Original: One-Inlier is Enough: Towards Efﬁcient Position Encoding for Point Cloud Registration
Revised: One-Inlier is First: Towards Efﬁcient Position Encoding for Point Cloud Registration

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress 
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Transformer architecture has shown great potential for many visual tasks, includingpoint cloud registration.
Revised: Transformer architecture has shown great potential for many visual tasks, includingpoint cloud registration.

Original: As an order-aware module, position encoding plays animportant role in Transformer architecture applied to point cloud registration task.
Revised: As an order-aware module, position encoding plays animportant role in Transformer architecture applied to point cloud registration task.

Original: In this paper, we propose a one-inlier based position encoding method for pointcloud registration network.
Revised: In this paper, we propose a one-inlier based position encoding method for pointcloud registration network.

Original: Speciﬁcally, we ﬁrst ﬁnd one correspondence by adifferentiable optimal transport layer, and use it to normalize each point for positionencoding.
Revised: Speciﬁcally, we ﬁrst ﬁnd one correspondence by adifferentiable optimal transport layer, and use it to normalize each point for positionencoding.

Original: It can eliminate the challenges brought by the different reference framesof two point clouds, and mitigate the feature ambiguity by learning the spatialconsistency.
Revised: It can eliminate the challenges brought by the different reference framesof two point clouds, and mitigate the feature ambiguity by learning the spatialconsistency.

Original: Then, we propose a joint approach for establishing correspondenceand position encoding, presenting an iterative optimization process.
Revised: Then, we propose a joint approach for establishing correspondenceand position encoding, presenting an iterative optimization process.

Original: Finally, wedesign a progressive way for point cloud alignment and feature learning to gradually optimize the rigid transformation.
Revised: Finally, wedesign a progressive way for point cloud alignment and feature learning to gradually optimize the rigid transformation.

Original: The proposed position encoding is veryefﬁcient, requiring only a small addition of memory and computing overhead.
Revised: The proposed position encoding is veryefﬁcient, requiring only a small addition of memory and computing overhead.

Original: Extensive experiments demonstrate the proposed method can achieve competitiveperformance with the state-of-the-art methods in both indoor and outdoor scenes.
Revised: Extensive experiments demonstrate the proposed method can achieve competitiveperformance with the state-of-the-art methods in both indoor and outdoor scenes.

Original:  Introduction
Revised:  Introduction

Original: Point cloud registration is a fundamental research topic in many applications including scene reconstruction, autonomous driving, robotics, etc.
Revised: Point cloud registration is a fundamental research topic in many applications including scene reconstruction, autonomous driving, robotics, etc.

Original: The goal of it is to align two partially overlapped pointclouds by estimating a rigid transformation between them.
Revised: The goal of it is to align two partially overlapped pointclouds by estimating a rigid transformation between them.

Original: The commonly used feature-based pointcloud registration involves two aspects: establishing point correspondences and estimating the rigidtransformation.
Revised: The commonly used feature-based pointcloud registration involves two aspects: establishing point correspondences and estimating the rigidtransformation.

Original: Although it has been widely studied in past decades, it is still challenging due to thedifﬁculties such as low-overlap, repetitive patterns.
Revised: Although it has been widely studied in past decades, it is still challenging due to thedifﬁculties such as low-overlap, repetitive patterns.

Original: In recent years, learning-based works [27, 28, 43, 49] have made great advances in 3D representation.
Revised: In recent years, learning-based works [27, 28, 43, 49] have made great advances in 3D representation.

Original: In addition to utilizing deep neural networks to learn better local feature descriptions [1, 8, 10], recentmethods [46, 47, 29, 17] try to introduce Transformer [40] architecture to establish correspondences.
Revised: In addition to utilizing deep neural networks to learn better local feature descriptions [1, 8, 10], recentmethods [46, 47, 29, 17] try to introduce Transformer [40] architecture to establish correspondences.

Original: One of the most important strategies of Transformer is position encoding, which has been provencrucial when applying Transformer to many computer vision tasks [21, 34, 35].
Revised: One of the most important strategies of Transformer is position encoding, which has been provencrucial when applying Transformer to many computer vision tasks [21, 34, 35].

Original: However, how toadopt the position encoding to the domain of 3D point cloud registration is still challenging.
Revised: However, how toadopt the position encoding to the domain of 3D point cloud registration is still challenging.

Original: Differentfrom other one-instance tasks (e.g., classiﬁcation, segmentation and detection), which handles oneimage or point cloud in an inference, point cloud registration is a two-instance task.
Revised: Differentfrom other one-instance tasks (e.g., classiﬁcation, segmentation and detection), which handles oneimage or point cloud in an inference, point cloud registration is a two-instance task.

Original: It needs to learnfeatures of two point clouds simultaneously and match them.
Revised: It needs to learnfeatures of two point clouds simultaneously and match them.

Original: Since the two point clouds are underuncorrelated reference frames, the straightforward position encoding is not a good idea [23].
Revised: Since the two point clouds are underuncorrelated reference frames, the straightforward position encoding is not a good idea [23].

Original: Recently, some methods attempt to design speciﬁc position encoding techniques [25, 23, 29] forpoint cloud registration.
Revised: Recently, some methods attempt to design speciﬁc position encoding techniques [25, 23, 29] forpoint cloud registration.

Original: Geotransformer [29] calculates the relative positions between all points, thencombines the position encoding with self-attention.
Revised: Geotransformer [29] calculates the relative positions between all points, thencombines the position encoding with self-attention.

Original: This approach is equivalent to treating all pointsas reference points.
Revised: This approach is equivalent to treating all pointsas reference points.

Original: It works well, but adds a lot of computational overhead due to the additional x y x y x  y calculations about the distance and the angle similarity betweenpoints.
Revised: It works well, but adds a lot of computational overhead due to the additional x y x y x  y calculations about the distance and the angle similarity betweenpoints.

Original: DoPE [25] presents an efﬁcient manner to learn a jointorigin as the reference point for point cloud position encoding.
Revised: DoPE [25] presents an efﬁcient manner to learn a jointorigin as the reference point for point cloud position encoding.

Original: However, it depends on the centroid of point cloud to initialize thejoint-origin, which is susceptible to noise in the low-overlappingscenes.
Revised: However, it depends on the centroid of point cloud to initialize thejoint-origin, which is susceptible to noise in the low-overlappingscenes.

Original: In this paper, we propose an efﬁcient one-inlier based positionencoding method for point cloud registration.
Revised: In this paper, we propose an efﬁcient one-inlier based positionencoding method for point cloud registration.

Original: We utilize a rigorously deﬁned property of rigid transformation, i.e., it does notchange the length of a vector (isometric isomorphism).
Revised: We utilize a rigorously deﬁned property of rigid transformation, i.e., it does notchange the length of a vector (isometric isomorphism).

Original: As illustrated in Fig.
Revised: As illustrated in Fig.

Original: 1, ( x 1 , y 1 ) and ( O x , O y )are two correct correspondences, and the lengths of vector (cid:126)x 1 ( x 1 → O x ) and (cid:126)y 1 ( y 1 → O y ) are equal.
Revised: 1, ( x 1 , y 1 ) and ( O x , O y )are two correct correspondences, and the lengths of vector (cid:126)x 1 ( x 1 → O x ) and (cid:126)y 1 ( y 1 → O y ) are equal.

Original: Suppose O x
Revised: Suppose O x

Original: and O y are selected as the reference points respectively, then we can ﬁnd that ( x 1 , y 1 ) ismore likely to be a correct correspondence than ( x 2 , y 1 ) only by position information without localfeature, because the length of (cid:126)x 1 is more close than (cid:126)x 2 to that of (cid:126)y 1 .
Revised: and O y are selected as the reference points respectively, then we can ﬁnd that ( x 1 , y 1 ) ismore likely to be a correct correspondence than ( x 2 , y 1 ) only by position information without localfeature, because the length of (cid:126)x 1 is more close than (cid:126)x 2 to that of (cid:126)y 1 .

Original: In this way, position informationcan beneﬁt the establishing of correspondences when the local feature is ambiguous.
Revised: In this way, position informationcan beneﬁt the establishing of correspondences when the local feature is ambiguous.

Original: This showsthat only one inlier is enough to preserve the spatial consistency [2, 7] of correct correspondencesand ensure the distinctiveness of point-wise features.
Revised: This showsthat only one inlier is enough to preserve the spatial consistency [2, 7] of correct correspondencesand ensure the distinctiveness of point-wise features.

Original: Based on this observation, we only expect toﬁnd one inlier for position encoding.
Revised: Based on this observation, we only expect toﬁnd one inlier for position encoding.

Original: Speciﬁcally, we adopt the two points of this inlier to normalizeeach point cloud respectively, then use the normalized points for position encoding to enhance thefeature of each point.
Revised: Speciﬁcally, we adopt the two points of this inlier to normalizeeach point cloud respectively, then use the normalized points for position encoding to enhance thefeature of each point.

Original: The selected inlier plays two roles in the position encoding: 1)
Revised: The selected inlier plays two roles in the position encoding: 1)

Original: The two pointsin the inlier see the same spot of the same scenario, so positional differences caused by referenceframes can be compensated based on their normalization.
Revised: The two pointsin the inlier see the same spot of the same scenario, so positional differences caused by referenceframes can be compensated based on their normalization.

Original: It is feasible to do so to ensure spatialconsistency, i.e., the relative spatial correlation between arbitrary point and the reference point ismaintained across the two point clouds.
Revised: It is feasible to do so to ensure spatialconsistency, i.e., the relative spatial correlation between arbitrary point and the reference point ismaintained across the two point clouds.

Original: However, it is not easy to ﬁnd an accurate inlier in advance as the reference point, which is achicken-and-egg problem because determining the reference point depends on discriminative pointwise features.
Revised: However, it is not easy to ﬁnd an accurate inlier in advance as the reference point, which is achicken-and-egg problem because determining the reference point depends on discriminative pointwise features.

Original: Thus, we propose a joint learning approach to optimize both position encoding andcorrespondences.
Revised: Thus, we propose a joint learning approach to optimize both position encoding andcorrespondences.

Original: It performs an iterative optimization process for ﬁnding good correspondencesand position encoding.
Revised: It performs an iterative optimization process for ﬁnding good correspondencesand position encoding.

Original: Speciﬁcally, we select multi-inliers by performing a differentiable optimaltransport layer to produce a virtual correspondence instead of directly ﬁnding one correspondence,avoiding the unstable caused by the arbitrary selection.
Revised: Speciﬁcally, we select multi-inliers by performing a differentiable optimaltransport layer to produce a virtual correspondence instead of directly ﬁnding one correspondence,avoiding the unstable caused by the arbitrary selection.

Original: Then the two points corresponding to thecorrespondence are considered as the reference points.
Revised: Then the two points corresponding to thecorrespondence are considered as the reference points.

Original: Next, the reference points are utilized tofurther encode point-wise geometric position features, and generate more discriminative features forpoint cloud.
Revised: Next, the reference points are utilized tofurther encode point-wise geometric position features, and generate more discriminative features forpoint cloud.

Original: The updated point cloud features are further used for ﬁnding one virtual correspondenceand performing position encoding again.
Revised: The updated point cloud features are further used for ﬁnding one virtual correspondenceand performing position encoding again.

Original: Such circular optimization leads to more explicit positionencoding.
Revised: Such circular optimization leads to more explicit positionencoding.

Original: Furthermore, we use a progressive approach to gradually align the two point clouds(reference points), which reduces dependency on initial settings.
Revised: Furthermore, we use a progressive approach to gradually align the two point clouds(reference points), which reduces dependency on initial settings.

Original: During each step of the aboveoptimization process, an estimated transformation is recovered from some correspondences to alignthe two point clouds.
Revised: During each step of the aboveoptimization process, an estimated transformation is recovered from some correspondences to alignthe two point clouds.

Original: The position gap between reference points is narrowed in the process ofalignment.
Revised: The position gap between reference points is narrowed in the process ofalignment.

Original: We repeat the above step several times, which presents a progressive optimization process.
Revised: We repeat the above step several times, which presents a progressive optimization process.

Original: Fig.
Revised: Fig.

Original: 2 shows the overview of the proposed method.
Revised: 2 shows the overview of the proposed method.

Original: Compared to the previous works on the position encoding for point cloud registration [25, 23, 29],this kind of strategy mainly has three advantages: 1) It is more reasonable for ﬁnding a correctcorrespondence than centroid estimation for position encoding, especially in the low-overlappingscenarios.
Revised: Compared to the previous works on the position encoding for point cloud registration [25, 23, 29],this kind of strategy mainly has three advantages: 1) It is more reasonable for ﬁnding a correctcorrespondence than centroid estimation for position encoding, especially in the low-overlappingscenarios.

Original: 2) It is succinct and efﬁcient, and competitive in terms of GPU memory usage and speed,which has been veriﬁed in the subsequent experiments.
Revised: 2) It is succinct and efﬁcient, and competitive in terms of GPU memory usage and speed,which has been veriﬁed in the subsequent experiments.

Original: We only need to ﬁnd an inlier to efﬁcientlyenhance the distinctiveness of features.
Revised: We only need to ﬁnd an inlier to efﬁcientlyenhance the distinctiveness of features.

Original: It is robust, which reduces the impact of outlier throughconstructing a virtual correspondence and avoids the dependency on the initialization through jointoptimization.
Revised: It is robust, which reduces the impact of outlier throughconstructing a virtual correspondence and avoids the dependency on the initialization through jointoptimization.

Original: In summary, our main contributions are as follows:
Revised: In summary, our main contributions are as follows:

Original: • An efﬁcient position encoding with few correspondences for point cloud registration, which
Revised: • An efﬁcient position encoding with few correspondences for point cloud registration, which

Original: is light-weight and only relies on a few points for accurate position encoding.
Revised: is light-weight and only relies on a few points for accurate position encoding.

Original: • A joint optimization strategy that, optimizes the correspondence establishment and positionencoding with point-wise feature encoding as the agent.
Revised: • A joint optimization strategy that, optimizes the correspondence establishment and positionencoding with point-wise feature encoding as the agent.

Original: By the way, the point cloud featuresare also continuously optimized.
Revised: By the way, the point cloud featuresare also continuously optimized.

Original: • A progressive point clouds alignment approach, which progressively updates the point cloudpositions so as to reduce dependency on initialization.
Revised: • A progressive point clouds alignment approach, which progressively updates the point cloudpositions so as to reduce dependency on initialization.

Original: Decoder
Revised: Decoder

Original: Coarse Correspondence
Revised: Coarse Correspondence

Original: Fine Correspondence
Revised: Fine Correspondence

Original: Y
Revised: Y

Original: Encoder
Revised: Encoder

Original: Y
Revised: Y

Original: X o
Revised: X o

Original: Y o 2.
Revised: Y o 2.

Original: Position Embedding
Revised: Position Embedding

Original: Vector
Revised: Vector

Original: S h a r e d ( ( ))
Revised: S h a r e d ( ( ))

Original: PV X ( ( ))
Revised: PV X ( ( ))

Original: P V Y
Revised: P V Y

Original: Y
Revised: Y

Original: Inlier Learning
Revised: Inlier Learning

Original: Position Embedding { , }R t
Revised: Position Embedding { C }

Original: Progressive Alignment 3.
Revised: Progressive Alignment 3.

Original: Progressive Alignment
Revised: Progressive Alignment

Original: W e i gh t e d S
Revised: W e i gh t e d S

Original: VD { , } R t
Revised: VD { , } R t

Original: M a t c
Revised: M a t c

Original: h e d
Revised: h e d

Original: C o
Revised: C o

Original: rr
Revised: rr

Original: e s pond
Revised: e s pond

Original: e n ce s ×L Cross Attention Self Attention
Revised: e n ce s ×L Cross Attention Self Attention

Original: C o a r s e
Revised: C o a r s e

Original: M a t c h i ng
Revised: M a t c h i ng

Original: Mutual Top-k
Revised: Mutual Top-k

Original: F i n e
Revised: F i n e

Original: M a t c h i ng
Revised: M a t c h i ng

Original: Local Patch
Revised: Local Patch

Original: C
Revised: C

Original: X
Revised: X

Original: X F
Revised: X F

Original: Y F
Revised: Y F

Original: Optimal Transport
Revised: Optimal Transport

Original: Matching similarity matrix dustbin
Revised: Matching similarity matrix dustbin

Original: Refrence Points
Revised: Refrence Points

Original: X o
Revised: X o

Original: Y o
Revised: Y o

Original: Y 1.
Revised: Y 1.

Original: Inlier Learning
Revised: Inlier Learning

Original: Y F
Revised: Y F

Original: X F
Revised: X F

Original: X X
Revised: X X

Original: Xrepeat
Revised: X

Original:  Related Work
Revised:  Related Work

Original: Local 3D feature descriptors.
Revised: Local 3D feature descriptors.

Original: Early traditional handcrafted descriptors widely summarize the localgeometry in the form of histogram.
Revised: Early traditional handcrafted descriptors widely summarize the localgeometry in the form of histogram.

Original: Histograms usually contain statistical information such as thespatial distribution and geometric properties of points [15].
Revised: Histograms usually contain statistical information such as thespatial distribution and geometric properties of points [15].

Original: The descriptors represented with the3D spatial distribution includes Spin Image [18], 3D Shape Context [12], etc.
Revised: The descriptors represented with the3D spatial distribution includes Spin Image [18], 3D Shape Context [12], etc.

Original: Another category ofdescriptors includes Fast Point Feature Histograms (FPFHs)
Revised: Another category ofdescriptors includes Fast Point Feature Histograms (FPFHs)

Original: [30], SHOT [38], etc.
Revised: [30], SHOT [38], etc.

Original: Recent worksattempt to encode 3D local descriptors in a data-driven way.
Revised: Recent worksattempt to encode 3D local descriptors in a data-driven way.

Original: Zeng et al.
Revised: Zeng et al.

Original: propose the pioneering work3DMatch [48], which utilizes 3D Convolutional Neural Networks to encode local descriptors fromlocal volumetric patches.
Revised: propose the pioneering work3DMatch [48], which utilizes 3D Convolutional Neural Networks to encode local descriptors fromlocal volumetric patches.

Original: PPFNet [10] introduces the rotation-invariant Point-Pair Features (PPF)[11] to encode local descriptors.
Revised: PPFNet [10] introduces the rotation-invariant Point-Pair Features (PPF)[11] to encode local descriptors.

Original: Follow-up work PPF-FoldNet [9] incorporates FoldingNet [44] asbackbone for unsupervised descriptor learning.
Revised: Follow-up work PPF-FoldNet [9] incorporates FoldingNet [44] asbackbone for unsupervised descriptor learning.

Original: 3DSN [14] leverages the voxelized Smoothed Density
Revised: 3DSN [14] leverages the voxelized Smoothed Density

Original: Value (SDV) to learn rotation-invariant local descriptors.
Revised: Value (SDV) to learn rotation-invariant local descriptors.

Original: FCGF [8] ﬁrst utilizes the sparse 3D fullyconvolutional network to extract dense feature descriptors.
Revised: FCGF [8] ﬁrst utilizes the sparse 3D fullyconvolutional network to extract dense feature descriptors.

Original: SpinNet
Revised: SpinNet

Original: [1] proposes a 3D cylindricalconvolution layer for learning rotation-invariant features.
Revised: [1] proposes a 3D cylindricalconvolution layer for learning rotation-invariant features.

Original: Learning-based point cloud registration.
Revised: Learning-based point cloud registration.

Original: Deep Closest Point [42] utilizes a graph neural networkto extract point cloud features and a pointer network to predict correspondences.
Revised: Deep Closest Point [42] utilizes a graph neural networkto extract point cloud features and a pointer network to predict correspondences.

Original: IDAM [22]proposes an iterative distance-aware similarity matrix convolution module to integrate the featureand Euclidean space into the point matching.
Revised: IDAM [22]proposes an iterative distance-aware similarity matrix convolution module to integrate the featureand Euclidean space into the point matching.

Original: Predator [17] leverages the attention mechanism todetect the overlapping region for sampling salient feature points.
Revised: Predator [17] leverages the attention mechanism todetect the overlapping region for sampling salient feature points.

Original: CoFiNet [47] utilizes a coarseto-ﬁne strategy for correspondence ﬁnding.
Revised: CoFiNet [47] utilizes a coarseto-ﬁne strategy for correspondence ﬁnding.

Original: Lepard [23] utilizes the Rotary positional encoding[34] to encode 3D positional information for feature learning.
Revised: Lepard [23] utilizes the Rotary positional encoding[34] to encode 3D positional information for feature learning.

Original: NgeNet [50] introduces a multi-scalearchitecture to extract features with multiple neighborhood sizes for adaptive selection of featurescale.
Revised: NgeNet [50] introduces a multi-scalearchitecture to extract features with multiple neighborhood sizes for adaptive selection of featurescale.

Original: REGTR [46] uses transformer to directly predict a set of point correspondences and estimatesthe rigid transformation using a variant of the Kabsch-Umeyama [19, 39] algorithm.
Revised: REGTR [46] uses transformer to directly predict a set of point correspondences and estimatesthe rigid transformation using a variant of the Kabsch-Umeyama [19, 39] algorithm.

Original: Position embedding.
Revised: Position embedding.

Original: The idea of relative positional encodings ﬁrst appeared in machine translation[32] and music generation [16].
Revised: The idea of relative positional encodings ﬁrst appeared in machine translation[32] and music generation [16].

Original: They produce learned positional embeddings in the self-attention.
Revised: They produce learned positional embeddings in the self-attention

Original: DETR [6] applies ﬁxed positional encodings [26, 4] to the input of the transformer architecturefor object detection. Swin Transformer [24] adds relative positionbiases in similarity computation of self-attention, improving performance over models without these biases.
Revised: DETR [ ﬁxed positional encodings [ the input of the transformer architecture position of self-attention, improving performance over models without these biases.

Original: LoFTR [35] uses the2D extension of the position encoding to produce position-dependent features for image matching.
Revised: LoFTR [35] uses the2D extension of the position encoding to produce position-dependent features for image matching.

Original: DoPE [25] proposes to calculate a joint-origin of all points for point cloud position encoding.
Revised: DoPE [25] proposes to calculate a joint-origin of all points for point cloud position encoding.

Original: Geotransformer [29] treats all points as reference points to calculate the relative positions betweenpoint pairs and combines them with self-attention to embed positional information into the features.
Revised: Geotransformer [29] treats all points as reference points to calculate the relative positions betweenpoint pairs and combines them with self-attention to embed positional information into the features.

Original:  Approach
Revised:  Approach

Original: 3.1 Problem Statement
Revised: 3.1 Problem Statement

Original: Suppose there are two point clouds: source point cloud X
Revised: Suppose there are two point clouds: source point cloud X

Original: = { x i ∈ R 3 , i = 1 , ..., M } and target pointcloud Y = { y j ∈ R 3 , j = 1 , ..., N } , where M and N are the point numbers of X and Y , respectively.
Revised: = { x i ∈ R 3 , i = 1 , ..., M } and target pointcloud Y = { y j ∈ R 3 , j = 1 , ..., N } , where M and N are the point numbers of X and Y , respectively.

Original: Our goal is to recover the rigid transformation T
Revised: Our goal is to recover the rigid transformation T

Original: =
Revised: =

Original: { R , t } that best aligns the two point clouds, where
Revised: { R , t } that best aligns the two point clouds, where

Original: R ∈ SO (3) represents the rotation component and t ∈ R 3 represents the translation component, asfollows:
Revised: R ∈ SO (3) represents the rotation component and t ∈ R 3 represents the translation component, asfollows:

Original: [Equation]
Revised: [Equation]

Original: ¯ C is a set of predicted correspondences between the source and target point clouds.
Revised: ¯ C is a set of predicted correspondences between the source and target point clouds.

Original: [Figure]
Revised: [Figure]

Original: Fig.
Revised: Fig.

Original: 2 illustrates our overview framework.
Revised: 2 illustrates our overview framework.

Original: Following [35, 47], our method employs a coarse-to-ﬁnemanner to ﬁnd correspondences.
Revised: Following [35, 47], our method employs a coarse-to-ﬁnemanner to ﬁnd correspondences.

Original: In the coarse stage, we ﬁrst use the KPConv [37] to down-sampleraw input points to uniformly distributed nodes ˜ X ∈ R ˜
Revised: In the coarse stage, we ﬁrst use the KPConv [37] to down-sampleraw input points to uniformly distributed nodes ˜ X ∈ R ˜

Original: M × 3 and ˜ Y ∈ R ˜ N × 3 , and learn the associatedfeatures F ˜ X ∈ R ˜ M × ˜ D and F ˜ Y ∈ R ˜ N × ˜ D .
Revised: M × 3 and ˜ Y ∈ R ˜ N × 3 , and learn the associatedfeatures F ˜ X ∈ R ˜ M × ˜ D and F ˜ Y ∈ R ˜ N × ˜ D .

Original: Then coarse correspondences are generated by matchingthe nodes.
Revised: Then coarse correspondences are generated by matchingthe nodes.

Original: After obtaining the coarse correspondences from the previous stage, we reﬁne them in ﬁnelevel.
Revised: After obtaining the coarse correspondences from the previous stage, we reﬁne them in ﬁnelevel.

Original: Speciﬁcally, we assign each point in the ﬁne level to its nearest nodes in geometry space.
Revised: Speciﬁcally, we assign each point in the ﬁne level to its nearest nodes in geometry space.

Original: Eachnode will be expanded into a local patch by the point-to-node grouping strategy [20, 47], and thepatch correspondences can be established by the matching of nodes.
Revised: Eachnode will be expanded into a local patch by the point-to-node grouping strategy [20, 47], and thepatch correspondences can be established by the matching of nodes.

Original: For each patch correspondence,we utilize an optimal transport layer [31] to calculate a soft assignment matrix between the ﬁne-levelpoints within this patch correspondence, then obtain the ﬁne point correspondences by the mutualtop-k selection.
Revised: For each patch correspondence,we utilize an optimal transport layer [31] to calculate a soft assignment matrix between the ﬁne-levelpoints within this patch correspondence, then obtain the ﬁne point correspondences by the mutualtop-k selection.

Original: The ﬁnal point correspondences are collected from all coarse correspondences.
Revised: The ﬁnal point correspondences are collected from all coarse correspondences.

Original: Since the results of ﬁne correspondences mainly depend on the coarse results, our method focuses onboosting the accuracy of the coarse results.
Revised: Since the results of ﬁne correspondences mainly depend on the coarse results, our method focuses onboosting the accuracy of the coarse results.

Original: Speciﬁcally, an inlier-based position learning approach isproposed to ﬁnd an inlier and normalize the point clouds for position embedding (Sec. 3.3).
Revised: Speciﬁcally, an inlier-based position learning approach isproposed to ﬁnd an inlier and normalize the point clouds for position embedding (Sec. 3.3).

Original: Next, weleverage the matched correspondences (the byproduct of the "one-inlier" establishing) to progressivelyalign point clouds (Sec. 3.4).
Revised: Next, weleverage the matched correspondences (the byproduct of the "one-inlier" establishing) to progressivelyalign point clouds (Sec. 3.4).

Original: After that, we propose to jointly optimize correspondence establishmentand position embedding bridged by feature reconstruction (Sec. 3.5).
Revised: After that, we propose to jointly optimize correspondence establishmentand position embedding bridged by feature reconstruction (Sec. 3.5).

Original: Finally, we introduce the lossfunction for training the network (Sec. 3.6).
Revised: Finally, we introduce the lossfunction for training the network (Sec. 3.6).

Original: 3.3 Inlier-based Position Learning
Revised: 3.3 Inlier-based Position Learning

Original: Inlier learning.
Revised: Inlier learning.

Original: As mentioned before, once a correspondence is an inlier, its corresponding points inthe two point clouds can be considered as the reference points respectively.
Revised: As mentioned before, once a correspondence is an inlier, its corresponding points inthe two point clouds can be considered as the reference points respectively.

Original: Since the two referencepoints are mapped from the same 3D point in different views, the matching points satisﬁed theground-truth rigid transformation can preserve the spatial consistency (e.g., the spatial distance andorientation) relative to the respective reference points across the two point clouds.
Revised: Since the two referencepoints are mapped from the same 3D point in different views, the matching points satisﬁed theground-truth rigid transformation can preserve the spatial consistency (e.g., the spatial distance andorientation) relative to the respective reference points across the two point clouds.

Original: Thus, only oneinlier is desired to compensate for the difference in 3D spatial position because the two point cloudsin different reference systems can be normalized by the two points of the inlier.
Revised: Thus, only oneinlier is desired to compensate for the difference in 3D spatial position because the two point cloudsin different reference systems can be normalized by the two points of the inlier.

Original: Given that the inlieris preemptively agnostic, our approach is mainly devoted to the mining of one correct correspondence.
Revised: Given that the inlieris preemptively agnostic, our approach is mainly devoted to the mining of one correct correspondence.

Original: The cross-attention [40, 31] can encode the geometric consistency across the two point clouds byintegrating the features of the other point cloud.
Revised: The cross-attention [40, 31] can encode the geometric consistency across the two point clouds byintegrating the features of the other point cloud.

Original: We use it to enhance the distinctiveness of features.
Revised: We use it to enhance the distinctiveness of features.

Original: Formally, given the coarse features F ˜ X and F ˜ Y , we use learnable matrices W q to project the F ˜ X toquery q , and use ( W k , W v ) to project F ˜ Y to key and value ( k , v ) as follows:
Revised: Formally, given the coarse features F ˜ X and F ˜ Y , we use learnable matrices W q to project the F ˜ X toquery q , and use ( W k , W v ) to project F ˜ Y to key and value ( k , v ) as follows:

Original: [Equation]
Revised: [Equation]

Original: where W q , W k , W v ∈ R ˜ D × ˜ D .
Revised: where W q , W k , W v ∈ R ˜ D × ˜ D .

Original: The coarse feature F ˜ X is updated by F (cid:48)
Revised: The coarse feature F ˜ X is updated by F (cid:48)

Original: ˜ X
Revised: ˜ X

Original: i = F ˜ X
Revised: i = F ˜ X

Original: i +
Revised: i +

Original: MLP( (cid:80) ˜
Revised: MLP( (cid:80) ˜

Original: Nj =

Revised: Nj =

Original: α ij v j ) , where α ij = softmax( q i k Tj / (cid:112) ˜ D ) is the feature similarity score between
Revised: α ij v j ) , where α ij = softmax( q i k Tj / (cid:112) ˜ D ) is the feature similarity score between

Original: F ˜ X i and F ˜ Y j .
Revised: F ˜ X i and F ˜ Y j .

Original: MLP( · ) denotes a fully connected network.
Revised: MLP( · ) denotes a fully connected network.

Original: The cross-attention features for F ˜ Y arecomputed and denoted in a similar way.
Revised: The cross-attention features for F ˜ Y arecomputed and denoted in a similar way.

Original: Following the cross-attention we use a self-attention toaggregate the global context, which updates features in a similar way like cross-attention, but isperformed inside the source and target point cloud independently.
Revised: Following the cross-attention we use a self-attention toaggregate the global context, which updates features in a similar way like cross-attention, but isperformed inside the source and target point cloud independently.

Original: After obtaining the strengthened features F (cid:48)
Revised: After obtaining the strengthened features F (cid:48)

Original: ˜ X and F (cid:48)
Revised: ˜ X and F (cid:48)

Original: ˜ Y , we leverage a differentiable optimal transportto ﬁnd good correspondences.
Revised: ˜ Y , we leverage a differentiable optimal transportto ﬁnd good correspondences.

Original: We compute the similarity matrix ˜ S ∈ R ˜ M × ˜ N between F (cid:48)
Revised: We compute the similarity matrix ˜ S ∈ R ˜ M × ˜ N between F (cid:48)

Original: ˜ X and F (cid:48)
Revised: ˜ X and F (cid:48)

Original: ˜ Y :
Revised: ˜ Y :

Original: [Equation]
Revised: [Equation]

Original: then we append a new row and a new column as in [31], ﬁlled with a learnable dustbin parameterα .
Revised: then we append a new row and a new column as in [31], ﬁlled with a learnable dustbin parameterα .

Original: The Sinkhorn algorithm [33] is adopted on ˜ S for searching the best solution of the optimaltransportation problem.
Revised: The Sinkhorn algorithm [33] is adopted on ˜ S for searching the best solution of the optimaltransportation problem.

Original: Finally, we obtain the soft matching scores ¯ S ∈ R ˜ M × ˜ N between the ˜ X and˜ Y by dropping the last row and the last column of ˜ S .
Revised: Finally, we obtain the soft matching scores ¯ S ∈ R ˜ M × ˜ N between the ˜ X and˜ Y by dropping the last row and the last column of ˜ S .

Original: Intuitively, the higher the matching score, thehigher the probability that the associated correspondence is an inlier.
Revised: Intuitively, the higher the matching score, thehigher the probability that the associated correspondence is an inlier.

Original: But it is not always true due tothe widely existing repetitive structures, textureless structures, etc.
Revised: But it is not always true due tothe widely existing repetitive structures, textureless structures, etc.

Original: In order to enhance the robustnessof the proposed algorithm and reduce the bias caused by the above issues, we propose to pick multiplereliable correspondences ¯ C topk with higher matching scores via top-k selection strategy.
Revised: In order to enhance the robustnessof the proposed algorithm and reduce the bias caused by the above issues, we propose to pick multiplereliable correspondences ¯ C topk with higher matching scores via top-k selection strategy.

Original: We use theselected correspondences to produce a virtual correspondence by an averaging operation of the pointcoordinates.
Revised: We use theselected correspondences to produce a virtual correspondence by an averaging operation of the pointcoordinates.

Original: The corresponding points of the virtual correspondence are regarded as the referencepoints ¯ o =
Revised: The corresponding points of the virtual correspondence are regarded as the referencepoints ¯ o =

Original: { ¯ o X , ¯ o Y } , where the ¯ o X , ¯ o Y denote the reference point in source and target point cloud,respectively.
Revised: { ¯ o X , ¯ o Y } , where the ¯ o X , ¯ o Y denote the reference point in source and target point cloud,respectively.

Original: Position embedding.
Revised: Position embedding.

Original: After obtaining the reference points ¯ o , they are utilized to normalize the twopoint clouds respectively.
Revised: After obtaining the reference points ¯ o , they are utilized to normalize the twopoint clouds respectively.

Original: Speciﬁcally, we use them to calculate point-wise position vector relative tothe respective reference point for each node in the two point clouds:
Revised: Speciﬁcally, we use them to calculate point-wise position vector relative tothe respective reference point for each node in the two point clouds:

Original: [Equation]
Revised: [Equation]

Original: Once the reference points and (˜ x i , ˜ y j ) are inliers, i.e., they both meet the ground-truth rigid transformation between the point clouds, then the two vector V (˜ x i ) , V (˜ y j ) satisfy the isometric isomorphismproperty.
Revised: Once the reference points and (˜ x i , ˜ y j ) are inliers, i.e., they both meet the ground-truth rigid transformation between the point clouds, then the two vector V (˜ x i ) , V (˜ y j ) satisfy the isometric isomorphismproperty.

Original: It means that the isomorphism of ˜ x i or ˜ y
Revised: It means that the isomorphism of ˜ x i or ˜ y

Original: j maintaining constant distance with respect to therespective reference point in different reference frames.
Revised: j maintaining constant distance with respect to therespective reference point in different reference frames.

Original: Based on this property, the point-wise vectorsnormalized by the reference points are fed into position embedding layer for feature reconstruction.
Revised: Based on this property, the point-wise vectorsnormalized by the reference points are fed into position embedding layer for feature reconstruction.

Original: Speciﬁcally, we use an MLP with 4 hidden layers of feature size being 256 on each individualvector, and output an embedded geometric position feature.
Revised: Speciﬁcally, we use an MLP with 4 hidden layers of feature size being 256 on each individualvector, and output an embedded geometric position feature.

Original: Formally, for vector V (˜ x i ) and V (˜ y j ) , thecorresponding feature is denoted as P ( V (˜ x i )) and P ( V (˜ y j )) , separately.
Revised: Formally, for vector V (˜ x i ) and V (˜ y j ) , thecorresponding feature is denoted as P ( V (˜ x i )) and P ( V (˜ y j )) , separately.

Original: Then we use the embeddedgeometric position features to enhance the distinctiveness of the node features.
Revised: Then we use the embeddedgeometric position features to enhance the distinctiveness of the node features.

Original: In a simple andeffective manner, we strengthen the node features by directly adding the geometric position featuresto the primal features, i.e., H ˜ X i = F (cid:48)
Revised: In a simple andeffective manner, we strengthen the node features by directly adding the geometric position featuresto the primal features, i.e., H ˜ X i = F (cid:48)

Original: ˜ X
Revised: ˜ X

Original: i + P ( V (˜ x i )) , H ˜ Y
Revised: i + P ( V (˜ x i )) , H ˜ Y

Original: j = F
Revised: j = F

Original: [Equation]
Revised: [Equation]

Original: Y
Revised: Y

Original: [Equation]
Revised: [Equation]

Original: + P ( V (˜
Revised: + P ( V (˜

Original: [Equation]
Revised: [Equation]

Original: 3.4 Progressive Point Cloud Alignment
Revised: 3.4 Progressive Point Cloud Alignment

Original: As mentioned before, the main challenge of position encoding is that the two point clouds aregenerally under irrelevant reference frames.
Revised: As mentioned before, the main challenge of position encoding is that the two point clouds aregenerally under irrelevant reference frames.

Original: We further solve this challenge by progressively aligningthe point clouds in coordinate space.
Revised: We further solve this challenge by progressively aligningthe point clouds in coordinate space.

Original: We expect to recover a rigid transformation with reliablecorrespondences ¯ C topk to make the position encoding easier.
Revised: We expect to recover a rigid transformation with reliablecorrespondences ¯ C topk to make the position encoding easier.

Original: Position update.
Revised: Position update.

Original: After obtaining the coarse correspondences ¯ C topk with relatively higher matchingscores, we use them to estimate the current optimal transformation ¯ T t .
Revised: After obtaining the coarse correspondences ¯ C topk with relatively higher matchingscores, we use them to estimate the current optimal transformation ¯ T t .

Original: Speciﬁcally, given the coarsecorrespondences ¯ C topk and corresponding matching scores, we use weighted SVD [5] to estimate thecurrent optimal transformation ¯ T t .
Revised: Speciﬁcally, given the coarsecorrespondences ¯ C topk and corresponding matching scores, we use weighted SVD [5] to estimate thecurrent optimal transformation ¯ T t .

Original: Then we use the estimated ¯ T t = { ¯ R t , ¯ t t } to update the relativeposition between two point clouds:
Revised: Then we use the estimated ¯ T t = { ¯ R t , ¯ t t } to update the relativeposition between two point clouds:

Original: [Equation]
Revised: [Equation]

Original: Naturally, the reference points, i.e., the centroids ¯ o of the selected patches in the two point cloudshave also been updated:
Revised: Naturally, the reference points, i.e., the centroids ¯ o of the selected patches in the two point cloudshave also been updated:

Original: [Equation]
Revised: [Equation]

Original: where ¯ o X t represents the transformed reference point of the source point cloud.
Revised: where ¯ o X t represents the transformed reference point of the source point cloud.

Original: As the two pointclouds are roughly aligned by the estimated transformation, the position vectors of the two points in each correct correspondence tend to be consistent.
Revised: As the two pointclouds are roughly aligned by the estimated transformation, the position vectors of the two points in each correct correspondence tend to be consistent.

Original: Thus, the encoded position feature can provide amore positive effect for establishing correspondences.
Revised: Thus, the encoded position feature can provide amore positive effect for establishing correspondences.

Original: Progressive alignment.
Revised: Progressive alignment.

Original: We repeat the above operation leveraging the enhanced features duringthe joint optimization (see Sec. 3.5) to progressively update the relative positions between thetwo point clouds.
Revised: We repeat the above operation leveraging the enhanced features duringthe joint optimization (see Sec. 3.5) to progressively update the relative positions between thetwo point clouds.

Original: Speciﬁcally, every time before the position encoding operation, we perform analignment between the two point clouds.
Revised: Speciﬁcally, every time before the position encoding operation, we perform analignment between the two point clouds.

Original: We utilize the enhanced features to rebuild some reliablecorrespondences ¯ C topk and re-estimate the current optimal transformation ¯ T t for the alignment.
Revised: We utilize the enhanced features to rebuild some reliablecorrespondences ¯ C topk and re-estimate the current optimal transformation ¯ T t for the alignment.

Original: Intuitively, the enhanced features are more distinctive than original features, so the latter estimatedtransformation is more accurate than that in previous stage.
Revised: Intuitively, the enhanced features are more distinctive than original features, so the latter estimatedtransformation is more accurate than that in previous stage.

Original: Thus, multiply transformations enable thetwo point clouds to be progressively aligned in coordinate space.
Revised: Thus, multiply transformations enable thetwo point clouds to be progressively aligned in coordinate space.

Original: In this way, the position encodingcan also be optimized in the progressive alignment process.
Revised: In this way, the position encodingcan also be optimized in the progressive alignment process.

Original: 3.5 Jointly Optimizing Position Encoding and Correspondences
Revised: 3.5 Jointly Optimizing Position Encoding and Correspondences

Original: Inlier learning and position encoding of the point clouds is a chicken-and-egg problem: the proposedposition encoding for learning distinctive node features would require the knowledge about at leastone correct correspondence; but meantime, establishing one accurate correspondence also relieson the correct position encoding of the point clouds for feature reconstruction.
Revised: Inlier learning and position encoding of the point clouds is a chicken-and-egg problem: the proposedposition encoding for learning distinctive node features would require the knowledge about at leastone correct correspondence; but meantime, establishing one accurate correspondence also relieson the correct position encoding of the point clouds for feature reconstruction.

Original: Either of themis hard to be solved in advance.
Revised: Either of themis hard to be solved in advance.

Original: In order to tackle this circular problem, we propose to jointlyoptimize the correspondences and position encoding by multiply feature rebuilding.
Revised: In order to tackle this circular problem, we propose to jointlyoptimize the correspondences and position encoding by multiply feature rebuilding.

Original: As shown in
Revised: As shown in

Original: Fig.
Revised: Fig.

Original: 2, we incorporate the position encoding into the process of the node feature learning.
Revised: 2, we incorporate the position encoding into the process of the node feature learning.

Original: Weconstruct it together with inlier learning as a joint optimization task, with a view to improvingthe accuracy of coarse correspondences by enhancing the discrimination of position encoding.
Revised: Weconstruct it together with inlier learning as a joint optimization task, with a view to improvingthe accuracy of coarse correspondences by enhancing the discrimination of position encoding.

Original: Speciﬁcally, the joint optimization is performed by enhancing the features of point clouds with 3Dpositional information.
Revised: Speciﬁcally, the joint optimization is performed by enhancing the features of point clouds with 3Dpositional information.

Original: The updated more accurate vectors V (˜ x i ) , V (˜ y j ) with respect to the selectedreference points facilitate better node feature encoding, which contributes to ﬁnding one more accuratecorrespondence.
Revised: The updated more accurate vectors V (˜ x i ) , V (˜ y j ) with respect to the selectedreference points facilitate better node feature encoding, which contributes to ﬁnding one more accuratecorrespondence.

Original: Meanwhile, the more accurate correspondence means better reference points ¯ o ,which enables that the updated vectors V (˜ x i ) , V (˜ y j ) gradually satisfy the isometric isomorphismproperty.
Revised: Meanwhile, the more accurate correspondence means better reference points ¯ o ,which enables that the updated vectors V (˜ x i ) , V (˜ y j ) gradually satisfy the isometric isomorphismproperty.

Original: It will result in more discriminative node-wise features.
Revised: It will result in more discriminative node-wise features.

Original: In the process, the features of thepoint clouds are continuously optimized.
Revised: In the process, the features of thepoint clouds are continuously optimized.

Original: 3.6 Loss Functions
Revised: 3.6 Loss Functions

Original: The training loss function L =
Revised: The training loss function L =

Original: L c + L f is composed of the coarse correspondence loss L c and ﬁnecorrespondence loss L f .
Revised: L c + L f is composed of the coarse correspondence loss L c and ﬁnecorrespondence loss L f .

Original: Coarse correspondence loss.
Revised: Coarse correspondence loss.

Original: To supervise the patch-wise feature descriptors for coarse correspondences, we follow [17, 29] and use a variant of the circle loss [36].
Revised: To supervise the patch-wise feature descriptors for coarse correspondences, we follow [17, 29] and use a variant of the circle loss [36].

Original: Considering again an point cloudpair X and Y , we ﬁrst pick the patches P X
Revised: Considering again an point cloudpair X and Y , we ﬁrst pick the patches P X

Original: i (denotes the patch corresponding to the node ˜ x i ∈ ˜ X )
Revised: i (denotes the patch corresponding to the node ˜ x i ∈ ˜ X )

Original: thathas at least one positive patch in Y to form a collection of patches P .
Revised: thathas at least one positive patch in Y to form a collection of patches P .

Original: For each patch P X i ∈ P , theset of its positive (share at least 10% overlap) and negative (do not share overlap) patches in Y aredenoted as ε ip and ε in , respectively.
Revised: For each patch P X i ∈ P , theset of its positive (share at least 10% overlap) and negative (do not share overlap) patches in Y aredenoted as ε ip and ε in , respectively.

Original: The loss of coarse correspondence on X is then calculated as:
Revised: The loss of coarse correspondence on X is then calculated as:

Original: [Equation]
Revised: [Equation]

Original: where d ji = (cid:13)(cid:13)(cid:13)
Revised: where d ji = (cid:13)(cid:13)(cid:13)

Original: ˜ F X i − ˜ F Y j (cid:13)(cid:13)(cid:13) 2 represents the distance in feature space, and λ ji = ( o ji ) 12 , and o ji denotesthe overlap ratio between P X i and P Y j .
Revised: ˜ F X i − ˜ F Y j (cid:13)(cid:13)(cid:13) 2 represents the distance in feature space, and λ ji = ( o ji ) 12 , and o ji denotesthe overlap ratio between P X i and P Y j .

Original: ∆ p , ∆ n are positive and negative margins, respectively.
Revised: ∆ p , ∆ n are positive and negative margins, respectively.

Original: Theweights for each positive and negative sample are calculated individually: β i,jp
Revised: Theweights for each positive and negative sample are calculated individually: β i,jp

Original: = γ ( d ji − ∆ p ) andβ i,kn = γ (∆ n − d ki ) , with the empirical margins ∆ p = 0.1, ∆ n = 1.4.
Revised: = γ ( d ji − ∆ p ) andβ i,kn = γ (∆ n − d ki ) , with the empirical margins ∆ p = 0.1, ∆ n = 1.4.

Original: γ is a hyper-parameter.
Revised: γ is a hyper-parameter.

Original: Theloss L Y c on Y is computed in the same way.
Revised: Theloss L Y c on Y is computed in the same way.

Original: The total loss is L c =
Revised: The total loss is L c =

Original: ( L X c + L Y c ) / 2 .
Revised: ( L X c + L Y c ) / 2 .

Original: Fine correspondence loss.
Revised: Fine correspondence loss.

Original: For ﬁne correspondence, we ﬁrst determine the ground-truth coarsecorrespondences using the ground-truth relative transformations.
Revised: For ﬁne correspondence, we ﬁrst determine the ground-truth coarsecorrespondences using the ground-truth relative transformations.

Original: Then we extract a set of groundtruth ﬁne correspondences M i in each correspondence patches.
Revised: Then we extract a set of groundtruth ﬁne correspondences M i in each correspondence patches.

Original: We label the unmatched points inthis two patches as I
Revised: We label the unmatched points inthis two patches as I

Original: i and J i .
Revised: i and J i .

Original: The ﬁne correspondence loss is computed as:
Revised: The ﬁne correspondence loss is computed as:

Original: [Equation]
Revised: [Equation]

Original: where C is the number of the patch pairs.
Revised: where C is the number of the patch pairs.

Original: ¯ P i ∈ R ( n i +1) ×
Revised: ¯ P i ∈ R ( n i +1) ×

Original: ( m i +1) is the soft assignment matrix of thei -th patch correspondence.
Revised: ( m i +1) is the soft assignment matrix of thei -th patch correspondence.

Original: n i and m i represent the number of the ﬁne level points in this patch pair.
Revised: n i and m i represent the number of the ﬁne level points in this patch pair.

Original: [Table]
Revised: [Table]

Original:  Experiments
Revised:  Experiments

Original: We evaluate our approach on indoor 3DMatch [48] and 3DLoMatch [17] benchmarks (Sec.
Revised: We evaluate our approach on indoor 3DMatch [48] and 3DLoMatch [17] benchmarks (Sec.

Original: 4.1),and outdoor KITTI odometry [13] benchmark (Sec. 4.2).
Revised: 4.1),and outdoor KITTI odometry [13] benchmark (Sec. 4.2).

Original: The point cloud pairs in 3DMatch have >30 % overlap, while those in 3DLoMatch have low overlap of 10 % ∼ 30 % .
Revised: The point cloud pairs in 3DMatch have >30 % overlap, while those in 3DLoMatch have low overlap of 10 % ∼ 30 % .

Original: KITTI odometry is anoutdoor driving scenario sparse point cloud dataset acquired by LiDAR.
Revised: KITTI odometry is anoutdoor driving scenario sparse point cloud dataset acquired by LiDAR.

Original: Details about the datasetsand implementation are provided in the supplementary material.
Revised: Details about the datasetsand implementation are provided in the supplementary material.

Original: 4.1 3DMatch and 3DLoMatch
Revised: 4.1 3DMatch and 3DLoMatch

Original: Metrics.
Revised: Metrics.

Original: Our main metric, related to the practical purpose of point cloud registration, is Registration
Revised: Our main metric, related to the practical purpose of point cloud registration, is Registration

Original: Recall (RR), i.e., the percentage of point cloud pairs whose transformation error (RMSE) is below0.2 m.
Revised: Recall (RR), i.e., the percentage of point cloud pairs whose transformation error (RMSE) is below0.2 m.

Original: It is the most important metric in point cloud registration task [17, 47], because it directlymeasures the ﬁnal registration performance.
Revised: It is the most important metric in point cloud registration task [17, 47], because it directlymeasures the ﬁnal registration performance.

Original: Following [3, 17], we also report Feature Matching
Revised: Following [3, 17], we also report Feature Matching

Original: Recall (FMR), i.e., the fraction of point cloud pairs whose Inlier Ratio is larger than 5% , and Inlier
Revised: Recall (FMR), i.e., the fraction of point cloud pairs whose Inlier Ratio is larger than 5% , and Inlier

Original: Ratio (IR), deﬁned as the fraction of correct correspondences (residuals are less than 0.1 m under theground truth transformation) among the putative matches.
Revised: Ratio (IR), deﬁned as the fraction of correct correspondences (residuals are less than 0.1 m under theground truth transformation) among the putative matches.

Original: Comparisons to the state-of-the-art approaches.
Revised: Comparisons to the state-of-the-art approaches.

Original: We compare the registration results of ourmethod with recent state-of-the-art approaches including: 3DSN [14], FCGF [8], D3Feat [3], SpinNet[1], YOHO [41], Predator [17] 1 , CoFiNet [47], REGTR [46] and GeoTransformer (GeoTrans in thetable) [29] in Tab. 1.
Revised: We compare the registration results of ourmethod with recent state-of-the-art approaches including: 3DSN [14], FCGF [8], D3Feat [3], SpinNet[1], YOHO [41], Predator [17] 1 , CoFiNet [47], REGTR [46] and GeoTransformer (GeoTrans in thetable) [29] in Tab. 1.

Original: We report the results under different numbers of sampled correspondences.
Revised: We report the results under different numbers of sampled correspondences.

Original: Since
Revised: Since

Original: REGTR [46] needs to take all the ﬁnal downsampled points for transformation estimation insteadof sampling, we only provide the results in the original paper.
Revised: REGTR [46] needs to take all the ﬁnal downsampled points for transformation estimation insteadof sampling, we only provide the results in the original paper.

Original: Following [3, 17], we run RANSAC50k to estimate the rigid transformation.
Revised: Following [3, 17], we run RANSAC50k to estimate the rigid transformation.

Original: When compared with local 3D descriptors, our methodoutperforms all the approaches on both datasets.
Revised: When compared with local 3D descriptors, our methodoutperforms all the approaches on both datasets.

Original: For end-to-end methods, the proposed method and
Revised: For end-to-end methods, the proposed method and

Original: GeoTransformer [29] outperform other methods by a large margin on all the metrics, especially in thelow-overlap scenarios (3DLoMatch).
Revised: GeoTransformer [29] outperform other methods by a large margin on all the metrics, especially in thelow-overlap scenarios (3DLoMatch).

Original: This is because our method and GeoTransformer both consider the position information in the network, which demonstrates the importance of position encoding forpoint cloud registration.
Revised: This is because our method and GeoTransformer both consider the position information in the network, which demonstrates the importance of position encoding forpoint cloud registration.

Original: It is worth noting that our method performs worse than GeoTransformer [29] on Inlier Ratio (IR).
Revised: It is worth noting that our method performs worse than GeoTransformer [29] on Inlier Ratio (IR).

Original: However, when referring to the most important metric Registration Recall (RR) which better reﬂectsthe ﬁnal performance on point cloud registration, the proposed method achieves the state-of-the-artperformance.
Revised: However, when referring to the most important metric Registration Recall (RR) which better reﬂectsthe ﬁnal performance on point cloud registration, the proposed method achieves the state-of-the-artperformance.

Original: It can be explained by the fact that the inlier rate is not total positively-correlated withthe registration recall.
Revised: It can be explained by the fact that the inlier rate is not total positively-correlated withthe registration recall.

Original: As pointed out in [17, 47, 29], besides the inlier rate, the other important issuefor registration is the sparsity of the correspondences.
Revised: As pointed out in [17, 47, 29], besides the inlier rate, the other important issuefor registration is the sparsity of the correspondences.

Original: Aggregated inliers may improve the inlierrate, but is not necessarily beneﬁcial for the registration.
Revised: Aggregated inliers may improve the inlierrate, but is not necessarily beneﬁcial for the registration.

Original: Since our position encoding can introducespatial consistency, it is possible to ﬁnd matching pairs that cannot be found by local features alone.
Revised: Since our position encoding can introducespatial consistency, it is possible to ﬁnd matching pairs that cannot be found by local features alone.

Original: Thus, although the inlier rate of our method is not the highest, our method still achieves the bestperformance in terms of Registration Recall.
Revised: Thus, although the inlier rate of our method is not the highest, our method still achieves the bestperformance in terms of Registration Recall.

Original: [Figure]
Revised: [Figure]

Original: Initialization and optimization.
Revised: Initialization and optimization.

Original: Our method performs an iterative optimization for ﬁnding reference points and reconstructing features.
Revised: Our method performs an iterative optimization for ﬁnding reference points and reconstructing features.

Original: In order to clearly show the effectiveness of the optimizationprocess, we count the scene frequency of different inlier ratios in the initial and optimized ¯ C topk(described in Sec. 3.3) and provide the histograms of scene distributions in Fig. 3.
Revised: In order to clearly show the effectiveness of the optimizationprocess, we count the scene frequency of different inlier ratios in the initial and optimized ¯ C topk(described in Sec. 3.3) and provide the histograms of scene distributions in Fig. 3.

Original: Speciﬁcally, wecompute the inlier ratio of the initial and optimized ¯ C topk respectively for each scene on 3DMatchand 3DLoMatch datasets.
Revised: Speciﬁcally, wecompute the inlier ratio of the initial and optimized ¯ C topk respectively for each scene on 3DMatchand 3DLoMatch datasets.

Original: Then we count the number of scenes with different inlier ratios of ¯ C topkon both datasets and present them in percentage form.
Revised: Then we count the number of scenes with different inlier ratios of ¯ C topkon both datasets and present them in percentage form.

Original: In the initial stage, we only use the featureslearned by the backbone to ﬁnd ¯ C topk .
Revised: In the initial stage, we only use the featureslearned by the backbone to ﬁnd ¯ C topk .

Original: On 3DMatch dataset, the inlier ratio is between 0.7 and 1.0 inmost scenarios, and that of 3DLoMatch dataset is about 0 to 0.8.
Revised: On 3DMatch dataset, the inlier ratio is between 0.7 and 1.0 inmost scenarios, and that of 3DLoMatch dataset is about 0 to 0.8.

Original: This shows that the initial backbonefeatures in most scenes are enough to provide acceptable initial reference points.
Revised: This shows that the initial backbonefeatures in most scenes are enough to provide acceptable initial reference points.

Original: After the jointoptimization, the node-wise features are reconstructed with the normalized 3D positional information.
Revised: After the jointoptimization, the node-wise features are reconstructed with the normalized 3D positional information.

Original: The inlier ratio is increased to 0 .
Revised: The inlier ratio is increased to 0 .

Original: 9 ∼ 1 . 0 on the 3DMatch dataset, and the inlier ratio in most scenesof 3DLoMatch is raised to 0 .
Revised: 9 ∼ 1 . 0 on the 3DMatch dataset, and the inlier ratio in most scenesof 3DLoMatch is raised to 0 .

Original: 6 ∼ 1 . 0 .
Revised: 6 ∼ 1 . 0 .

Original: It demonstrates that the proposed optimization method cansigniﬁcantly improve the matching accuracy of the reference points.
Revised: It demonstrates that the proposed optimization method cansigniﬁcantly improve the matching accuracy of the reference points.

Original: Furthermore, we visualize theiterative alignment of the reference points (red and green patches) in Fig.
Revised: Furthermore, we visualize theiterative alignment of the reference points (red and green patches) in Fig.

Original: This indicates that thealignment error between two reference points gradually decreases during the optimization process.
Revised: This indicates that thealignment error between two reference points gradually decreases during the optimization process.

Original: [Table]
Revised: [Table]

Original: Computing overhead and runtime.
Revised: Computing overhead and runtime.

Original: Weevaluate the parameters, GPU memory usage and runtime of our method.
Revised: Weevaluate the parameters, GPU memory usage and runtime of our method.

Original: Tab. 2shows the comparison results with othermethods.
Revised: Tab. 2shows the comparison results with othermethods.

Original: Considering the practical application, we provide the peck of GPU memory usage.
Revised: Considering the practical application, we provide the peck of GPU memory usage.

Original: On parameters, GeoTransformer[29] is 27% more than ours, but our methodstill achieves a competitive registration accuracy compared to it and signiﬁcantly outperforms other methods.
Revised: On parameters, GeoTransformer[29] is 27% more than ours, but our methodstill achieves a competitive registration accuracy compared to it and signiﬁcantly outperforms other methods.

Original: On GPU memory usage, our method ranks 2nd only behind CoFiNet [47],and is only about 40% of GeoTransformer’s [29].
Revised: On GPU memory usage, our method ranks 2nd only behind CoFiNet [47],and is only about 40% of GeoTransformer’s [29].

Original: With such low memory usage, our method stillachieves the state-of-the-art registration performance.
Revised: With such low memory usage, our method stillachieves the state-of-the-art registration performance.

Original: On runtime, our method achieves the fastestregistration speed.
Revised: On runtime, our method achieves the fastestregistration speed.

Original: This demonstrates the superiority of our method in both accuracy and speed.
Revised: This demonstrates the superiority of our method in both accuracy and speed.

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: Metrics.
Revised: Metrics.

Original: Following [17], we evaluate ourmethod with three metrics: 1) Relative Rotation
Revised: Following [17], we evaluate ourmethod with three metrics: 1) Relative Rotation

Original: Error (RRE), the geodesic distance between estimated and ground-truth rotation matrices, 2) Relative Translation Error (RTE), the Euclidean distance between estimated and ground-truth translation vectors, and 3) Registration Recall (RR),which has mentioned before, the thresholds areset as RRE < 5 ◦ and RTE < 2m.
Revised: Error (RRE), the geodesic distance between estimated and ground-truth rotation matrices, 2) Relative Translation Error (RTE), the Euclidean distance between estimated and ground-truth translation vectors, and 3) Registration Recall (RR),which has mentioned before, the thresholds areset as RRE < 5 ◦ and RTE < 2m.

Original: Registration results.
Revised: Registration results.

Original: In Tab. 3, we compareour method with state-of-the-art approaches, including: 3DFeat-Net [45], FCGF [8], D3Feat [3],
Revised: In Tab. 3, we compareour method with state-of-the-art approaches, including: 3DFeat-Net [45], FCGF [8], D3Feat [3],

Original: SpinNet [1], Predator [17], CoFiNet [47] and GeoTransformer [29].
Revised: SpinNet [1], Predator [17], CoFiNet [47] and GeoTransformer [29].

Original: Our method performs the best forall metrics, which proves that our method is scene-agnostic and maintains good registration accuracyin strongly differing scenarios.
Revised: Our method performs the best forall metrics, which proves that our method is scene-agnostic and maintains good registration accuracyin strongly differing scenarios.

Original: 4.3 Ablation study
Revised: 4.3 Ablation study

Original: Table 4 shows the results of the ablation studies on the 3DMatch and 3DLoMatch datasets.
Revised: Table 4 shows the results of the ablation studies on the 3DMatch and 3DLoMatch datasets.

Original: Weremove one component at each time from the complete pipeline (Full) to measure the correspondingcontribution.
Revised: Weremove one component at each time from the complete pipeline (Full) to measure the correspondingcontribution.

Original: Removing the progressive alignment leads to lower Registration Recall on both datasets,which means that aligning reference points in 3D space is beneﬁcial for more accurate positionalembedding.
Revised: Removing the progressive alignment leads to lower Registration Recall on both datasets,which means that aligning reference points in 3D space is beneﬁcial for more accurate positionalembedding.

Original: Without the joint optimization, the error of the estimated rigid transformation drasticallyincreases and the matching inlier ratio decreases signiﬁcantly.
Revised: Without the joint optimization, the error of the estimated rigid transformation drasticallyincreases and the matching inlier ratio decreases signiﬁcantly.

Original: The joint optimization plays animportant role in learning discriminative features.
Revised: The joint optimization plays animportant role in learning discriminative features.

Original: It avoids a lot of incorrect matches arising from3D structural ambiguity.
Revised: It avoids a lot of incorrect matches arising from3D structural ambiguity.

Original: Finally, we replace the proposed one-inlier based position encoding with thecentroid based position encoding, i.e. using the centroids as reference points instead of learning aninlier.
Revised: Finally, we replace the proposed one-inlier based position encoding with thecentroid based position encoding, i.e. using the centroids as reference points instead of learning aninlier.

Original: Since the centroids do not constitute a correct correspondence for a pair of partially overlappingpoint clouds, this alternative results in worse performance.
Revised: Since the centroids do not constitute a correct correspondence for a pair of partially overlappingpoint clouds, this alternative results in worse performance.

Original: [Table]
Revised: [Table]

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we have introduced a simple but efﬁcient position encoding for point cloud registration.
Revised: In this paper, we have introduced a simple but efﬁcient position encoding for point cloud registration.

Original: In contrast to previous methods that rely on a lot of computing resources or need to meet stringentconditions, our method is light-weight and easy to implement.
Revised: In contrast to previous methods that rely on a lot of computing resources or need to meet stringentconditions, our method is light-weight and easy to implement.

Original: Our method only relies on a fewcorrespondences: we show that one inlier is enough to realize accurate position encoding forlearning distinguishable features.
Revised: Our method only relies on a fewcorrespondences: we show that one inlier is enough to realize accurate position encoding forlearning distinguishable features.

Original: We utilize a differentiable optimal transport layer to ﬁnd a goodcorrespondence, which normalizes each point for position encoding.
Revised: We utilize a differentiable optimal transport layer to ﬁnd a goodcorrespondence, which normalizes each point for position encoding.

Original: It builds a reliable bridgebetween point clouds in different reference frames, and preserves spatial consistency to reducefeature ambiguity.
Revised: It builds a reliable bridgebetween point clouds in different reference frames, and preserves spatial consistency to reducefeature ambiguity.

Original: Besides, we propose a joint optimization between correspondence establishmentand position encoding, and along with that, the point cloud features are also iteratively optimized.
Revised: Besides, we propose a joint optimization between correspondence establishmentand position encoding, and along with that, the point cloud features are also iteratively optimized.

Original: Furthermore, a progressive alignment approach is proposed to update point cloud positions andfeatures alternately.
Revised: Furthermore, a progressive alignment approach is proposed to update point cloud positions andfeatures alternately.

Original: The proposed method achieves a competitive registration performance with morecomplex state-of-the-art pipelines on indoor and outdoor datasets.
Revised: The proposed method achieves a competitive registration performance with morecomplex state-of-the-art pipelines on indoor and outdoor datasets.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] See the supplemental material.
Revised: [Yes] See the supplemental material.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [Yes] See thesupplemental material.
Revised: [Yes] See thesupplemental material.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [N/A]
Revised: [N/A]

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main ex-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main ex-

Original: perimental results (either in the supplemental material or as a URL)?
Revised: perimental results (either in the supplemental material or as a URL)?

Original: [Yes] See the
Revised: [Yes] See the

Original: supplemental material.
Revised: supplemental material.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?

Original: [Yes] See the supplemental material.
Revised: [Yes] See the supplemental material.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi-
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experi-

Original: ments multiple times)?
Revised: ments multiple times)?

Original: [No]
Revised: [No]

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: of GPUs, internal cluster, or cloud provider)?
Revised: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See the supplemental material.
Revised: [Yes] See the supplemental material.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] See Section 4.
Revised: [Yes] See Section 4.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] See the supplemental material.
Revised: [Yes] See the supplemental material.

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No]
Revised: [No]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]