shifting the output feature map.
Revised: For example, applying a translation on a 2D image and then going through a convolution layer is identical to processing the original image with a convolution layer and then shifting the output feature map.

Original: Therefore convolution layers are translationequivariant.
Revised: Therefore convolution layers are translationequivariant.

Original: Vector Neurons extend the equvariance property to SO(3) rotations for point cloud neural networks.
Revised: Vector Neurons extend the equvariance property to SO(3) rotations for point cloud neural networks.

Original: The key idea is to augment the scalar feature in each feature dimension to a vector in R 3 and redesignthe linear, nonlinear, pooling, and normalization layers accordingly.
Revised: The key idea is to augment the scalar feature in each feature dimension to a vector in R 3 and redesign the linear, nonlinear, pooling, and normalization layers accordingly.

Original: In a regular network, the featurematrix with feature dimension C corresponding to a set of N points is V ∈ R N × C .
Revised: In a regular network, the feature matrix with feature dimension C corresponding to a set of N points is V ∈ R N × C .

Original: In Vector Neuronnetworks, the feature matrix is of form V ∈ R N × C × 3 .
Revised: In Vector Neuron networks, the feature matrix is of form V ∈ R N × C × 3 .

Original: The mapping between layers can be writtenas f : R N × C l × 3 → R N × C l +1 × 3 , where l is the layer index.
Revised: The mapping between layers can be written as f : R N × C l × 3 → R N × C l +1 × 3 , where l is the layer index.

Original: Following this design, the representationof SO(3) rotations in feature space is straight-forward: g ( R ) · V :=
Revised: Following this design, the representation of SO(3) rotations in feature space is straight-forward: g ( R ) · V :=

Original: V R , where g ( R ) denotes the rotation operation in the feature space, parameterized by the 3-by-3 rotation matrix R ∈ SO(3) .
Revised: V R , where g ( R ) denotes the rotation operation in the feature space, parameterized by the 3-by-3 rotation matrix R ∈ SO(3) .

Original: Inthe following, we ignore the ﬁrst dimension N of V for simplicity.
Revised: In the following, we ignore the ﬁrst dimension N of V for simplicity.

Original: The linear layer for Vector Neurons is similar to a traditional MLP:
Revised: The linear layer for Vector Neurons is similar to a traditional MLP:

Original: f lin ( V ) =
Revised: f lin ( V ) =

Original: WV , where
Revised: WV , where

Original: W ∈ R C l +1 × C l .
Revised: W ∈ R C l +1 × C l .

Original: It is easy to see that such a mapping is SO(3) -equivariant:
Revised: It is easy to see that such a mapping is SO(3) -equivariant:

Original: [Equation]
Revised: [Equation]

Original: Designing nonlinearities for Vector Neuron networks is less trivial.
Revised: Designing nonlinearities for Vector Neuron networks is less trivial.

Original: A naive ReLu on all elements inthe feature matrix will break the equivariance.
Revised: A naive ReLu on all elements in the feature matrix will break the equivariance.

Original: Instead, a vectorized version of ReLu is designed asthe truncation of vectors along a hyperplane.
Revised: Instead, a vectorized version of ReLu is designed as the truncation of vectors along a hyperplane.

Original: Speciﬁcally, given an input feature matrix V , predict acanonical direction k = UV ∈ R 1 × 3 where U ∈ R 1 × C .
Revised: Speciﬁcally, given an input feature matrix V , predict a canonical direction k = UV ∈ R 1 × 3 where U ∈ R 1 × C .

Original: Then the ReLU is deﬁned as:
Revised: Then the ReLU is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: otherwise

Original: [Equation]

Original: for each row v ∈ R 1 × 3 of V .
Revised: for each row v ∈ R 1 × 3 of V .

Original: This design preserves SO(3) equivariance because k and V willrotate together, and
Revised: This design preserves SO(3) equivariance because k and V will rotate together, and

Original: (cid:104) V R, kR (cid:105) =
Revised: (cid:104) V R, kR (cid:105) =

Original: (cid:104) V, k (cid:105) ,
Revised: (cid:104) V, k (cid:105) ,

Original: meaning that rotations do not change the activation.
Revised: meaning that rotations do not change the activation.

Original: Forthe design of pooling layers and normalization layers of Vector Neurons, please refer to the originalpaper for details [5].
Revised: For the design of pooling layers and normalization layers of Vector Neurons, please refer to the original paper for details [8].

Original:  Methodology
Revised:  Methodology

Original: We employ a SO(3) -equivariant neural network to construct a feature space preserving the samerotation representations as to the input space, thus simplifying solving the rotation for feature alignment.
Revised: We employ a SO(3) -equivariant neural network to construct a feature space preserving the same rotation representations as to the input space, thus simplifying solving the rotation for feature alignment.

Original: Furthermore, deep implicit representation learning and end-to-end training of registrationimproves the robustness of the features against noise and allows registration between two differentscans of the same geometry.
Revised: Furthermore, deep implicit representation learning and end-to-end training of registration improve the robustness of the features against noise and allow registration between two different scans of the same geometry.

Original: 3.1 SO(3) -equivariant point cloud global feature extractor
Revised: 3.1 SO(3) -equivariant point cloud global feature extractor

Original: We choose PointNet as the feature extractor backbone.
Revised: We choose PointNet as the feature extractor backbone.

Original: The permutation invariance property impliesthat two point clouds correspond to the same feature embeddings if they only differ in the permutation of points.
Revised: The permutation-invariant property implies that two point clouds correspond to the same feature embeddings if they only differ in the permutation of points.

Original: We replace the layers in PointNet with the corresponding Vector Neuron version [5].
Revised: We replace the layers in PointNet with the corresponding Vector Neuron version [8].

Original: Denote the feature extraction network as f , a point cloud P ∈ R N × 3 , and its rotated and permutedcopy P (cid:48) =
Revised: Denote the feature extraction network as f , a point cloud P ∈ R N × 3 , and its rotated and permuted copy P (cid:48) =

Original: MPR , where M ∈ R N × N is a permutation matrix and R ∈ SO(3) is a 3-by-3 rotationmatrix.
Revised: MPR , where M ∈ R N × N is a permutation matrix and R ∈ SO(3) is a 3-by-3 rotation matrix.

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: where Q := f ( P )
Revised: where Q := f ( P )

Original: ∈ R C × 3 is the feature of P extracted by the encoder.
Revised: ∈ R C × 3 is the feature of P extracted by the encoder.

Original: Equation (3) is guaranteedby the SO(3) -equivariance and permutation-invariance properties, which is essential for our featurespace registration.
Revised: Equation (3) is guaranteed by the SO(3) -equivariance and permutation-invariance properties, which is essential for our featurespace registration.

Original: A Vector Neuron network is designed to take inputs of R N × C 0 × 3 , while the point cloud input is
Revised: A Vector Neuron network is designed to take inputs of R N × C 0 × 3 , while the point cloud input is

Original: P ∈ R N × 3 .
Revised: P ∈ R N × 3 .

Original: Simply reshaping P to R N × 1 × 3 with C 0 = 1 does not work, because a linear layerwith weight W ∈ R C 1 × 1 will make the resulted feature
Revised: Simply reshaping P to R N × 1 × 3 with C 0 = 1 does not work, because a linear layer with weight W ∈ R C 1 × 1 will make the resulted feature

Original: WP ∈
Revised: WP ∈

Original: R N × C 1 × 3 linearly dependent in thethird dimension for each point.
Revised: R N × C 1 × 3 linearly dependent in the third dimension for each point.

Original: Following [5], we use one DGCNN edge-convolution [4] layer toinitialize the R N × C 0 × 3 feature from raw R N × 3 point cloud using the neighboring points.
Revised: Following [8], we use one DGCNN edge-convolution [7] layer to initialize the R N × C 0 × 3 feature from raw R N × 3 point cloud using the neighboring points.

Original: See [5]for more details.
Revised: See [8] for more details.

Original: Furthermore, we explore replacing the k-nearest neighbor operation with randomsampling of k points in a ball with  certain radius in the edge convolution layer.
Revised: Furthermore, we explore replacing the k-nearest neighbor operation with a random sampling of k points in a ball with a certain radius in the edge convolution layer.

Original: The motivation isto alleviate the sensitivity of the feature against point cloud density (see Sec. 4.2.3).
Revised: The motivation is to alleviate the sensitivity of the feature against point cloud density (see Sec. 4.2.3).

Original: However, theevaluations in this paper are using k-NN by default unless speciﬁed.
Revised: However, the evaluations in this paper use k-NN by default unless speciﬁed.

Original: 3.2 Deep implicit representation learning
Revised: 3.2 Deep implicit representation learning

Original: Being able to register two rotated copies of the same point cloud is not enough for practical use.
Revised: Being able to register two rotated copies of the same point cloud is not enough for practical use.

Original: Inreal application, the measurements have noise, and the individual points captured by different scansgenerally do not correspond to the same physical points.
Revised: In real applications, the measurements have noise, and the individual points captured by different scans generally do not correspond to the same physical points.

Original: Our solution is to repurpose the global feature (for registration ultimately) for implicit shape representation.
Revised: Our solution is to repurpose the global feature (for registration ultimately) for implicit shape representation.

Original: Following the Occupancy Network [6], we build an encoder-decoder network.
Revised: Following the Occupancy Network [9], we build an encoder-decoder network.

Original: The encoder is the aforementioned SO(3) -equivariant feature extractor f , taking a point cloud P ∈ R N × 3 as input.
Revised: The encoder is the aforementioned SO(3) -equivariant feature extractor f , taking a point cloud P ∈ R N ×as input.

Original: The decoder takes the encoded shape feature Q = f ( P ) ∈ R C × 3 and a queried positionp ∈ R 3 as input, predicting the occupancy value of that position v ( p ; Q ) ∈
Revised: The decoder takes the encoded shape feature Q = f ( P ) ∈ R C × 3 and a queried position p ∈ R 3 as input, predicting the occupancy value of that position v ( p ; Q ) ∈

Original: Notice that p neednot to be coming from P .
Revised: Notice that p need not be coming from P .

Original: In practice, we sample n querying points for each input to calculate theimplicit shape reconstruction (occupancy value prediction) loss:
Revised: In practice, we sample n querying points for each input to calculate the implicit shape reconstruction (occupancy value prediction) loss:

Original: [Equation]
Revised: [Equation]

Original: where v gt ( p ) ∈ { 0 , 1 } is the ground truth occupancy value of a query point p .
Revised: where v gt ( p ) ∈ { 0 , 1 } is the ground truth occupancy value of a query point p .

Original: The sampling strategyof the querying points is consistent with the Occupancy Network [6].
Revised: The sampling strategy of the querying points is consistent with the Occupancy Network [9].

Original: In this way, we implicitlyencourage the feature f ( P 1 ) , f ( P 2 ) , ... to be close as long as P 1 , P 2 , ... are point clouds of the samegeometry.
Revised: In this way, we implicitly encourage the feature f ( P 1 ) , f ( P 2 ) , ... to be close as long as P 1 , P 2 , ... are point clouds of the same geometry.

Original: 3.3 Point cloud registration through feature alignment
Revised: 3.3 Point cloud registration through feature alignment

Original: With the above network design, we can further relax the connection between the two point clouds
Revised: With the above network design, we can further relax the connection between the two point clouds

Original: P (cid:48) and P in (3).
Revised: P (cid:48) and P in (3).

Original: The features Q and Q (cid:48) are approximately rotation-equivariant even when P and P (cid:48)are not permutations of each other, as long as they are sampled on the same geometry.
Revised: The features Q and Q (cid:48) are approximately rotation-equivariant even when P and P (cid:48) are not permutations of each other, as long as they are sampled on the same geometry.

Original: We say it isapproximate because a trained implicit shape reconstruction network can only represent the implicitshape approximately.
Revised: We say it is approximate because a trained implicit shape reconstruction network can only represent the implicit shape approximately.

Original: Therefore, we also include the registration step in the training loop, directlyencouraging the accurate registration of point clouds with noise.
Revised: Therefore, we also include the registration step in the training loop, directly encouraging the accurate registration of point clouds with noise.

Original: As shown below, the registration isof closed form and  differentiable, allowing end-to-end training ofbetter features.
Revised: As shown below, the registration has a closed-form solution and is differentiable, allowing end-to-end training of features.

Original: One can see that (3) is exactly in the form of  orthogonal Procrustes problem.
Revised: One can see that (3) is exactly in the form of an orthogonal Procrustes problem.

Original: The learned features
Revised: The learned features

Original: Q and Q (cid:48) can be regarded as two pseudo point clouds and each row of them are automaticallymatched .
Revised: Q and Q (cid:48) can be regarded as two pseudo point clouds, and each row of them is automatically matched .

Original: This problem has a standard closed-form solution.
Revised: This problem has a standard closed-form solution.

Original: First calculate the cross covariancematrix H = Q T Q (cid:48) . Then conduct Singular Value Decomposition (SVD) on H : H = USV T .
Revised: First, calculate the cross-covariance matrix H = Q T Q (cid:48) followed by its Singular Value Decomposition (SVD) as H = USV T .

Original: Theoptimal rotation matrix ˆ R is given as:
Revised: The optimal rotation matrix ˆ R is given as

Original: [Equation]
Revised: [Equation]

Original:  where Λ = diag (1 , 1 , det ( V U T ))
Revised: (5) where Λ = diag (1 , 1 , det ( V U T ))

Original: ∈ R 3 × 3 .
Revised: ∈ R 3 × 3 .

Original: We design the registration loss as the mean square error between the residual rotation matrix and theidentity matrix:
Revised: We design the registration loss as the mean squared error between the residual rotation matrix and the identity matrix, i.e., chordal distance, as

Original: [Equation]
Revised: [Equation]

Original:  where R est is calculated from (5), and (cid:107)(cid:107) F denotes the Frobenius norm.
Revised: (6) where R est is calculated from (5), and (cid:107)·(cid:107) F denotes the Frobenius norm.

Original:  Experiments 
Revised:  Experiments 4.1 Overview

Original: 4.1 Overview

Original: Our network is trained on the ModelNet40 dataset [7].
Revised: Our network is trained on the ModelNet40 dataset [10].

Original: It is composed of 12311 CAD models from40 categories of objects.
Revised: It is composed of 12311 CAD models from 40 categories of objects.

Original: Following the ofﬁcial split, 9843 models are used for training, and 2468models are used for testing.
Revised: Following the ofﬁcial split, 9843 models are used for training, andmodels are used for testing.

Original: The models are preprocessed using the tool provided by Stutz and
Revised: The models are preprocessed using the tool provided by Stutz and

Original: Geiger [8] to obtain watertight and simpliﬁed models that are centered and scaled to a unit cube.
Revised: Geiger [11] to obtain watertight and simpliﬁed models that are centered and scaled to a unit cube.

Original: Given watertight models, the occupancy value of any 3D coordinate can be determined dependingon whether it is inside or outside the model.
Revised: Given watertight models, the occupancy value of any 3D coordinate can be determined depending on whether it is inside or outside the model.

Original: We also evaluate our method on a real-world  dataset: 7Scenes [9], which is a subset of 3DMatch[10].
Revised: We also evaluate our method on a real-world indoor dataset: 7Scenes [12], which is a subset of 3DMatch [13].

Original: We choose itin order to be consistent and comparable with one of our baselines FMR [11].7Scenes consists of 7 sequences of RGBD scans of indoor environments.
Revised: We choose it to be consistent and comparable with one of our baselines, FeatureMetric Registration (FMR) [14]. 7Scenes consists of 7 sequences of RGBD scans of indoor environments.

Original:  RGB information is notused in the experiments.
Revised: The RGB information is not used in the experiments.

Revised: [Table]

Original: The training of the network is composed of two stages.
Revised: The training of the network is composed of two stages.

Original: The network is ﬁrst trained for occupancyvalue prediction only, after which the network is trained for the occupancy prediction and registrationtasks jointly.
Revised: The network is ﬁrst trained for occupancy value prediction only, after which the network is trained for the occupancy prediction and registration tasks jointly.

Original: In the ﬁrst stage, 1024 points are randomly sampled on each mesh model as the inputpoint cloud. with batch size 24.
Revised: In the ﬁrst stage, 1024 points are randomly sampled on each mesh model as the input point cloud with batch size 24.

Original: In the second stage, a pair of point clouds are sampled on the sameshape model with  different pose and number of points.
Revised: In the second stage, a pair of point clouds are sampled on the same shape model with a different pose and number of points.

Original: The registration loss between the pair iscalculated, together [Table] [Table]
Revised: The registration loss between the pair is calculated, together with the occupancy loss of each of the point clouds.

Original: [Table]
Revised: The occupancy value is queried at 2048 points in the unit cube.

Original: The  dimension of the learned global feature is set as C = 342 , corresponding to 342 × 3 = 1026dimensions for a traditional feature vector, close to the dimension chosen by previous literature including PointNetLK [12], PCRNet [13] [Table] and Feature-Metric Registration (FMR) [11].
Revised: The batch size of such pairs is 10 in this stage. The dimension of the learned global feature is set as C = 342 , corresponding to 342 × 3 =dimensions for a traditional feature vector, close to the dimension chosen by previous literature including PointNetLK [3], PCRNet [15], and FMR [14].

Original: The networkis trained on a single Tesla V100 GPU.
Revised: The network is trained on a single Tesla V100 GPU.

Original: 4.2 Evaluation on synthetic data: ModelNet
Revised: Representative correspondence-free methods PCRNet and FMR are our baselines. PointNetLK also falls in this category, which we omit here because FMR follows a very similar approach to PointNetLK and can be viewed as its improved version. We also included results from some representative correspondence-based methods to give readers a better context of both types of approaches. The experiments on the baselines are based on their ofﬁcial open-source implementation and pretrained weights.

Revised: In all following experiments, random initial rotations are generated using the axis-angle representation, of which the angle is sampled from a uniform distribution with the speciﬁed maximum, and the rotation axis is also randomly picked.

Revised: The evaluation metric is the mean isotropic rotation error as in [16].

Original:  4.2.1 Registration with rotated copies of point clouds
Revised: 4.2 Evaluation on synthetic data: ModelNet 4.2.1 Registration with rotated copies of point clouds

Original: We ﬁrst tested the registration of two points clouds that are only different by rotation and permutation.
Revised: We ﬁrst tested the registration of two points clouds that are only different by rotation and permutation.

Original: Both point clouds have 1000 points sampled from the mesh model.
Revised: Both point clouds have 1024 points sampled from the mesh model.

Original: The initial rotationsare generated randomly using axis-angle representation to specify the maximum magnitude.

Original: Theerror is also measured as the angle in axis-angle representations.

Original: We compare with two representative correspondence-free deep registration methods as baselines: PCR-Net [13] and FMR [11].

Original: The experiments on the baseline methods are based on their ofﬁcial open-source implementationand pretrained weights.

Original: The results are shown in Table 1.
Revised: The results are shown in Table 1.

Original: We can see that  both PCR-Net and FMRare sensitive to the initial rotation angle and only work properly when the initial rotation is smallbecause they rely on iterative reﬁnements.
Revised: We can see that as local methods, both PCR-Net and FMR only work properly when the initial rotation is small because they rely on iterative reﬁnements.

Original: In comparison,the registration accuracy of our methodis independent of the initial rotation angle, and it provides almost perfect rotational registration.
Revised: In comparison, our method provides consistent and almost perfect registration independent of the initial rotation angle.

Revised: Both correspondence-based methods, RPM-Net and DeepGMR, work well in all initial conditions.

Revised: DeepGMR delivers similarly perfect accuracy as ours, both of which are global methods.

Revised: The error of RPM-Net slightly increases as the initial rotation gets larger while remaining small overall, probably because the inputs are noise-free, and perfectly matching pairs of points exist in the data, allowing the correspondence-ﬁnding process to work nicely.

Original: 4.2.2 Registration with rotated copies of point clouds corrupted with Gaussian noise
Revised: 4.2.2 Registration with rotated copies of point clouds corrupted with Gaussian noise

Original: In practice, the pair of point clouds for registration will not be identical to each other.
Revised: In practice, the pair of point clouds for registration will not be identical to each other.

Original: Thereforewe test the case where the source and target point clouds are both corrupted by noise.
Revised: Therefore we test the case where the source and target point clouds are both corrupted by noise.

Original: We add a
Revised: We add a

Original: Gaussian noise with σ = 0 .
Revised: Gaussian noise with σ = 0 .

Original: 01 to all points before putting them through the networks.
Revised: 01 to all points before putting them through the networks.

Original: The rest ofthe setup is the same as in Sec.
Revised: The rest of the setup is the same as in Sec.

Original: As shown in Table 2, PCR-Net and FMR show a similartrend of increasing error when the initial angle gets larger.
Revised: As shown in Table 2, PCR-Net and FMR show a similar trend of increasing error when the initial angle gets larger.

Original: Our method, though showing a slightlylarger registration error, still outperforms the baseline in most columns.
Revised: Our method, though showing a slightly larger registration error, still outperforms the baseline in most columns.

Original: Most importantly, the errorremains consistent across different initialization.
Revised: Most importantly, the error remains consistent across different initialization.

Revised: [Table]

Original: The result shows that  our  method not only works inan ideal situation where the point clouds are noise-free but also delivers accurate rotation estimationunder noise corruption.
Revised: [Table] As the point-wise correspondence is corrupted by noise, RPM-Net deteriorates fast as the initial rotation grows. DeepGMR performs the best, showing the beneﬁt of global methods and that its component-wise correspondence is more robust to noise. It also shows a small gap between our correspondence-free method and the state-of-the-art correspondence-based global method, motivating future improvements.

Original: 4.2.3 Registration with point clouds with different densities
Revised: 4.2.3 Registration with point clouds with different densities

Original: To further test the robustness of the proposed method against variations in the point clouds, wetested the registration performance when given two point clouds of different densities.
Revised: To further test the robustness of the proposed method against variations in the point clouds, we tested the registration performance when given two point clouds of different densities.

Revised: It is a practical challenge when the sensor changes or one conducts frame-to-model alignments.

Original: Here wesample 1000 and 500 points for the pair of point clouds, respectively.
Revised: Here we sample 1024 and 512 points for the pair of point clouds, respectively.

Original: The registration accuracyis shown in Table 3.The trend of the three methods is similar to what is shown in Sec. 4.2.1 and Sec.
Revised: The result is shown in Table 3.

Original: We do observe an increase of error in our method in this experiment.
Revised: We do observe an increase of error in our method in this experiment.

Original: A possible reasonis that the DGCNN layer for feature initialization at the beginning of the encoder is sensitive to thepoint cloud density, since it takes the relative coordinate of neighboring points as inputs.
Revised: A possible reason is that the DGCNN layer for feature initialization at the beginning of the encoder is sensitive to the point cloud density since it takes the relative coordinate of neighboring points as inputs.

Original: Therefore, we  experimented  replacing the k-nearest neighbor with random sampling in a ball centered at each point when collecting neighboring points for the edge-convolution of that point.
Revised: Therefore, we also experimented with replacing the k-nearest neighbor with random sampling in a ball centered at each point when collecting neighboring points for the edge-convolution of that point.

Revised: A more detailed discussion is in the supplementary material.

Original: Inpractice, we sample 20 neighboring points for both strategy, and the radius for ball sampling is 0.2. [Table] Since our method is invariant to [Table]
Revised: With either of the strategies, the registration error is slightly larger than that of the baselines when the initial rotation angle is small but is much smaller when the initial rotation is arbitrary.

Original: , in this only [Table] [Table] of Table
Revised: We notice a similar trend in the correspondence-based methods.

Original: [Table] The result is in Table 4. One can see that the ball sampling strategy is able to improve the registration performance for point clouds of different density. However, the error is slightly highercompared with k-NN when the point clouds are rotated copies or with Gaussian noise with the samenumber of points.
Revised: RPM-Net outperforms DeepGMR if the initial rotation is small but performs much worse otherwise.

Original: It is understandable since the ball sampling introduces randomness in the featureinitialization, and could slightlyworsen the registration when the inputs are originally ideal. With either of the strategies, under the condition of density difference, the registration error isslightly larger  than that of the baselines when the initial rotation angle is small, but is much smallerwhen the initial rotation is arbitrary. A possible way for further improvement is to use our methodas initialization, and then run an iterative-optimization-based method such as FMR or PointNetLKfor reﬁnement. It is not closely related to our key innovation, therefore not further discussed in thispaper.
Revised: Between the two global methods, DeepGMR shows a slightly larger error than ours.

Original: 4.3 Evaluation on real-world data: 7Scenes
Revised: 4.3 Evaluation on real-world data: 7Scenes

Original: The model evaluated on 7Scenes is ﬁrst trained on ModelNet40 with the two stages mentionedin Sec. 4.1, and then ﬁnetuned on 7Scenes with the registration loss only.
Revised: The model evaluated on 7Scenes is ﬁrst trained on ModelNet40 with the two stages mentioned in Sec. 4.1, and then ﬁnetuned on 7Scenes with the registration loss only.

Original: The train/test split isconsistent with [11].
Revised: The train/test split is consistent with the work of Huang et al.

Original: 4.3.1 Registration with point clouds that are sampled differently

Original: The experiments on 7Scenes consists of two parts.
Revised: The experiments on 7Scenes consist of two parts.

Original: In the ﬁrst part, the pair of point clouds are fromthe same frame, but sampled differently.
Revised: In the ﬁrst part, the pair of point clouds are from the same frame but sampled differently.

Revised: In the second part, the pair of point clouds are from adjacent frames and only overlaps partially.

Revised: They reﬂect challenges in real-world applications since any change of the viewpoint of the sensor could result in different samples of points and visible parts.

Revised: Due to the page limit, we only show the result of the ﬁrst part here, and the second part is in the supplementary material.

Revised: 4.3.1 Registration with point clouds that are sampled differently

Original: The original dense RGBD scan has more than 100,000points.
Revised: The original dense RGBD scan has more than 100,000 points.

Original: 1000 points are sampled randomly to form an input point cloud.
Revised: 1024 points are sampled randomly to form an input point cloud.

Original: The pair of point cloudsthen go through random rotation to form the ﬁnal input.
Revised: The pair of point clouds then go through random rotation to form the ﬁnal input.

Original: In this case, the points in one point cloudis neither a noisy version nor a subset ofthe other point cloud.
Revised: In this case, the points in one point cloud are neither a noisy version nor a subset of

Original: To our knowledge, we are one of theﬁrst to report registration results under this setting in correspondence-free methods [12, 11, 13, 14]. Table 5 shows that our method is working properly on real world scan of point clouds that aredifferent samplings of the underlying geometry. 4.3.2 Registration with point clouds that are partially overlapping In the second part, we report registration results of point clouds coming from adjacent frames in thesequence.
Revised: [Table]

Original: Because of camera movement, the pair of point clouds only overlaps partially. Since ourmethod only estimates rotations, we ﬁrst align the adjacent frames using ground truth pose, thenrandomly rotate two point clouds separately in a common frame, so that the input point clouds arerelated by a rotation. To our knowledge, we are also one of the ﬁrst to show quantitative results onpartially overlapping point clouds in correspondence-free methods [12, 11, 13, 14]. The results are in Table 6.
Revised: the other.

Original: Both our method and the baseline deteriorate in this setting. For FMR,the system completely fails when taking partially-overlapping inputs. Our method consistently givesbetter rotation estimation under all initial conditions. Still, the accuracy has a lot of room for improvement, indicating that there is still a long way to go for correspondence-free registration methods.
Revised: To our knowledge, we are one of the ﬁrst to report registration results under this setting in correspondence-free methods [3, 14, 15].

Original: For example, recent work by Huang et al. [15] predicting the overlapping part of point cloudpairs may inspire correspondence-free methods to better handle partial overlapping inputs by predicting features focused on the overlapping parts. [Table] [Table]
Revised: Table 4 shows that our method is working properly on real-world scans of point clouds that are different samplings of the underlying geometry.

Original: Overall, our proposed rotational registration method can estimate consistent results independent ofinitial rotations.
Revised: Overall, our proposed rotational registration method can estimate consistent results independent of initial rotations.

Original: It outperforms the correspondence-free baselines in all evaluation settings wheninitial rotational angle is arbitrary.
Revised: It outperforms the correspondence-free baselines in all evaluation settings when the initial rotational angle is arbitrary.

Original: Experiments onboth syntheticdata and real-world data areconducted, covering conditions with noisy inputs, varying density, different sampling, and partialoverlapping.
Revised: Experiments on synthetic and real-world data are conducted, covering conditions with noisy inputs, varying density, different sampling, and partial overlapping.

Original: Although outperforming the baseline, the partial overlapping case is not perfectlyhandled by our method yet, and we brieﬂy discussed possible future directions.
Revised: However, our method still shows some error increase when challenging inputs are presented, motivating future improvements.

Original:  Related Work and Discussion 
Revised:  Related Work and Discussion 5.1 Correspondence-based point cloud registration

Original: 5.1 Correspondence-based point cloud registration

Original: A major challenge here is to recover the corresponding pairs of points from a pair of point clouds.
Revised: A major challenge here is to recover the corresponding pairs of points from a pair of point clouds.

Original: ICP simply matches the closest points together, solves the transformation, and iteratively rematchesthe closest points after aligning the two point clouds using the estimated transformation [16].
Revised: ICP simply matches the closest points together, solves the transformation, and iteratively rematches the closest points after aligning the two point clouds using the estimated transformation [2].

Original: Pointto-line [17], point-to-plane [18], plane-to-plane [19], and Generalized-ICP [20] build local geometricstructures to the loss formulation.
Revised: Point-toline [17], point-to-plane [18], plane-to-plane [19], and Generalized-ICP [20] build local geometric structures to the loss formulation.

Original: Finding correspondences require strong feature descriptor for points, on which deep learning approaches show expertise.
Revised: Finding correspondences require strong feature descriptor for points, on which deep learning approaches show expertise.

Original: Through metric learning, a feature descriptor can be learned such thatmatching points are close to each other in the feature space, while non-matching points are far away.
Revised: Through metric learning, a feature descriptor can be learned such that matching points are close to each other in the feature space, while non-matching points are far away.

Original: Approaches following this idea include PPFNet [21], 3DSmoothNet [22], SpinNet [23], and FCGF[24].
Revised: Approaches following this idea include PPFNet [21], 3DSmoothNet [22], SpinNet [23], and FCGF [24].

Original: Good point matching leads to accurate pose estimation when solving the orthogonal Procrustesproblem.
Revised: Good point matching leads to accurate pose estimation when solving the orthogonal Procrustes problem.

Original: Therefore, the error in pose estimation can be used to supervise point matching and feature learning.
Revised: Therefore, the error in pose estimation can be used to supervise point matching and feature learning.

Original: DCP [25], RPM-Net [26], DGR [27], and 3DRegNet [28] are some of the methodsleveraging ground truth pose to supervise point feature learning.
Revised: DCP [25], RPM-Net [16], DGR [26], and 3DRegNet [27] are some of the methods leveraging ground truth pose to supervise point feature learning.

Original: Neural networks are designed tobetter pick the keypoints (e.g., USIP [29] and SampleNet [30]).
Revised: Neural networks are designed to better pick the keypoints (e.g., USIP [28] and SampleNet [29]).

Original: Keypoint selection and featurelearning may also be considered jointly.
Revised: Keypoint selection and feature learning may also be considered jointly.

Original: Representative works include 3DFeat-Net  [31],D3Feat[32], DeepICP [33], and PRNet [34].
Revised: Representative works include 3DFeat-Net [30], D3Feat [31], DeepICP [32], and PRNet [33].

Original: The main remaining challenge is that such matching pairs may not exist in the input point cloudsin the ﬁrst place, because point clouds are sparse and a point may not be captured repeatedly bydifferent scans.
Revised: The main remaining challenge is that such matching pairs may not exist in the input point clouds in the ﬁrst place, because point clouds are sparse and a point may not be captured repeatedly by different scans.

Original: Soft matching (or many to many) [35, 36] is proposed to address this problem,but it is at best an approximation of the underlying true matching using sparse samples, which candeteriorate the performance when the sparsity increases.
Revised: Soft matching (or many to many) [34, 35] is proposed to address this problem, but it is at best an approximation of the underlying true matching using sparse samples, which can deteriorate the performance when the sparsity increases.

Revised: DeepGMR [5] learns latent Gaussian Mixture Models (GMMs) for point clouds and establishes component-to-component correspondences.

Original: 5.2 Correspondence-free point cloud registration
Revised: 5.2 Correspondence-free point cloud registration

Original: Correspondence-free registration treats a point cloud as a whole rather than a collection of points,requiring a global representation for an entire point cloud.
Revised: Correspondence-free registration treats a point cloud as a whole rather than a collection of points, requiring a global representation for an entire point cloud.

Original: In this way, the limitation brought by thesparsity as mentioned in Sec.
Revised: In this way, the limitation brought by the sparsity as mentioned in Sec.

Original: 5.1 is circumvented.
Revised: 5.1 is circumvented.

Original: CVO [37, 38] represents a point cloud as a function in a reproducing kernel Hilbert space, transforming the registration problem to maximizing theinner product of two functions.
Revised: CVO [36, 37] represents a point cloud as a function in a reproducing kernel Hilbert space, transforming the registration problem to maximizing the inner product of two functions.

Original: A series of deep-learning-based methods attempt to extract a globalfeature embedding for a point cloud and solve the registration by aligning the global features.
Revised: A series of deep-learning-based methods attempt to extract a global feature embedding for a point cloud and solve the registration by aligning the global features.

Original: Examples include PointNetLK [12], PCRNet [13], and Feature-Metric Registration [11].
Revised: Examples include PointNetLK [3], PCRNet [15], and Feature-Metric Registration [14].

Original: A limitationof global-feature-based registration is that the nonlinearity of the feature extraction networks leavesus few clues about the structure and properties of the feature space.
Revised: A limitation of global-feature-based registration is that the nonlinearity of the feature extraction networks leaves us few clues about the structure and properties of the feature space.

Original: Therefore these methods relyon iterative local optimization such as the Gauss-Newton algorithm.
Revised: Therefore these methods rely on iterative local optimization such as the Gauss-Newton algorithm.

Original: Consequently, these methodsrequire good initialization to achieve decent results.
Revised: Consequently, these methods require good initialization to achieve decent results.

Original: Our method differentiates from previous work in that the feature extraction network is SO(3) equivariant, enabling well-behaved optimization inthe feature space (detailed in Sec. 3).
Revised: Our method differentiates from previous work in that the feature extraction network is SO(3) equivariant, enabling well-behaved optimization in the feature space (detailed in Sec. 3).

Original: 5.3 Group equivariant neural networks
Revised: 5.3 Group equivariant neural networks

Original: Neural networks today mainly preserve symmetry against translation, limiting the capacity of networks to deal with broader transformations presented in data.
Revised: Neural networks today mainly preserve symmetry against translation, limiting the capacity of networks to deal with broader transformations presented in data.

Original: One line of work forms kernels assome steerable functions so that the rotations in the input space can be transformed into rotationsin the output space [39, 40, 41].
Revised: One line of work forms kernels as some steerable functions so that the rotations in the input space can be transformed into rotations in the output space [38, 39, 40].

Original: However, this strategy can limit the expressiveness of the featureextractor since the form of the kernels is constrained.
Revised: However, this strategy can limit the expressiveness of the feature extractor since the form of the kernels is constrained.

Original: Another strategy is to lift the input space to a higher-dimensional space where the group is contained so that equivariance is obtained naturally [42, 43].
Revised: Another strategy is to lift the input space to a higher-dimensional space where the group is contained so that equivariance is obtained naturally [41, 42].

Original: However, for each lifted input to K groupelements, we need to integrate over the entire group for computing its convolution.
Revised: However, for each lifted input to K group elements, we need to integrate over the entire group for computing its convolution.

Original: While beingmathematically sound, this approach involves the use of group exp and log maps (for mapping fromand to the Lie algebra) and a Monte Carlo approximation to compute the convolution integral overa ﬁxed set of group elements [43, Sections 4.2-4.4].
Revised: While being mathematically sound, this approach involves the use of group exp and log maps (for mapping from and to the Lie algebra) and a Monte Carlo approximation to compute the convolution integral over a ﬁxed set of group elements [42, Sections 4.2-4.4].

Original: Cohen and Welling [44] studied the problem of learning the irreducible representations of commutative Lie groups.
Revised: Cohen and Welling [43] studied the problem of learning the irreducible representations of commutative Lie groups.

Original: Kondor and Trivedi [42] studied the generalization of equivariance and convolutionin neural networks to the action of compact groups.
Revised: Kondor and Trivedi [41] studied the generalization of equivariance and convolution in neural networks to the action of compact groups.

Original: The latter also discussed group convolutions andtheir connection with Fourier analysis [45, 46].
Revised: The latter also discussed group convolutions and their connection with Fourier analysis [44, 45].

Original: In this context, the convolution theorem naturallyextends to the group-valued functions and the representation theory [47].
Revised: In this context, the convolution theorem naturally extends to the group-valued functions and the representation theory [46].

Original: A new design of SO(3) equivariant neural networks is proposed by Deng et al.
Revised: A new design of SO(3) equivariant neural networks is proposed by Deng et al.

Original: This approachallows incorporating the SO(3) equivariance property into existing network architectures such as
Revised: This approach allows incorporating the SO(3) equivariance property into existing network architectures such as

Original: PointNet and DGCNN by replacing the linear, nonlinear, pooling, and normalization layers withtheir vector neuron version.
Revised: PointNet and DGCNN by replacing the linear, nonlinear, pooling, and normalization layers with their vector neuron version.

Original: Our equivariant feature learner builds on this simple and useful idea.
Revised: Our equivariant feature learner builds on this simple and useful idea.

Original: 5.4 Deep implicit shape 
Revised: 5.4 Deep implicit shape modeling

Original: [Figure]

Original: Implicit shape modeling represents a shape as a ﬁeld on which a function such as an occupancy orsigned distance function is deﬁned.
Revised: Implicit shape modeling represents a shape as a ﬁeld on which a function such as an occupancy or signed distance function is deﬁned.

Original: The surface is typically deﬁned by the equation f ( x, y, z )
Revised: The surface is typically deﬁned by the equation f ( x, y, z )

Original: = cfor some constant c .
Revised: = c for some constant c .

Original: In recent years, deep models are developed to model such implicit functionﬁelds, where a network is trained to predict the function value given a query input point coordinate.
Revised: In recent years, deep models are developed to model such implicit function ﬁelds, where a network is trained to predict the function value given a query input point coordinate.

Original: Some representative works include DeepSDF [48], OccNet [6], NeRF [49].
Revised: Some representative works include DeepSDF [47], OccNet [9], NeRF [48].

Original: Implicit functions model a shape continuously, up to the inﬁnite resolution, which offers opportunities in registration to circumvent the challenge in data association due to sparsity in point clouds.
Revised: Implicit functions model a shape continuously, up to the inﬁnite resolution, which offers opportunities in registration to circumvent the challenge in data association due to sparsity in point clouds.

Original: DI-Fusion [50] leverages deep implicit models to do surface registration and mapping by optimizingthe position of the query points to minimize the absolute SDF value.
Revised: DI-Fusion [49] leverages deep implicit models to do surface registration and mapping by optimizing the position of the query points to minimize the absolute SDF value.

Original: iNeRF [51] performs RGBonly registration through deep implicit models by minimizing the pixel residual between observedimages and neural-rendered images.
Revised: iNeRF [50] performs RGBonly registration through deep implicit models by minimizing the pixel residual between observed images and neural-rendered images.

Original: In this work, we also exploit the continuous nature of the deepimplicit model but by using the learned latent feature instead of the decoded function ﬁeld.
Revised: In this work, we also exploit the continuous nature of the deep implicit model but by using the learned latent feature instead of the decoded function ﬁeld.

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we presented a correspondence-free rotational registration method for point clouds.
Revised: In this paper, we presented a correspondence-free rotational registration method for point clouds.

Original: The method is built upon the developments in equivariant neural networks and implicit shape representation learning.
Revised: The method is built upon the developments in equivariant neural networks and implicit shape representation learning.

Original: We construct a feature space where the rotational relations in Euclidean space arepreserved, in which registration can be done efﬁciently.
Revised: We construct a feature space where the rotational relations in Euclidean space are preserved, in which registration can be done efﬁciently.

Original: We circumvent the need for data associationand solve the rotation in closed form, achieving low errors independent of initial rotation angles.
Revised: We circumvent the need for data association and solve the rotation in closed form, achieving low errors independent of initial rotation angles.

Original: Furthermore, we leverage implicit representation learning and end-to-end registration to enhancethe network robustness against non-ideal cases where the points are not exactly corresponding.
Revised: Furthermore, we leverage implicit representation learning and end-to-end registration to enhance the network robustness against non-ideal cases where the points are not exactly corresponding.

Original: We conducted experiments on synthetic and real-world data, showing the network’s ability to handlevarious input imperfections. Nevertheless, the robustness of our method against partial overlappinginputs still has room for improvement.
Revised: We conducted experiments on synthetic and real-world data, showing the network’s ability to handle various input imperfections while also motivating future work for improved performance.

Original: Furthermore, there are two open issues unsolved.
Revised: Furthermore, there are two open issues unsolved.

Original: First,generalizing the SO(3) -equivariant neural network to SE(3) so that the registration of rotation andtranslation can be solved together.
Revised: The ﬁrst problem is on generalizing the SO(3) -equivariant neural network to SE(3) so that the registration of rotation and translation can be solved together.

Original: Secondly, generalizing the registration to handle outdoor LiDARscans.
Revised: The second problem is on generalizing the registration to handle outdoor LiDAR scans.

Original: These problems are left for future studies.
Revised: These problems are left for future studies.

Revised: Acknowledgment

Revised: Toyota Research Institute provided funds to support this work.

Revised: Funding for M. Ghaffari was in part provided by NSF Award No. 2118818.