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

Original: Abstract
Revised: Abstract

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

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

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

Original: Speciﬁcally, we ﬁrst ﬁnd one correspondence by a differentiable optimal transport layer, and use it to normalize each point for position encoding.
Revised: Speciﬁcally, we ﬁrst ﬁnd one correspondence by a differentiable optimal transport layer, and use it to normalize each point for position encoding.

Original: It can eliminate the challenges brought by the different reference frames of two point clouds, and mitigate the feature ambiguity by learning the spatial consistency.
Revised: It can eliminate the challenges brought by the different reference frames of two point clouds, and mitigate the feature ambiguity by learning the spatial consistency.

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

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

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

Original: Extensive experiments demonstrate the proposed method can achieve competitive performance with the state-of-the-art methods in both indoor and outdoor scenes.
Revised: Extensive experiments demonstrate the proposed method can achieve competitive performance 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 point clouds by estimating a rigid transformation between them.
Revised: The goal of it is to align two partially overlapped point clouds by estimating a rigid transformation between them.

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

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

Original: In recent years learning-based works [28, 44, 50, 8] have made great advances in 3D representation.
Revised: In recent years learning-based works [28, 44, 50, 8] have made great advances in 3D representation.

Original: In addition to utilizing deep neural networks to learn better local feature descriptions [1, 9, 11], recent methods [18, 48, 29, 47] try to introduce Transformer [41] architecture to establish correspondences.
Revised: In addition to utilizing deep neural networks to learn better local feature descriptions [1, 9, 11], recent methods [18, 48, 29, 47] try to introduce Transformer [41] architecture to establish correspondences.

Original: One of the most important strategies of Transformer is position encoding, which has been proven crucial when applying Transformer to many computer vision tasks [22, 34, 35].
Revised: One of the most important strategies of Transformer is position encoding, which has been proven crucial when applying Transformer to many computer vision tasks [22, 34, 35].

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

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

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

Original: Since the two point clouds are under uncorrelated reference frames, the straightforward position encoding is not a good idea [24].
Revised: Since the two point clouds are under uncorrelated reference frames, the straightforward position encoding is not a good idea [24].

Original: Recently, some methods attempt to design speciﬁc position encoding techniques [26, 24, 29] for point cloud registration.
Revised: Recently, some methods attempt to design speciﬁc position encoding techniques [26, 24, 29] for point cloud registration.

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

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

Original: It works well, but adds a lot of computational overhead due to the additional
Revised: It works well, but adds a lot of computational overhead due to the additional

Original: [Figure] calculations about the distance and the angle similarity between points.
Revised: 1 x 1 y 2 x 2 y x  y  calculations about the distance and the angle similarity between points.

Original: DoPE [26] presents an efﬁcient manner to learn a jointorigin as the reference point forpoint cloud position encoding.
Revised: DoPE [26] presents an efﬁcient manner to learn a jointorigin as the reference point for position encoding.

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

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

Original: We utilize a rigorously deﬁned property of rigid transformation, i.e., it does not change the length of a vector (isometric isomorphism).
Revised: We utilize a rigorously deﬁned property of rigid transformation, i.e., it does not change 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( 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( 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 ) is more likely to be a correct correspondence than ( x 2 , y 1 ) only by position information without local feature, 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 ) is more likely to be a correct correspondence than ( x 2 , y 1 ) only by position information without local feature, 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 information can beneﬁt the establishing of correspondences when the local feature is ambiguous.
Revised: In this way, position information can beneﬁt the establishing of correspondences when the local feature is ambiguous.

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

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

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

Original: The selected inlier plays two roles in the position encoding: 1) The two points in the inlier see the same spot of the same scenario, so positional differences caused by reference frames can be compensated based on their normalization.
Revised: The selected inlier plays two roles in the position encoding: 1) The two points in the inlier see the same spot of the same scenario, so positional differences caused by reference frames can be compensated based on their normalization.

Original: 2) It is feasible to do so to ensure spatial consistency, i.e., the relative spatial correlation between arbitrary point and the reference point is maintained across the two point clouds.
Revised: 2) It is feasible to do so to ensure spatial consistency, i.e., the relative spatial correlation between arbitrary point and the reference point is maintained across the two point clouds.

Original: However, it is not easy to ﬁnd an accurate inlier in advance as the reference point, which is a chicken-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 a chicken-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 and correspondences.
Revised: Thus, we propose a joint learning approach to optimize both position encoding and correspondences.

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

Original: Speciﬁcally, we select multi-inliers by performing a differentiable optimal transport 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 optimal transport 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 the correspondence are considered as the reference points.
Revised: Then the two points corresponding to the correspondence are considered as the reference points.

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

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

Original: Such circular optimization leads to more explicit position encoding.
Revised: Such circular optimization leads to more explicit position encoding.

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 above optimization process, an estimated transformation is recovered from some correspondences to align the two point clouds.
Revised: During each step of the above optimization process, an estimated transformation is recovered from some correspondences to align the two point clouds.

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

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 [26, 24, 29], this kind of strategy mainly has three advantages: 1) It is more reasonable for ﬁnding a correct correspondence than centroid estimation for position encoding, especially in the low-overlapping scenarios.
Revised: Compared to the previous works on the position encoding for point cloud registration [26, 24, 29], this kind of strategy mainly has three advantages: 1) It is more reasonable for ﬁnding a correct correspondence than centroid estimation for position encoding, especially in the low-overlapping scenarios.

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 efﬁciently enhance the distinctiveness of features.
Revised: We only need to ﬁnd an inlier to efﬁciently enhance the distinctiveness of features.

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

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 is light-weight and only relies on a few points for accurate position encoding.
Revised: • An efﬁcient position encoding with few correspondences for point cloud registration, which 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 position encoding with point-wise feature encoding as the agent.
Revised: • A joint optimization strategy that, optimizes the correspondence establishment and position encoding with point-wise feature encoding as the agent.

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

Revised: • A progressive point clouds alignment approach, which progressively updates the point cloud positions 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 { C }
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: X
Revised: X

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

Original:  Related Work
Revised:  Related Work

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

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

Original: Histograms usually contain statistical information such as the spatial distribution and geometric properties of points [16].
Revised: Histograms usually contain statistical information such as the spatial distribution and geometric properties of points [16].

Original: The descriptors represented with the 3D spatial distribution include Spin Image [19], 3D Shape Context [13], etc.
Revised: The descriptors represented with the 3D spatial distribution include Spin Image [19], 3D Shape Context [13], etc.

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

Original: [30], SHOT [39], etc.
Revised: [30], SHOT [39], etc.

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

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

Original: propose the pioneering work 3DMatch [49], which utilizes 3D Convolutional Neural Networks to encode local descriptors from local volumetric patches.
Revised: propose the pioneering work 3DMatch [49], which utilizes 3D Convolutional Neural Networks to encode local descriptors from local volumetric patches.

Original: PPFNet [11] introduces the rotation-invariant Point-Pair Features (PPF)
Revised: PPFNet [11] introduces the rotation-invariant Point-Pair Features (PPF)

Original: [12] to encode local descriptors.
Revised: [12] to encode local descriptors.

Original: Follow-up work PPF-FoldNet [10] incorporates FoldingNet [45] as backbone for unsupervised descriptor learning.
Revised: Follow-up work PPF-FoldNet [10] incorporates FoldingNet [45] as backbone for unsupervised descriptor learning.

Original: 3DSN [15] leverages the voxelized Smoothed Density
Revised: 3DSN [15] 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 [9] ﬁrst utilizes the sparse 3D fully convolutional network to extract dense feature descriptors.
Revised: FCGF [9] ﬁrst utilizes the sparse 3D fully convolutional network to extract dense feature descriptors.

Original: SpinNet
Revised: SpinNet

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

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

Original: Deep Closest Point [43] utilizes a graph neural network to extract point cloud features and a pointer network to predict correspondences.
Revised: Deep Closest Point [43] utilizes a graph neural network to extract point cloud features and a pointer network to predict correspondences.

Original: IDAM [23] proposes an iterative distance-aware similarity matrix convolution module to integrate the feature and Euclidean space into the point matching.
Revised: IDAM [23] proposes an iterative distance-aware similarity matrix convolution module to integrate the feature and Euclidean space into the point matching.

Original: Predator [18] leverages the attention mechanism to detect the overlapping region for sampling salient feature points.
Revised: Predator [18] leverages the attention mechanism to detect the overlapping region for sampling salient feature points.

Original: CoFiNet [48] utilizes a coarseto-ﬁne strategy for correspondence ﬁnding.
Revised: CoFiNet [48] utilizes a coarseto-ﬁne strategy for correspondence ﬁnding.

Original: Lepard [24] utilizes the Rotary positional encoding [34] to encode 3D positional information for feature learning.
Revised: Lepard [24] utilizes the Rotary positional encoding [34] to encode 3D positional information for feature learning.

Original: NgeNet [51] introduces a multi-scale architecture to extract features with multiple neighborhood sizes for adaptive selection of feature scale.
Revised: NgeNet [51] introduces a multi-scale architecture to extract features with multiple neighborhood sizes for adaptive selection of feature scale.

Original: REGTR [47] uses transformer to directly predict a set of point correspondences and estimates the rigid transformation using a variant of the Kabsch-Umeyama [20, 40] algorithm.
Revised: REGTR [47] uses transformer to directly predict a set of point correspondences and estimates the rigid transformation using a variant of the Kabsch-Umeyama [20, 40] algorithm.

Original: Position embedding.
Revised: Position embedding.

Original: The idea of relative positional encodings ﬁrst appeared in machine translation [32] and music generation [17].
Revised: The idea of relative positional encodings ﬁrst appeared in machine translation [32] and music generation [17].

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 [27, 4] to the input of the transformer architecture for object detection.
Revised: DETR [6] applies ﬁxed positional encodings [27, 4] to the input of the transformer architecture for object detection.

Original: Swin Transformer [25] adds relative position biases in similarity computation of self-attention, improving performance over models without these biases.
Revised: Swin Transformer [25] adds relative position biases in similarity computation of self-attention, improving performance over models without these biases.

Original: LoFTR [35] uses the 2D extension of the position encoding to produce position-dependent features for image matching.
Revised: LoFTR [35] uses the 2D extension of the position encoding to produce position-dependent features for image matching.

Original: DoPE [26] proposes to calculate a joint-origin of all points for point cloud position encoding.
Revised: DoPE [26] 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 between point 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 between point pairs and combines them with self-attention to embed positional information into the features.

Original:  Approach 3.1 Problem Statement
Revised:  Approach 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 point cloud Y =
Revised: = { x i ∈ R 3 , i = 1 , ..., M } and target point cloud Y =

Original: { y j ∈ R 3 , j = 1 , ..., N } , where M and N are the point numbers of X and Y , respectively.
Revised: { 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, as follows:
Revised: R ∈ SO (3) represents the rotation component and t ∈ R 3 represents the translation component, as follows:

Original: [Equation]
Revised: [Equation]

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

Original: 3.2 Pipeline
Revised: 3.2 Pipeline

Original: Fig.
Revised: Fig.

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

Original: Following [35, 48, 36], our method employs a coarse-toﬁne manner to ﬁnd correspondences.
Revised: Following [35, 48, 36], our method employs a coarse-toﬁne manner to ﬁnd correspondences.

Original: In the coarse stage, we ﬁrst use the KPConv [38] to down-sample raw input points to uniformly distributed nodes ˜ X ∈ R ˜
Revised: In the coarse stage, we ﬁrst use the KPConv [38] to down-sample raw input points to uniformly distributed nodes ˜ X ∈ R ˜

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

Original: Then coarse correspondences are generated by matching the nodes.
Revised: Then coarse correspondences are generated by matching the nodes.

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

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: Each node will be expanded into a local patch by the point-to-node grouping strategy [21, 48], and the patch correspondences can be established by the matching of nodes.
Revised: Each node will be expanded into a local patch by the point-to-node grouping strategy [21, 48], and the patch 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-level points within this patch correspondence, then obtain the ﬁne point correspondences by the mutual top-k selection.
Revised: For each patch correspondence, we utilize an optimal transport layer [31] to calculate a soft assignment matrix between the ﬁne-level points within this patch correspondence, then obtain the ﬁne point correspondences by the mutual top-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 on boosting the accuracy of the coarse results.
Revised: Since the results of ﬁne correspondences mainly depend on the coarse results, our method focuses on boosting the accuracy of the coarse results.

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

Original: Next, we leverage the matched correspondences (the byproduct of the "one-inlier" establishing) to progressively align point clouds (Sec. 3.4).
Revised: Next, we leverage the matched correspondences (the byproduct of the "one-inlier" establishing) to progressively align point clouds (Sec. 3.4).

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

Original: Finally, we introduce the loss function for training the network (Sec. 3.6).
Revised: Finally, we introduce the loss function 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 in the two point clouds can be considered as the reference points respectively.
Revised: As mentioned before, once a correspondence is an inlier, its corresponding points in the two point clouds can be considered as the reference points respectively.

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

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

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

Original: The cross-attention [41, 31] can encode the geometric consistency across the two point clouds by integrating the features of the other point cloud.
Revised: The cross-attention [41, 31] can encode the geometric consistency across the two point clouds by integrating 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 to query 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 to query 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
Revised: i

Original: [Equation]
Revised: [Equation]

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

Original: Following the cross-attention we use a self-attention to aggregate the global context, which updates features in a similar way like cross-attention, but is performed inside the source and target point cloud independently.
Revised: Following the cross-attention we use a self-attention to aggregate the global context, which updates features in a similar way like cross-attention, but is performed 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 transport to ﬁnd good correspondences.
Revised: ˜ Y , we leverage a differentiable optimal transport to 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 parameter α .
Revised: then we append a new row and a new column as in [31], ﬁlled with a learnable dustbin parameter α .

Original: The Sinkhorn algorithm
Revised: The Sinkhorn algorithm

Original: [33] is adopted on ˜ S for searching the best solution of the optimal transportation problem.
Revised: [33] is adopted on ˜ S for searching the best solution of the optimal transportation 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, the higher the probability that the associated correspondence is an inlier.
Revised: Intuitively, the higher the matching score, the higher the probability that the associated correspondence is an inlier.

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

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

Original: We use the selected correspondences to produce a virtual correspondence by an averaging operation of the point coordinates.
Revised: We use the selected correspondences to produce a virtual correspondence by an averaging operation of the point coordinates.

Original: The corresponding points of the virtual correspondence are regarded as the reference points ¯
Revised: The corresponding points of the virtual correspondence are regarded as the reference points ¯

Original: o = { ¯ o X , ¯ o Y } , where the ¯ o X , ¯ o Y denote the reference point in source and target point cloud, respectively.
Revised: o = { ¯ 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 two point clouds respectively.
Revised: After obtaining the reference points ¯ o , they are utilized to normalize the two point clouds respectively.

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

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 the respective reference point in different reference frames.
Revised: j maintaining constant distance with respect to the respective reference point in different reference frames.

Original: Based on this property, the point-wise vectors normalized by the reference points are fed into position embedding layer for feature reconstruction.
Revised: Based on this property, the point-wise vectors normalized 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 individual vector, and output an embedded geometric position feature.
Revised: Speciﬁcally, we use an MLP with 4 hidden layers of feature size being 256 on each individual vector, and output an embedded geometric position feature.

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

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

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

Original: ˜
Revised: ˜

Original: Y
Revised: Y

Original: j + P ( V (˜ y j )) .
Revised: j + P ( V (˜ y j )) .

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

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

Original: We expect to recover a rigid transformation with reliable correspondences ¯ C topk to make the position encoding easier.
Revised: We expect to recover a rigid transformation with reliable correspondences ¯ 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 matching scores, we use them to estimate the current optimal transformation ¯ T t .
Revised: After obtaining the coarse correspondences ¯ C topk with relatively higher matching scores, we use them to estimate the current optimal transformation ¯ T t .

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

Original: Then we use the estimated ¯ T t = { ¯ R t , ¯ t t } to update the relative position between two point clouds:
Revised: Then we use the estimated ¯ T t = { ¯ R t , ¯ t t } to update the relative position 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 clouds have also been updated:
Revised: Naturally, the reference points, i.e., the centroids ¯ o of the selected patches in the two point clouds have 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 point clouds 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 point clouds 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 a more positive effect for establishing correspondences.
Revised: Thus, the encoded position feature can provide a more positive effect for establishing correspondences.

Original: Progressive alignment.
Revised: Progressive alignment.

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

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

Original: We utilize the enhanced features to rebuild some reliable correspondences ¯ C topk and re-estimate the current optimal transformation ¯ T t for the alignment.
Revised: We utilize the enhanced features to rebuild some reliable correspondences ¯ 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 estimated transformation is more accurate than that in previous stage.
Revised: Intuitively, the enhanced features are more distinctive than original features, so the latter estimated transformation is more accurate than that in previous stage.

Original: Thus, multiple transformations enable the two point clouds to be progressively aligned in coordinate space.
Revised: Thus, multiple transformations enable the two point clouds to be progressively aligned in coordinate space.

Original: In this way, the position encoding can also be optimized in the progressive alignment process.
Revised: In this way, the position encoding can 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 proposed position encoding for learning distinctive node features would require the knowledge about at least one correct correspondence; but meantime, establishing one accurate correspondence also relies on 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 proposed position encoding for learning distinctive node features would require the knowledge about at least one correct correspondence; but meantime, establishing one accurate correspondence also relies on the correct position encoding of the point clouds for feature reconstruction.

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

Original: In order to tackle this circular problem, we propose to jointly optimize the correspondences and position encoding by multiple feature rebuilding.
Revised: In order to tackle this circular problem, we propose to jointly optimize the correspondences and position encoding by multiple 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: We construct it together with inlier learning as a joint optimization task, with a view to improving the accuracy of coarse correspondences by enhancing the discrimination of position encoding.
Revised: We construct it together with inlier learning as a joint optimization task, with a view to improving the 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 3D positional information.
Revised: Speciﬁcally, the joint optimization is performed by enhancing the features of point clouds with 3D positional information.

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

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 isomorphism property.
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 isomorphism property.

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

Original: During the above process, the features of the point clouds are continuously optimized.
Revised: During the above process, the features of the point 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 ﬁne correspondence loss L f .
Revised: L c + L f is composed of the coarse correspondence loss L c and ﬁne correspondence loss L f .

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

Original: To supervise the patch-wise feature descriptors for coarse correspondences, we follow [18, 29] and use a variant of the circle loss [37].
Revised: To supervise the patch-wise feature descriptors for coarse correspondences, we follow [18, 29] and use a variant of the circle loss [37].

Original: Considering again an point cloud pair X and Y , we ﬁrst pick the patch P X i (denotes the patch corresponding to the node ˜ x i ∈ ˜ X ) that has at least one positive patch in Y to form a collection of patches P .
Revised: Considering again an point cloud pair X and Y , we ﬁrst pick the patch P X i (denotes the patch corresponding to the node ˜ x i ∈ ˜ X ) that has at least one positive patch in Y to form a collection of patches P .

Original: For each patch P X i ∈ P , the set of its positive (share at least 10% overlap) and negative (do not share overlap) patches in Y are denoted as ε i p and ε i n , respectively.
Revised: For each patch P X i ∈ P , the set of its positive (share at least 10% overlap) and negative (do not share overlap) patches in Y are denoted 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 denotes the 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 denotes the 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: The weights for each positive and negative sample are calculated individually: β i,jp
Revised: The weights for each positive and negative sample are calculated individually: β i,jp

Original: = γ ( d ji − ∆ p ) and β i,kn = γ (∆ n − d ki ) , with the empirical margins ∆ p = 0.1, ∆ n = 1.4.
Revised: = γ ( d ji − ∆ p ) and β 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: The loss L Y c on Y is computed in the same way.
Revised: The loss 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 coarse correspondences using the ground-truth relative transformations.
Revised: For ﬁne correspondence, we ﬁrst determine the ground-truth coarse correspondences 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 in this two patches as I
Revised: We label the unmatched points in this 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: L f = − 1 C

Original: C (cid:88) i =

Original: ( (cid:88) ( x

Original: [Equation]

Original: y ) ∈M i log

Original: [Equation]

Original: x ∈I i log

Original: [Equation]

Original: y ∈J i log

Original: [Equation]
Revised: [Equation]

Original: (8) where C is the number of the patch pairs.
Revised: where C is the number of the patch pairs.

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

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

Original: m i and n i represent the number of the ﬁne level points in this patch pair.
Revised: m i and n 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 [49] and 3DLoMatch [18] benchmarks (Sec. 4.1), and outdoor KITTI odometry [14] benchmark (Sec. 4.2).
Revised: We evaluate our approach on indoor 3DMatch [49] and 3DLoMatch [18] benchmarks (Sec. 4.1), and outdoor KITTI odometry [14] 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 an outdoor driving scenario sparse point cloud dataset acquired by LiDAR.
Revised: KITTI odometry is an outdoor driving scenario sparse point cloud dataset acquired by LiDAR.

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

Original: [Table] e., the percentage of point cloud pairs whose transformation error (RMSE) is below 0.2 m. It is the most important metric in point cloud registration task [18, 48] [Table] because it directly measures the ﬁnal registration performance.
Revised: Recall (RR), i.e., the percentage of point cloud pairs whose transformation error (RMSE) is below 0.2 m. It is the most important metric in point cloud registration task [18, 48], because it directly measures the ﬁnal registration performance.

Original: Following [3, 18], we also report Feature Matching
Revised: Following [3, 18], 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 the ground truth transformation) among the putative matches.
Revised: Ratio (IR), deﬁned as the fraction of correct correspondences (residuals are less than 0.1 m under the ground 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 our method with recent state-of-the-art approaches including: 3DSN [15], FCGF [9], D3Feat [3], SpinNet [1], YOHO [42], Predator [18] 2 , CoFiNet [48], REGTR [47] and GeoTransformer (GeoTrans in the table) [29] in Tab. 1.
Revised: We compare the registration results of our method with recent state-of-the-art approaches including: 3DSN [15], FCGF [9], D3Feat [3], SpinNet [1], YOHO [42], Predator [18] 2 , CoFiNet [48], REGTR [47] and GeoTransformer (GeoTrans in the table) [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 [47] needs to take all the ﬁnal downsampled points for transformation estimation instead of sampling, we only provide the results in the original paper.
Revised: REGTR [47] needs to take all the ﬁnal downsampled points for transformation estimation instead of sampling, we only provide the results in the original paper.

Original: Following [3, 18], we run RANSAC50k to estimate the rigid transformation.
Revised: Following [3, 18], we run RANSAC50k to estimate the rigid transformation.

Original: When compared with local 3D descriptors, our method outperforms all the approaches on both datasets.
Revised: When compared with local 3D descriptors, our method outperforms 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 the low-overlap scenarios (3DLoMatch).
Revised: GeoTransformer [29] outperform other methods by a large margin on all the metrics, especially in the low-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 for point 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 for point 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 reﬂects the ﬁnal performance on point cloud registration, the proposed method achieves the state-of-the-art performance.
Revised: However, when referring to the most important metric Registration Recall (RR) which better reﬂects the ﬁnal performance on point cloud registration, the proposed method achieves the state-of-the-art performance.

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

Original: As pointed out in [18, 48, 29], besides the inlier rate, the other important issue for registration is the sparsity of the correspondences.
Revised: As pointed out in [18, 48, 29], besides the inlier rate, the other important issue for registration is the sparsity of the correspondences.

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

Original: Since our position encoding can introduce spatial consistency, it is possible to ﬁnd matching pairs that cannot be found by local features alone.
Revised: Since our position encoding can introduce spatial 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 best performance in terms of Registration Recall.
Revised: Thus, although the inlier rate of our method is not the highest, our method still achieves the best performance 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 optimization process, 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 optimization process, 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, we compute the inlier ratio of the initial and optimized ¯ C topk respectively for each scene on 3DMatch and 3DLoMatch datasets.
Revised: Speciﬁcally, we compute the inlier ratio of the initial and optimized ¯ C topk respectively for each scene on 3DMatch and 3DLoMatch datasets.

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

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

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

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

Original: After the joint optimization, the node-wise features are reconstructed with the normalized 3D positional information.
Revised: After the joint optimization, 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 scenes of 3DLoMatch is raised to 0 .
Revised: 9 ∼ 1 . 0 on the 3DMatch dataset, and the inlier ratio in most scenes of 3DLoMatch is raised to 0 .

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

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

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

Original: This indicates that the alignment error between two reference points gradually decreases during the optimization process.
Revised: This indicates that the alignment 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: We evaluate the parameters, GPU memory usage and runtime of our method.
Revised: We evaluate the parameters, GPU memory usage and runtime of our method.

Original: Tab.shows the comparison results with other methods.
Revised: Tab.shows the comparison results with other methods.

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 method still 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 method still 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 [48], and is only about 40% of GeoTransformer’s [29].
Revised: On GPU memory usage, our method ranks 2nd only behind CoFiNet [48], and is only about 40% of GeoTransformer’s [29].

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

Original: On runtime, our method achieves the fastest registration speed.
Revised: On runtime, our method achieves the fastest registration 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: 4.2 KITTI
Revised: 4.2 KITTI

Original: [Table]
Revised: [Table]

Original: Metrics.
Revised: Metrics.

Original: Following [18], we evaluate our method with three metrics: 1) Relative Rotation
Revised: Following [18], we evaluate our method 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 are set as RRE < 5 ◦
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 are set as RRE < 5 ◦

Original: and RTE < 2m.
Revised: and RTE < 2m.

Original: Registration results.
Revised: Registration results.

Original: In Tab. 3, we compare our method with state-of-the-art approaches, including: 3DFeat-Net [46], FCGF [9], D3Feat [3], SpinNet [1], Predator [18], CoFiNet [48] and
Revised: In Tab. 3, we compare our method with state-of-the-art approaches, including: 3DFeat-Net [46], FCGF [9], D3Feat [3], SpinNet [1], Predator [18], CoFiNet [48] and

Original: GeoTransformer [29].
Revised: GeoTransformer [29].

Original: Our method performs the best for all metrics, which proves that our method is scene-agnostic and maintains good registration accuracy in strongly differing scenarios.
Revised: Our method performs the best for all metrics, which proves that our method is scene-agnostic and maintains good registration accuracy in 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: We remove one component at each time from the complete pipeline (Full) to measure the corresponding contribution.
Revised: We remove one component at each time from the complete pipeline (Full) to measure the corresponding contribution.

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 positional embedding.
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 positional embedding.

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

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

Original: It avoids a lot of incorrect matches arising from 3D structural ambiguity.
Revised: It avoids a lot of incorrect matches arising from 3D structural ambiguity.

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

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

Original: [Table]
Revised: [Table]

Original: 4.4 Relationship between the registration and correspondences
Revised: 4.4 Relationship between the registration and correspondences

Original: In order to analyze the relationship between the registration results and the established correspondences, we count the scene frequency of different inlier ratios for Geotransformer and our method, and present the curve charts of scene distributions in Fig.
Revised: In order to analyze the relationship between the registration results and the established correspondences, we count the scene frequency of different inlier ratios for Geotransformer and our method, and present the curve charts of scene distributions in Fig.

Original: Speciﬁcally, we count the number of scenes with different inlier ratios on 3DMatch and 3DLoMatch datasets, and present them in frequency form.
Revised: Speciﬁcally, we count the number of scenes with different inlier ratios on 3DMatch and 3DLoMatch datasets, and present them in frequency form.

Original: Besides, we also calculate the scene registration recall of different inlier ratios and provide the curve charts in Fig.
Revised: Besides, we also calculate the scene registration recall of different inlier ratios and provide the curve charts in Fig.

Original: On 3DMatch dataset, the inlier ratio of our method is between 0.
Revised: On 3DMatch dataset, the inlier ratio of our method is between 0.

Original: and 0.9 in most scenarios, while the inlier ratio of Geotransformer is about 0.5 to 1.0.
Revised: and 0.9 in most scenarios, while the inlier ratio of Geotransformer is about 0.5 to 1.0.

Original: On 3DLoMatch, the inlier ratio of most scenarios for our method is between 0.05 and 0.6.
Revised: On 3DLoMatch, the inlier ratio of most scenarios for our method is between 0.05 and 0.6.

Original: For Geotransformer, it has more scenarios where the inlier ratio is 0.5 ∼ 0.9.
Revised: For Geotransformer, it has more scenarios where the inlier ratio is 0.5 ∼ 0.9.

Original: In fact, the inlier ratio is counted by averaging all the point cloud pairs.
Revised: In fact, the inlier ratio is counted by averaging all the point cloud pairs.

Original: Geotransformer [29] has higher inlier ratio in some scenarios, which causes the mean inlier ratio to be higher than ours.
Revised: Geotransformer [29] has higher inlier ratio in some scenarios, which causes the mean inlier ratio to be higher than ours.

Original: However, higher inlier ratio is not a necessary condition for high registration recall.
Revised: However, higher inlier ratio is not a necessary condition for high registration recall.

Original: As show in the Fig.
Revised: As show in the Fig.

Original: 6, we can observe that our method can achieve better registration recall than Geotransformer with the same inlier ratio (especially the interval [0.15, 0.5] on 3DLoMatch dataset), which shows that the inliers of our method may be better distributed for model estimation
Revised: 6, we can observe that our method can achieve better registration recall than Geotransformer with the same inlier ratio (especially the interval [0.15, 0.5] on 3DLoMatch dataset), which shows that the inliers of our method may be better distributed for model estimation

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original:  Conclusion
Revised:  Conclusion

Original: In this paper
Revised: In this paper

Original: [Figure]
Revised: [Figure]

Original: we have introduced a simple but efﬁcient position encoding for point cloud registration.
Revised: 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 stringent conditions, 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 stringent conditions, our method is light-weight and easy to implement.

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

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

Original: It builds a reliable bridge between point clouds in different reference frames, and preserves spatial consistency to reduce feature ambiguity.
Revised: It builds a reliable bridge between point clouds in different reference frames, and preserves spatial consistency to reduce feature ambiguity.

Original: Besides, we propose a joint optimization between correspondence establishment and position encoding, and along with that, the point cloud features are also iteratively optimized.
Revised: Besides, we propose a joint optimization between correspondence establishment and 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 and features alternately.
Revised: Furthermore, a progressive alignment approach is proposed to update point cloud positions and features alternately.

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

Original: Limitations.
Revised: Limitations.

Original: Our method adopts a coarse-to-ﬁne manner to ﬁnd correspondences.
Revised: Our method adopts a coarse-to-ﬁne manner to ﬁnd correspondences.

Original: However, the ﬁnal point correspondences are largely subject to the coarse correspondences.
Revised: However, the ﬁnal point correspondences are largely subject to the coarse correspondences.

Original: Once a patch correspondence is incorrect, it would result in incorrect point correspondences.
Revised: Once a patch correspondence is incorrect, it would result in incorrect point correspondences.

Original: These incorrect point correspondences would have a potential negative impact on the ﬁnal registration accuracy.
Revised: These incorrect point correspondences would have a potential negative impact on the ﬁnal registration accuracy.

Original: Besides, our method would fail in the case of the overlapping region between two point clouds is too small or the overlapping region is ambiguous structure.
Revised: Besides, our method would fail in the case of the overlapping region between two point clouds is too small or the overlapping region is ambiguous structure.

Original: A possible solution is to expand the overlapping region via point cloud completion.
Revised: A possible solution is to expand the overlapping region via point cloud completion.

Original:  Acknowledgment
Revised:  Acknowledgment

Original: This work was in part supported by the National Natural Science Foundation of China under Grants 62176096.
Revised: This work was in part supported by the National Natural Science Foundation of China under Grants 62176096.

Original: Checklist
Revised: Checklist

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work?
Revised: [Yes] (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 the supplemental material.
Revised: [Yes] See the supplemental material.

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

Original: [Yes] 2.
Revised: [Yes] 2.

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

Original: [N/A] (b) Did you include complete proofs of all theoretical results?
Revised: [N/A] (b) Did you include complete proofs of all theoretical results?

Original: [N/A] 3.
Revised: [N/A] 3.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

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

Original: (b) Did you specify all the training details (e
Revised: (b) Did you specify all the training details (e

Original: g., data splits, hyperparameters, how they were chosen)?
Revised: g., data splits, hyperparameters, how they were 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 experiments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [No] (d) Did you include the total amount of compute and the type of resources used (e
Revised: [No] (d) Did you include the total amount of compute and the type of resources used (e

Original: g., type of GPUs, internal cluster, or cloud provider)?
Revised: g., type 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
Revised: If you are using existing assets (e

Original: g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: g., code, data, models) or curating/releasing new assets... (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] (d) Did you discuss whether
Revised: [No] (d) Did you discuss whether

Original: and how consent was obtained from people whose data you’re using/curating?
Revised: and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?

Original: [N/A] 5.
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

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