A robust inlier identification algorithm for point cloud registration via $\mathbf{\ell_0}$-minimization
Authors: Yinuo Jiang
Tang Xiuchuan
Cheng Cheng
Ye Yuan
Venue: NeurIPS 2024 poster (2024)
Keywords: Point cloud registration
Inlier identification
Optimization
Primary area: machine_vision
TL;DR: We propose an effective $\ell_0$-norm based inlier identification algorithm for robust point cloud registration.

Abstract: Correspondences in point cloud registration are prone to outliers, significantly reducing registration accuracy and highlighting the need for precise inlier identification. In this paper, we propose a robust inlier identification algorithm for point cloud registration by reformulating the conventional registration problem as an alignment error $\ell_0$-minimization problem. The $\ell_0$-minimization problem is formulated for each local set, where those local sets are built on a compatibility graph of input correspondences. To resolve the $\ell_0$-minimization, we develop a novel two-stage decoupling strategy, which first decouples the alignment error into a rotation fitting error and a translation fitting error. Second, null-space matrices are employed to decouple inlier identification from the estimation of rotation and translation respectively, thereby applying Bayesian theory to $\ell_0$-minimization problems and solving for fitting errors. Correspondences with the smallest errors are identified as inliers to generate a transformation hypothesis for each local set. The best hypothesis is selected to perform registration. We demonstrate that the proposed inlier identification algorithm is robust under high outlier ratios and noise through experiments. Extensive results on the KITTI, 3DMatch, and 3DLoMatch datasets demonstrate that our method achieves state-of-the-art performance compared to both traditional and learning-based methods in various indoor and outdoor scenes.

A robust inlier identification algorithm for point cloud registration via lo-minimization
Yinuo Jiang!*
Xiuchuan Tang²*
Cheng Cheng'
Ye Yuan
Sccooo of Artificial Intelligence and Automation, Huazhong University of Science and Technology, China
'Department of Automation, Tsinghua University, China
{j (jiangyinoo,c.o c_ cheenn,, yye edu cn; tangxiuchuan Smail tsinghua. edu,
https://githut: http:/ttww..coc/aaaaaaaa//aaeeea http:///mmmeeedeetiiiaaaattaaaae identification 1ia-10
Abstract
Correspondences in point cloud registration are prone to outliers, significantly reducing registration accuracy and highlighting the need for precise inlier iden- tification. this paper, we propose a robust inlier identification algorithm for point cloud registration by reformulating the conventional registration problem as an alignment error la fo-minimization problem. The fg-minimization problem is formulated for each local set, where those local sets are built compatibility graph of input correspondences. To resolve the fn-minimization, we develop a novel two-stage decoupling strategy, which first decouples the alignment error into a rotation fitting error and a translation fitting error. Second, null-space matrices are employed decouple inlier identification from the estimation of rotation and trans- lation respectively, thereby applying Bayes Theorem to tg fo-minimization problems and solving for fitting errors. Correspondences with the smallest errors are identi- fied as inliers to generate a transformation hypothesis for each local set. The best hypothesis is selected perform registration. We demonstrate that the proposed inlier identification algorithm is robust under high outlier ratios and noise through experiments. Extensive results on the KITTI, 3DMatch, and 3DLoMatch datasets demonstrate that our method achieves state-of-ther performance compared to both traditional and learning- based methods various indoor and outdoor scenes.
Introduction
Point cloud registration a fundamental task vision and robotics, playing an important role in many applications such 3D perception and reconstruction, simultancous localization and mapping (SLAM), and autonomous driving [8 45 33). aims to align two partially overlapping point clouds by estimating rigid transformation between them. A common registration pipeline involves extracting features through 3D local descriptors, establishing correspondences based on feature matching, and estimating the rigid transformation (8,44] 41. However, due to the less effectiveness of 3D local descriptors in feature extraction (39]. correspondences established through feature matching prone outliers, resulting in inaccurate registration.
Recent works in point could registration with outliers can generally categorized into three groups: leasning-based, geometry-only, and optimization-base methods. Learning- baaed methods [ 8 19
"Equal contribution.
i Corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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use networks to estimate confidence for correspondences and select those with high confidence for transformation estimation. These networks, however, are typically trained on specific scenarios, leading to limited generalization for outlier removal across various datasets 45] Geometry-one methods [ 45), such as SC²-PCR (6] and MAC 455, filter out outliers using geometric relations between correspondences. Such methods 6 45 rely on effective geometric features and may not produce acceptable inlier ratios in complex scenes or noisy environments [8].
On the other hand, optimization-basel methods formulating some non-convex objectives [8] The Branch-and- and--ound (BnB) algorithm is widely used to solve non-convex objectives 4 5 40 due to its ability to guarantee global optimality. However, the efficiency of BnB is affected by the dimensions of search space and the bounds on objectives [8], which may lead to worst-case exponential time [8, 39]. An alternative approach is to relax the non-convex registration problem into a convex semidefinite program K 39 However, semidefinite relaxation is computationally expensive and may introduce outliers or noise, leading to poor estimation results. Therefore, achieving robust and efficient registration in scenarios with high outlier ratios and noise remains challenging problem.
8
40 46] solve the registration problem by
To address these challenges, we propose a robust inlier identification algorithm for point cloud registration, which reformulates the conventional registration problem as an alignment error tu- minimization problem. More specifically, we define the alignment error and formulate an éu- minimization problem for each local set, where these sets are built from the compatibility graph of input correspondences. To resolve the non-convex Eg mi-miimmiztiinn problem effectively, we design a two-stage decoupling strategy. First, the alignment error is decoupled into a rotation fitting error and a translation fitting error by calculating the relative positions between points. This decoupling results in two fitting error fo-minimization problems with respect to rotation and translation, respectively. Second, null-spaces are introduced to remove rotation or translation from the constraints of fitting error fo-minimization problems, thereby decoupling inlier identification from the estimation of rotation or translation. The final decoupled & f6-minimization problems are solved for fitting errors through Bayes Theorem. For each local set, correspondences with the smallest errors are identified as inliers to generate a transformation hypothesis. The best hypothesis is selected to perform registration.
To the best of our knowledge, we are the first to propose a fo-norm based approach to solve the registration problem. We experimentally demonstrate that the proposed algorithm robust to high outlier ratios and noise, and is efficient with varying numbers of correspondences. Extensive results on the KITTI, 3DMatch, and 3DLoMMtth datasets also demonstrate that our method achieves the highest registration accuracy while being competitive in time efficiency compared to state-of-the-art methods. summary, our main contributions are as follows:
A novel robust inlier identification algorithm is proposed by reformulating the conventional registration as alignment error fo-minimization problem, which can effectively identify inliers and perform accurate registration under high outlier ratios and noise.
A two-stage decoupling strategy is designed for the proposed Ey miniiiization problem. This strategy first decouples rotation and translation, and then decouples inlier identification from rotation or translation estimation.
A robust Bayesian-based approach is proposed to solve the decoupled fn-minimization problem and identify inliers, enhancing the algorithm's 's performance on noisy data.
2 Related Work
3D local descriptors. Early handcrafted descriptors like PFH (77 and FPFH (6) mainly represent local fcatures by encoding geometric histograms (38). More recent works attempt to encode 3D local descriptors in a data-driven way. FCGF [9] extracts features through a fully convolutional neural network. Predator [17] applies an attention mechanism extract salient points in overlapping regions of point clouds. 3DMatch [44] and 3DSmoothNet [4] build a Siamese deep learning architecture for extracting local information. Although these feature descriptors achieve significant performance improvements, it is difficult to establish correspondences that are completely free of outliers [6] Therefore, robust registration is very important for accurate registration.
dearning-based methods. Inspired by the success of deep learning in 3D perception recent works have adopted learning networks for point cloud registration [
34
32
19
20
44
Deep

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global registration [8 utilizes sparse convolution and point-by-pons po-nt MLPs to classify input correspondences. [ explores a spatial consistency-guins non-local inlier classifier to remove outliers. VBReg 19 introduces variational non-local network for outlier rejection and learns features with Bayesian-driven long-range contextual dependencies. Despite significant advancements in learning-based registration, these methods are designed for specific scenarios and lack generalization across different datasets, which limits their applicability.
Geometry-only and optimization- based methods. Traditional methods have shown great value in practical applications because they are generalized and require no training. They are primarily classified into two categories: geometry-oni, and optimization- -ased methods. Geometry-only meth- ods 6 45 12 rely on geometric relations and graph-theory frameworks estimate transformations. 6 uses a second order spatial compatibility measure to compute the similarity between correspondences. MAC [45] loosens the maximum clique constraint to mine more local consensus information in a graph. However, these methods can not guarantee global optimality and may fail under high outlier ratios or noise. Optimization-ben based methods [46 39
5 19 aim to estimate
FGR [46] employs the Geman-McClure cost function and graduated non-convexity to solve the resulting non-convex optimization. Although it is fast and simple, it performs poorly with high outlier ratios [3 45). There are also some methods [4] 40 relying on the cranch-and-boun (BnB) algorithm for global registration. However, they suffer from high computational complexity and may require exponential time in the worst case [45]. Therefore, achieving robust and efficient registration scenarios with high outlier ratios and noise remains challenging.
optimal solutions (in the maximum likelihood sense) for transformations [39
3 Methods
3.1
Conventional Point Cloud Registration Problem Statement
Given the source point cloud P {p, E R
1,..., . N} and target point cloud Q {q, RR i= AM ). the objective of point cloud registration to align these two point clouds by estimating an optimal rigid transformation T {R. t]. where R SO(3) denotes the rotation matrix and t e & denotes the translation vector. The transformation is then solved by the following registration problem 25):
min R,t
Σ (P...)E
Rp, +t +
(1)
where C fc; | 1,..,, No) is the initial correspondence set. Each correspondence c; (P;, q:) formed through feature matching, using descriptors extracted from both point clouds.
3.2 Inlier Identification via l miminimizationn
However, the initial correspondence set C contains a large proportion of outliers due to incorrect feature matching, leading inaccurate registration. We aim to identify inliers within C by solving the proposed alignment error En-minimization problem. The pipeline of our method is shown in Fig. D
For the input correspondences, compatibility measure (6), where correspondences are represented as nodes and edges link geometri- cally compatible nodes 45). Based on the compatibility scores, we select N correspondences as reliable seeds, denoted as C, = {c; [ 1..... N1.. For each seed, we identify N₂ compatible correspondences form a local set [1] (refer to Appendix [A.] for details). The alignment error lo dy-minimization problem formulated for each local set. Specifically, for correspondences in the k-th local set (CR, (Pk.: qs.)
construct global compatibility graph using the second-order
I,..., 1,...,NN), the fo-minimization problem is defined follows:
Of = arg min ||Oxlle. O
22)
subject to: O = Q P&R - t Ex.
where PA = (px} E Nax3 and Q = {qx) E N2x3 denote the source and target points in the k-th local set. O₁ represents the introduced alignment error, E, represents the Gaussian noise, and I is a column vector of ones, ensuring the translation vector applied to each point in Pg. The alignment error O also serves inlier indicator. If the i-th correspondence Ck:- (p Ikr is
3

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Input Correspondences [c. I....AA
Hypothesis Selection
Inlier identification via lo f-miniimzaiionn
Two-stage Decoupling Strategy
Inlier Identification via o mimmimmmization
Two-stage Decoupling Strategy
Define alignment error for correspondence in each local set
Stage 1: Decouple alignment error rotation fitting error 0
Dag 625
Grg JAANS
Stage
Bayesian based infier identiication
Ri R
Cs O
Stage 1: Decouple alignment error into translation fitting error ō
o fomiiniiizttion problem
Stage 2:
Bayesian based idletification
min [[[02, Og. ozllle, min l|[og. Og. og]lle,
Figure I: Pipeline of our method. 1. Define alignment errors and formulate the Lo fo-minimization problem for each local set. 2. Decouple alignment error into rotation and translation fitting errors and decouple inlier identification from the estimation of rotation or translation through the Bayes Theorem. 3. Select the best hypothesis for registration.
an inlier (i.e., it satisfies RRps, - 9k, S Es. with the Gaussian noise Ek, ). Os, should idcally be a zero vector. Consequently, the indices of zero vectors in the solution Oj of Eq. (2) correspond to the inlier indices in the k-th set. The formulations for other local sets are defined in a similar way.
key insight into our approach is the of ln norm optimize alignment errors. This is based on the principle that only inliers can be fitted by the same transformation (5), and the optimal transformation is estimated the that fits the largest number of inlier correspondences. Therefore, our optimization objective to maximize the count of zero vectors in the alignment error. Compared the common formulations for point cloud registration [18], such as consensus maximization 7 and truncated least-squares [39], our formulation reduces the impact of outliers through Eo norm. The focus this norm is to minimize the number of non-zero vectors rather than their magnitudes, thereby enhancing the robustness of our method to outliers and noise.
3.3 Two-stage Decoupling Strategy
To resolve the proposed fo-minimization problem, we design a two-stage decoupling strategy. The solution process is described for the k-th local and similarly applied to other local sets.
Decoupling the alignment error into rotation and translation fitting errors. Simultancously estimating the rigid transformation with 6 degrees of freedom (DOF) time-consuming due to the high-dimensional parameter space [5] 39]. To effectively resolve the fn-minimization problem proposed in Eq. (2) for each local set, decouple the 6-DOF transformation into 3-DOF rotation and 3-DOF translation by computing the relative positions between point pairs. For any two given points Pı, and 9k, in the k-th local set, the translation vector ta cancels out in the subtraction [39
yky
qk.
R& (Pk, Pk.) + (Ok, - ok.) + (Ek, Ck.)
(3)
Based on Eq. (3. we define qk, = qk, qk, and Pk., = Pky Pk, the relative positions. Ōke Ok Où; represents the rotation fitting error to minimize, unaffected by translation. Ekty Ex, Ex. is the Gaussian noise. If both the i-th and j-th correspondences are inliers, Ōk, should ideally be zero vector. Therefore, the rotation fitting error Ōk, for the correspondence pair Cı. (Pk,, qk:)) and Ck2 (Pky qk,) is formulated as:
Ok:
9kss
R&Pkis
Exes
(4)

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Having decoupled rotation from translation, we can now formulate the fo-minimization problem for the rotation fitting error O2 in the k-th local set, focusing on the 3-DOF rotation Rg:
O arg min O
O
(5)
subject to: O₂ P - =
where Q2 E Rx3 and Ps E TR.x3 are relative positions between all point pairs in Qs and Pk- respectively. Here, N Na(Na=1) is the number of relative point pairs in a local set. The Gaussian 2
noise E is modeled as (O,AARI) where ÅR indicates the variance.
Once obtaining the rotation estimate R by solving Eq. (5), we substitute it back into Eq. to estimate the translation. The fg-minimization problem for the translation fitting error Og formulated as follows, focusing on the 3-DOF translation ty:
O arg min ||O O
subject to: Ö₂ = Qs 1e-PAe P t²1² Ex
where Eg modeled as (O, A, where de indicates the variance of Gaussian noise. The translation 1% is estimated by solving Eq. (6).
Decoupling rotation estimation from fo-minimization. Optimizing the estimation of rotation while simultancously identifying inliers is a chicken-and-oge problem, because reliable identification of inliers depends on the precise rotation estimation (as shown in Eq. (5)). To address this, we further decouple inlier identification from the estimation of rotation. The inliers that can be fitted by the same rotation are identified through Bayes Theorem and used for the subsequent rotation estimation.
We incorporate a robust Bayesian approach to solve Eq. (5). improving the algorithm's 's robustness to noisy data [42] The key step is to define null-space matrix +g whose rows form basis for the left null space of PA- By left-multiplying each term in the constraint of Eq. (5) with O₁ the component associated with the rotation R₃ is eliminated:
0₂ Q2 -
where Og represents the transformed rotation fitting error. Given that E Gaussian noise and the left-multillication by θ is a linear operation, O, also follows a Gaussian distribution with covariance matrix of Ag θ OT The likelihood formulated as:
P( (x) ( Oz ARlls) x exp
OxO,) 0.0.....I -
OxOx))
2AR
where Q Q and 6.O% n.. Based the Bayes Theorem and Maximum A Posteriori (MAP) estimate, the unconstrained optimization for rotation fitting error fg-minimization Eq. (5) redefined as:
min | ( -0. Yo,) ( 0.00:) -Q.0,))) An | (Q.
1
(9)
O. 2
where AR is the regularization parameter that trades off the fitting error and model complexity. However, since the formulation incorporating the to norm is known to be computationally expensive, we use the following convex relaxation:
min 1 ( O 2
1
' " o, 0.0z) An | f
(10)
where || | is the Frobenius norm (F norm), which is both differentiable and convex. To find the optimal solution, we the gradient of the objective function with respect to O₂ zero:
(å. +₂O+) + 2A = 0.
(11)
The optimal explicit solution Og can be directly calculated as:
o ( II O + 2ARI), I Q
(12)
5

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Based on Og we identify t correspondences with minimal rotation fitting error for accurate rotation estimation. These correspondences, indexed by Zn. provide the basis for estimating rotation from the SVD) decomposition of the matrix H U.VT E R3x3 23
For the k-th local set, the
rotation hypothesis is estimated as [5 :
H
P Pky (i.))IR
E
R U diag (1, 1, det (UV"))
(13)
Decoupling translation estimation from fo-mininization. Employing a strategy similar to that used for rotation estimation, utilize null-space matrix Q₁ that satisfies θ 1 = 0 to isolate the translation. By applying θ to the transpose the constraint Eq. ( we eliminate the components associated with translation l::
O₂ o = ( -.
(14)
Incorporating the Bayes Theorem, we formulate the following convex relaxation for the unconstrained optimization problem:
min _ (X, 0₂0O)) (X, - 0x, + A 10:0
Ô2 2
(15)
where X O Q (PxR))*) and II O The explicit solution Ō is obtained by solving the gradient of the objective function with respect to O₃:
O = ((2AT 00 '
(16)
where I denotes the identity matrix. Using Öj, top-K, correspondences with the smallest errors identified as inliers for translation estimation. Their index set is denoted as I, The translation hypothesis tğ for the k-th local set is estimated based on these inliers (Pk,, qky), with (1,5) E I.
3.4 Hypothesis Selection
Finally, evaluate and select the best estimation from the transformation hypotheses computed for all local sets:
N
RR't")
max R.
RAP. + tx - -9:l12 < <]
(17)
where N₂ is the number of input initial correspondences and 7 is predefined error threshold. For each transformation hypothesis, we quantify its effectiveness by counting the number of correspondences that satisfy the constraints within The transformation with the highest inlier count is selected for registration.
Experiments
4.1
Datasets and Experimental Setup
Synthetic dataset. We evaluate the accuracy, robustness, and efficiency of our algorithm using the
Bunny point cloud from the Stanford 3D Scan Repository [10). Similar to 5 39]. the Bunny model downsampled to N₂ points and resized to fit a [0, 1]3 cube, creating the source point cloud P. To generate the target point cloud Q, random transformation (R, t) applied P and then Gaussian noise E; . (O, o r²I3) ) is added. A pair of the original and moved points defines an inlier. The inliers contaminated with outliers generated by random transformations.
Outdoor seenes. For evaluations on outdoor scenes, we conduct experiments on the KITTI dataset 13]. Following [5] 6) use 555 pairs of point clouds from scenes 8 to 10 for test- ing. We construct a 30cm voxel grid to downsample point clouds and form correspondences using handcrafted FPFH 266 and learned FCGF [9] descriptors.
Indoor scenes. We conduct experiments on the 3DMatch dataset to evaluate performance on indoor scenes. Following [5, 6. 45), use RGB-D scans from 8 real indoor scenes for testing. The

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-+-GRR
RANSAC MAC
GORE Ours
FGR
RANSAC
GORE
*-TEASER+++
- TEASER++
MAC
Ours
Rotation Error (degrees)
21
(cm)
18
3
10
30
40
50
60
80
20
40
60
80
90
Outlier Ratio (%6
Outlier Ratio (96
FGR
RANSAC
GORE
Rotation Error (degrees)
-* TEASER+
MAC
Ours
(am)
* TEASER++
MAC
Ours
100
10
0.1
91
92 93 94
95
96
97
98
99
91
92
93 94
95
96 97
98 99
Outller Ratio (%)
Outlier Ratio (96
Figure 2: Robustness to outliers. The first row compares the rotation and translation errors as the outlier ratio increases from 10% to 90% the Bunny dataset [10], while the second row focuses
on the scenarios extreme outliers, i.e., the outlier ratio varies from 91% to 99%. Our method
demonstrates to be more robust to outliers compared to other methods 4
39 45 46
RANSAC
Rotation Error (degrees)
200
- -
MAC
Ours
Translation Error (cm)
500
-X
TEASER+
MAC
Ours
200
50
100
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Noise
0.05
Noise
Figure 3: Robustness to noise. Comparison results with 4 12
39
45
as the noise standard
deviation increases from 0.01 0.09 on the Bunny dataset 10
point clouds are downsampled using a 5em voxel grid. We use the hand-crafted FPFH 26 along with two learned descriptors, FCGF [26) and 3DSmoothNet [14], for feature extraction. To evaluate our method more challenging scenarios, we conduct experiments on 3DLoMatch (17) (overlap rate between scenes < 30%).. Following
[ [9] the Predator descriptor
[7 is used 3DLoMatch.
Evaluation criteria. Following registration recall (RR) evaluation metrics. The registration is considered successful when the RE < 13", TE < 30cm on 3DMatch & 3DLoMatch datasets, and RE < 5 TE < 60cm on KITTI dataset. Average RE and TE computed only on the successfully registered pairs [5]
we use the rotation error (RE), translation error (TE), and
Implementation details. We implement our method in PyTorch 224. All the experiments are conducted on machine with Intel Xeon Gold 6134 CPU and a single NVIDIA GTX3090.
4.2 Evaluation on Synthetic Dataset
Robustness outliers. We evaluate the robustness to outliers by increasing the outlier ratio from 10% to 90%. The Bunny point cloud is downsampled to N_ = 500. We add zero-mean Gaussian 0.01. For each outlier ratio, we conduct 50 independent
noise with standard deviation set trials and report the average rotation еттоr (RE) and translation error (TE). We compare our method with state-of-the-at traditional methods
46). As shown in the first row of Fig. 2 the 45
rotation and translation errors of FGR [46 increase sharply as the proportion of outliers increases. RANSAC [12 and GORE 4 start failing at an outlier ratio of 60%. Our method remains robust to outliers up 90% and produces more accurate estimates than all other methods. We further
7

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100
*+-EASER+++
Rotation (degreas)
1
FGR
RANSAC
GORE
TEASER+
TDS
16
Translation Error (cm)
12
0.0001
250 500
1000
2500
5000
Number correspondences Efficiency comparisons
RANSAC
GORE TEASER++
TDS
(b) Effectiveness TDS
Figure 4: Efficiency and effectiveness. The experiment results on the Bunny dataset
10
(a)
Efficiency comparison with other methods 12
39
45
46 with respect to the number of correspon-
dences. (b) Comparison of the proposed two-stage decoupling strategy (TDS) with optimization
methods outlier ratio of 90%.
Table 1: Comparison results KITTI dataset
13
using the FPFH
26 and FCGF 9 descriptors.
FPFH RE(O)
FCGF RE( )
Time(s)
RR( %6)
TE(em)
RR(%)1
TE(em)
Traditional
FGR [4
5.23
0.86
43.84
89.54
0.46
25.72
3.88
RANSAC 12
74.41
1.55
30.20
80.36
0.73
26.79
5.43
TEASER+ 39
91.17
1.03
17.98
95.51
0.33
22.38
0,03
SC²-PCR 6
99.46
0.35
7.87
98.02
0.33
20.69
0.31
MAC 45
97.66
0.41
8.61
97.84
0.34
19.34
3.29
TR-DE 5
96.76
0.90
15.63
98.20
0.38
18.00
TEAR 18
99.10
0.39
8.62
Deep learned
DGR S
77.12
1.64
33.10
96.90
0.34
21.70
2.29
PoinDDCC
98.92
0.38
8.35
97.84
0.33
20.32
0.45
VBReg 19
98.92
0.45
8.41
98.02
0.32
20.91
0.24
Ours
99.56
0.34
7.85
98.20
0.32
20.73
0.54
compare the performance different methods under extreme outlier ratios, i.e., when the outlier ratio increases from 91% 99%. The second row of Fig. shows that with outlier ratios as high as 99%, our method continues to perform well, consistently producing lower transformation errors than other methods.
Robustness to noise. We further evaluate robustness against Gaussian noise with different variances. As the noise standard deviation increases from 0.01 to 0.1, the geometric structure of Bunny model is completely destroyed (39] (refer to Appendix [A.5). Fig. shows the comparison results
increases from 0.01 to 0.09. When the noise variance reaches 0.03, the translation errors of geometric-only method MAC [5] significantly increase. Both FGR [46] and RANSAC [22 show large rotation errors when increases to 0.05. In contrast, method achieves the lowest rotation and translation errors under high noise, demonstrating its robustness to noise.
Efficiency and accuracy. We increase the number of correspondences N from 250 to 5000 to compare efficiency and accuracy. We set the noise standard deviation
0.01 and the outlier ratio to 50%. The comparison results are shown in Fig- 4 a)). As the number of correspondences increases, running time of GORE 4 and TEASER++ when N, grows to 2500, the running time of GORE is about 10a times longer than that of our method. Our method solves the fg-minimization problems with explicit solutions, significantly enhancing efficiency through parallel matrix computations and GPU execution. The curves of FGR, RANSAC, MAC, and our method in Fig. 4 a) are flat and difficult to visually distinguish, indicating the efficiency of these methods. However,
39 increases significantly. Notably,
shown in Appendix A.6, our method outperforms FGR [46], RANSAC [12. and MAC 45 in registration accuracy. Therefore, our inlier identification algorithm via Lu- fu-minimization is efficient while maintaining high accuracy.
Effectiveness of the two-stage decoupling strategy. We evaluate the effectiveness the two- stage decoupling strategy (TDS) by formulating the fg-minimization problem directly on the Bunny data instead of local sets and estimating the final rotation and translation. Specifically, we set
N. 100 and 0.01. As shown in Fig, b), we compare the TDS with optimization-

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(a) Input
(b) MAC
(c) SC2-PCR
(d) Ours
(e) Ground-truth
Figure 5: Qualitative comparisons with other methods. Qualitative comparisons on the 3DMatch
(the first row) and 3DLoMatch (the second row) datasets.
Table 2: Comparisons results on 3DMatch using FPFH, FCGF, and 3DSmoothNet descriptors.
FPFH RE(").
FCGF RE(")
3DSmoothNet RE(")
Time(s)
RR(%))
TEtem)]
RR
TE(em)
RR(%)
TE(mm)
Traditional
FGR AA
40.91
4.96
10.25
78.93
2.90
8.41
73.26
2.51
7.45
0.89
RANSAC
66.10
11.03
91.44
2.86
TEASER+ 79
75.48
2.48
7.31
85.71
2.73
8.66
92.05
2.23
6.62
0.03
SC--PCRI
83.90
2.12
6.69
93.16
2.06
6.53
94.82
1.76
5.98
0.12
MAC 45
2.11
6.80
93.72
2.04
6.51
94.57
2.21
6.52
5.51
TR-DE 1
91.37
2.71
7.62
TEAR 18]
94.52
2.06
6.55
Deep learnd
DGR 8
32.84
2.45
7.53
88.85
2.28
7.02
1.53
72.95
2.18
6.75
VBReg
19
82.57
2.14
6.77
93.53
2.04
6.49
37.09
6.15
15.65
0.20
Ours
83.92
2.12
6.64
93.28
2.04
6.48
95.07
1.75
5.97
0.36
39 46 at an outlier ratio of 90%. Our TDS achieves the highest registration accuracy, demonstrating its inlier identification capability. Additional competitive results as the outlier ratio increases from 0% to 90% are provided Appendix
methods 4 12
Evaluation on Outdoor Seenes
To evaluate our algorithm on real outdoor scenes, we conduct experiments on the KITTI dataset 13
The comparison results with state-of-the-arr traditional [5] I2
46 and learning- based [ 8 [9 methods are reported in Table We first use the FPFH 26 descriptor to generate initial correspondences. As shown
18
45
left column of Table 2 our method outperforms traditional and learning-batt based methods on all metrics. For the most important criterion of registration recall (RR), our method improves by about 2% over the nearest competitor MAC [45]. Following [6]. the average RE and TE only caleulated registration recall more likely to have larger average errors. Nonetheless, our method still achieves the best results RE and TE. Besides, report the comparison results with the FCGF [ descriptor in the right column of Table 2 Our method achieves the highest RR and the lowest RE due to its effective inlier identification algorithm. The superior performance demonstrates the ability of method to align sparse and non-uniformly distributed data in outdoor scenes. In addition to its high accuracy, our method also achieves comparable efficiency, making it highly competitive for practical applications. The visualizations of registration results on KITTI are provided in Appendix A.12
successfully registered pairs, which makes methods with high
4.4 Evaluation on Indoor Scenes
We further conduct experiments on the 3DMatch 44 and 3DLoMatch [17] datasets to evaluate the performance in real indoor scenes. The comparison results are reported in Table 2 and Table
Combined with FPFH, FCGF, and 3DSmoothNet descriptors. As shown in the left column of Table 2 compared to both traditional and learning-based methods, our method achieves the highest RR with the handcrafted FPFH 26 descriptor. The middle column of Table repports the comparison results with the learned FCGF [9] descriptor. Our method achieves the lowest RE and TE. Compared to SC2_ (6), our method improves the RR, RE, and TE by 0.13%, 0.97%, and 0.77% respectively,

---

True: 45 False: 371
True: 326 False: 90
True: 358 False: 58
True: 385 False: 31
Alignment
TEASET++
SC--PCR
MAC
Ours
Figure 6: Comparison results on output inlier ratio. We compare the predicted inlier counts of correct and incorrect correspondences in 3DLoMatch [17]. The first column provides the ground truth alignment, which shows that overlap is very limited. The significantly larger inlier ratio can be observed from the incorrect (red lines) and correct (green lines) correspondences.
which benefit from our fn-minimization formulation for inlier identification. Since TR-DE S and TEAR [8] have not made their code or results for FPFH and FCGF publicly available, their results are excluded in the left and middle columns of Table 2
Following [8]. we also compare the registration accuracy using the learned 3DSmoothNet [14] to extract features. The results the right column of Table 2 show that method outperforms all other methods across all evaluation metrics, demonstrating the robustness of our method to different local descriptors. We show the results of qualitative comparisons in Fig. 5 and Appendix A.12 Methods such as MAC may fail in scenes with ambiguous features or limited overlap. Our algorithm still achieves satisfactory alignment and is close to the ground truth.
Robust lower overlap ratios. Fur- thermore, we report results for more challenging dataset: 3DLoMatch [17 (overlap rate < 30%). Following [5] 19,, we use the Predator [[7 descrip- tor to generate the initial correspon- dences. We compare the registra- tion recall (RR) under different num- bers of correspondences. As shown in Table 3 the proposed method im- proves the average RR by 7% over TR-DE [5] demonstrating the effec- tiveness our method in dealing with
Table 3: Registration rate the 3DLoMatch dataset
with different number of correspondences.
5000
2500
1000
500
250
Predator
FGR 46
36.4
38.2
39.7
39.6
38.0
RANSAC
12
62.3
62.8
62.4
61.5
58.2
TEASER++ 39
62.9
62.6
61.9
59.0
56.7
SC²-PCR 6
68.9
68.4
68.7
67.1
64.9
MAC 45
69.4
69.3
68.4
67.7
64.6
TR-DE 5
64.0
64.8
61.7
58.8
56.5
PointDSC
68.1
66.5
63.4
60.5
VBReg 19
69.9
69.8
68.7
66.4
63.0
Ours
69.9
69.9
69.2
67.7
65.0
low-overlap scenarios. In Fig. 6 compare the output inlier ratio with traditional methods
6
39
45
in the low overlap scenario. Our method is more effective with more correct predicted inliers.
5 Conclusion
In this paper, we propose robust inlier identification algorithm by reformulating the conventional registration problem as alignment error fo-minimization problem. For each local set, resolve the fo-minimization problem using a designed two-stage decoupling strategy. First, the alignment error decoupled rotation fitting error and a translation fitting error, formulating two decoupled 6o-minimization problems. Second, null-space matrices are introduced to decouple the inlier identifi- cation from the estimation of rotation or translation respectively, there by applying robust Bayesian approach to decoupled f fg-minimization problems and solving for fitting errors. Correspondences with the smallest errors are identified inliers to generate a transformation hypothesis for each local set. We experimentally demonstrate that the proposed algorithm is robust to high outlier ratios and noise. Extensive results on the KITTI, 3DMatch, and 3DLoMatch datasets also demonstrate that our method achieves state-of-the-art registration accuracy while being comparable in efficiency in both indoor and outdoor scenes. Limitations and broader impact are discussed Appendix A.10
6 Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant numbers 92167201, 52 888 102, 62373160).
10

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A Appendix
In the appendix, we first provide the detailed construction for local (Sec. A.1 the rigorous definitions of evaluation metrics (Sec. [A.)). then describe the pseudocode for key parts (Sec. A.3)
and the hyper-par peeem-erreeter selection (See. A.44. We further provide additional experimental results (Sec. AA.5 Sec. A.6, Sec. A.7]] and Sec. (A8), ablation studies on parameters (Sec.[A.9) and discuss the limitations (Sec. A. and scalability (Sec. (Sec.(A.IT) of our work. Finally, we show more qualitative results of registration on 3DMatch, 3DthMatch, and KITTI (Sec. A.12) and provide the detailed information for these datasets (Sec. A.13)
A.1 Local set construction
In this section, we provide the detailed construction for local sets. We first construct a global compatibility graph for input correspondences. Specifically, we calculate the Euclidean distance between the correspondence pair (c,, C₁) as follows:
d(c,,e,) |ll: - Pjll -qq -yal-
(18)
where and Pj denote points in the source point cloud and q; and qj are the corresponding points the target point cloud. The first order compatibility score for each pair (ci, C₁) calculated based on the Euclidean distance, follows:
sic..,)= (00)=11-- (1-
(19)
de
where de is the threshold for distance. When the distance difference between two correspondences is less than d, they are considered compatible due to the length consistency of rigid transformations 6 The hard compatibility matrix can be formulated as:
{
1: dic,,, I de
Sh (ca,os)
=
0; dic..,) de
(20)
However, the first order compatibility measure suffers from outliers due to locality and ambiguity [6] Following [45), we calculate the second order compatibility scores [6] as edges in the graph. The second order compatibility score between the correspondence pair (c;, Cj) is computed based on the hard compatibility matrix:
N sh
S Sc,.es)
Sh (c),e))
(c..o))
Sh
(cv.e)))
(21)
where Ne is the number of input correspondences. Based on the compatibility graph, select K reliable correspondences as seeds and construct local sets for each seed. Specifically, following [.
we use first-order compatibility scores compute the leading eigenvectors via the power iteration method (22). These leading feature vectors serve confidence scores for reliable seed selection. For each seed, we explore top-N, neighbors in the second order measure space. Then, within each neighbor set, we recompute the second-order compatibility score and select the top-N2 (N2 < N2) correspondences as the local set for the i-th seed
A.2 Evaluation Metrics
Rotation Error (RE) measures the geometric distance in degrees between the estimated and ground- truth rotation matrices:
trace (R"Rgg)) -
RE arccos
(22)
where R. denotes the estimated rotation matrix and Rg: denotes the ground-truth rotation matrix.
Translation Error (TE) measures the Euclidean distance between the estimated and ground-truth translation vectors:
TE ||t - tt where t denotes the estimated translation vector and tyr denotes the tround-truth translation vector.
(23)
14

---

Registration Recall (RR) measures the fraction of correctly registered point cloud pairs whose RE and TE both below certain thresholds:
3D JDLMMuchh
RE; < 15 A TE, < 30 m]
(24)
RRKITTI
1 N
Σ
[RE < A TE₁- < 60 m]
Following cloud pairs
S 6 39 we compute the mean RE and TE only with the correctly registered point
15

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A.3 Pseudocode for key parts of our algorithm
There are two key parts in our algorithm: Inlier Identification via fg-minimization and two-stage decoupling strategy.
Algorithm 1: Inlier Identification via fg-minimization
Data: Source point cloud (p: E R i I....,,, N} and target point cloud
{q. E R i ,,...,,. M
Result: to minimization problem of alignment error for each local set
Establish correspondences C through feature matching
Calculate the compatibility score (e,,e,) according to Eq. (18)-Eq. (21)
3 Select N1 seed correspondences with high compatibility scores.
For i = 1,2, N1
5 Construct i-th local set, which containing N2 compatible correspondences.
Formulate the &o minimization problem of alignment error for the i-th local set according
Eq. (2)
EndFor
16

---

Algorithm 2: Two-stage decoupling strategy
Kg Data: Given point pairs in the k-th local set Pk: = (Pk. 1 and Q2 {yhu i=1 parameters AR. At, KR. K, Result: Estimated rotation Ri and translation to hypothesis for the k-th local set
% Decoupling the alignment error into rotation fitting error and translation fitting error
Calculate relative positions Px and Q2 for all pairs: Pky = Pe; - Pay and Ohs = qR 4
3 % Decoupling the inlier identification from the rotation estimation
Formulate the fo-minimization problem for the rotation fitting error Os:
0 = arg min |Oelee O
(25)
subject to: O₂ = Q2 - PAR Eg.
% Bayesian-based inlier identification and rotation estimation
Construct θ₁ from the left null-space of Py: O₂P₁ =
Eliminate the components related to R, in the constraints of rotation fitting
Zu-minimization: O,O, O Q O
8 Define Q₂ = ① Qu and II, = O G and formulate the unconstrained problem for rotation fitting error:
Q - θ " - 6,OO)) An 16
min 2
(26)
The explicit solution can be calculated directly:
0 = I O₁ + 22ARI) 'Q.
(27)
10 Solve R using SVD on the identified top KR pairs with the smallest error P7ạ and Q
% Decoupling the inlier identification from the translation estimation
12 Based on the estimated R the fo-minimization problem for the translation fitting error Ö is formulated as:
o arg min | O
(28)
subject to: Ó₂ = Q₁ - PAR - -
13 Define @ satisfying @1 = 0. Eliminate the components associated with translation:
O Ox (Qx
-
(29)
% Bayesian-based inlier identification and translation estimation
15 Define X, = O₂ QT - (PxRt) and II, = O Formulate the unconstrained optimization problem:
[X - Ox ' - 0,0 + |077
min 6
(30)
16 The explicit solution can be calculated directly:
O ((2 I O -
(31)
Solve to based on the identified top-K, correspondences with the smallest emror Pz, and Qz,
A.4 Hyper-parameter selection
The K₁ and K2 are set to and 20 all experiments. For other hyper-parameters (A X Kn and K2), employ grid search strategy with criterion of maximizing inliers. For given set of
17

---

parameters (Ap. KR), the optimization criterion for R₃ is shown as:
max R
[RR]
(32)
subject to: 4k.
Pk, = Eksy VA E In,
where I is the index set of the inlier correspondence pairs and the operation denotes the cardinality of the set. Similarly, the selection criterion for the translation vector tę established as follows:
max [e
(33)
subject to: 9k.
RiP:; - tt = Eke Vi 1I.
where Ug is the index of inlier correspondences.
A.5 Impact noise standard deviation
In this section, visually illustrate the impact of noise standard deviation the point cloud. As shown in Fig. 7 compared with (a) the clean Bunny model, when the noise standard deviation is increased 0.01, the geometric structure of the model in (b) remains mostly recognizable. Therefore, 0.01 is often used as the noise standard. However, as the noise standard deviation increases up to 0.1, the geometric structure of the Bunny is severely degraded, going beyond the noise levels typically encountered in robotics and computer vision applications.
(a) Bunny dataset
(b) Bunny dataset 0.01
(c) Bunny dataset 0.1
Figure The impact of Gaussian noise changes on the scanning model. Bunny point cloud scaled inside unit cube (0, 1]3 and corrupted by different levels of noise and outliers, all viewed from the same perspective angle. (a) Clean Bunny model point cloud. (b) Bunny dataset, generated from (a) 0.01. (c) Bunny dataset, generated
by adding isotropic Gaussian noise with standard deviation from (a) by adding isotropic Gaussian noise with a 0.1.
A.6 Efficiency and accuracy.
In this section, we report the inference time, rotation error, and translation error by increasing the corresponding number N. from 250 5000. The curves of FGR, RANSAC, MAC, and our method are flat and difficult to distinguish visually, demonstrating their efficiency. In addition, when there are fewer inputs, the influence outliers more obvious. The registration accuracy of FGR and RANSAC decreases significantly the number of points decreases. The rotation and translation errors of our method are less affected by the number of correspondences. Compared with other methods, our method achieves the most accurate and fast registration for each number of input correspondences.
A.7 Effectiveness of the two-stage decoupling strategy.
We evaluate the effectiveness our proposed two-stage decoupling strategy (TDS) by formulating the alignment етrоr & fg-minimization problem directly on the Bunny data instead of local sets. The rotation and translation are estimated without hypotheses. We provide comparison with other optimization-based methods 46 12
39 as the outlier ratio increases from 0% to 90%. Our
TDS consistently achieves the highest registration accuracy, demonstrating its inlier identification capability.
18

---

RANSAC
GORE
-X-TEASER+
MAC
Ours
-W- -
MAC
Ours
20
Rotation (degroes)
Translation Error (cm)
15
2
10
250
500
1000
2500
5000
250
500
1000
2500
5000
Number of correspondences
Number of correspondences
RANSAC
GORE
+ TEASER++
MAC
Ours
10000
Time (4)
0.0001
250
500
1000
2500
5000
Number of correspondences
Figure 8: Comparison results with respect to the number of correspondences.
RANSAC TEASER+
Rotation Error (degrees)
1000
12
100
Translation Emar
1
10
20
30
40
50
60
70
80
90
Outlier Ratio (%)
Outlier Ratio (%6)
Figure 9: Comparison results of our two-stage decoupling strategy with optimization-
methods. We compare the rotation error and translation error of proposed two-stage decoupling strategy (TDS) with optimization-based methods 46 12
A.8 Additional comparisons.
We also provide a comparison with learning-basee registration method EGST [33 we re-evaluate our method under the same dataset settings and metrics EGST. The comparison results on KITTI and 3DMatch are shown in beloo. The results of EGST reported in the table follow its published paper. Our method shows better performance in rotation error and comparable results in translation error.
Table 4: Comparison results with EGST.
Method
KITTI
3DMatch
Eror(()
Error(t)
Error(R)
Errr(())
EGST
0.0168
0.0018
0.2086
0.0087
Ours
0.0059
0.0078
0.0305
0.0059
19

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A.9 Sensitivity to parameters.
We conduct ablation studies on the KITTI dataset to evaluate the sensitivity of our algorithm various parameters. Firstly, we ablate the number of local sets N1 and correspondences N₂ in each local set. As shown in tables below, our method insensitive to N₁ and N2, achieving high registration rates (RR) and low errors (RE and TE). Then, we evaluate the impact of rotation estimation threshold K'z and translation estimation threshold K₁. shown in Fig. [0] the curves of registration metries (RR, RE and ТЕ) remain stable when KR and K, increase, indicating the insensitivity of our method to these parameters.
Table Ablation of the number of local sets.
RR(%)T f
REC*)
TE(cm)4
20
98.02
0.33
20.68
25
98.02
0.32
20.62
30
98.20
0.32
20.73
35
98.02
0.33
20.60
Table 6: Ablation of the number of correspondences in each local set.
RR(%)T
RE( X
TE(em).
5
98.02
0.33
20.72
10
98.02
0.33
20.69
15
98.02
0.33
20.68
20
98.20
0.32
20.73
-a- RR(%)
RE(")
TE(cm)
- RR(%)
RE(")
TE(cm)
100
10
1
Kp
(a)
(b)
Figure 10:
Ablation of inlier thresholds KR and K₃.
The results demonstrate that our method is parameter insensitive, making it reliable in practical implementations.
A.10 Limitations and broader impact.
We propose fo-norm based method to solve the point cloud registration problem. The method is robust to high outlier ratios and noise, and effective for different numbers of correspondences. introduces novel perspective to achieve accurate point cloud registration in practical applications. Our algorithm most likely to be used in quality inspection and autonomous driving. It can provide fast and accurate alignment between workpieces and templates, well enhance localization and scene perception for autonomous vehicles. Furthermore, we wish to test the effectiveness of our method in other areas involving registration tasks, including multimodal medical image registration. One possible situation is a quality inspection scenario, where our algorithm may fail when dealing with large workpiece surfaces without obvious features. Future research will focus on enhancing the robustness of our algorithm to featureless data.
A.11 Scalability of our algorithm.
Exploring the scalability of our algorithm and suitability for real-time applications is important for practical deployment. Existing algorithms struggle to achieve both fast speed and high accuracy. Our experiments demonstrate that our algorithm not only achieves high accuracy and robustness but also remains competitive in terms of efficiency, highlighting its potential for real-time applications. The
20

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speed of our method can be further improved through techniques such as parallel computing and C++ implementation. Notably, the two-stage decoupling strategy (TDS) consumes most of the running time (95% of the total), and thus, it could particularly benefit from parallelization. In the first stage of TDDS, we compute the relative positions for point pairs. In the second stage, the computation of null-space matrices also requires substantial processing time. Therefore, these two components are the primary targets for acceleration. Regarding scalability, the proposed two-stage decoupling strategy is a crucial step for inlier identification and can be flexibly combined with other methods to improve accuracy.
A.12 Qualitative results.
We show qualitative results on 3DMatch 44 and 3DLoMatch 17 in Fig. [2] The yellow and blue point clouds represent the source and target point clouds, respectively. The first column represents input point clouds and the second column represents the aligned point clouds transformed with the ground-truth transformations. Compared to other methods [6] 45]. our approach achieves better alignment results. We also provide registration results on the KITTI [13] dataset Fig. T3 The input source and target point clouds are in different poses, and the point clouds transformed using estimated transformations are successfully registered.
The visualization of failure cases is provided in Fig. 11 We observe that when there are repeated patterns (e.g., similar chairs appearing in different locations) or textureless structures (e.g., walls, floors), failure cases may occur due to the feature matching ambiguity. These remain challenging problems point cloud registration and have not yet been effectively addressed. Potential solutions include improving feature extraction applying point cloud completion based on the scene context.
(a) Input
(b) Ground Truth
(c) Ours
(a) Input
(b) Ground Truth
(c) Ours
Figure 11: Failure cases on the 3DMatch dataset.
A.13 Datasets.
Stanford Bunny The Bunny model from the Stanford 3D Scanning Repository [10] is scanned with a Cyberware 3030 MS scanner, with licensing restrictions against commercial Each takes the form of a range image, described in the local coordinate system of the scanner. These range images merged using modified ICP algorithm 30
Odometry KITTI KITTI [15] published under the NonCommercial -iireikkkk 3.0 License. contains 11 sequences scanned by Velodyne HDL-64 3D laser scanner in outdoor driving scenarios. Following 65, we use sequences 8-10 for testing.
3DMatch and 3DLoMated 3DMatch 47 comprises 62 scenes from SUN3D 36 7-Scenes 28 RGB-D Scenes v.2 (21). mnalysis-b--Sytheee
[3I), BundleFusion [[]. and Halbel al. (Table. 7. These scenes are captured from diverse indoor environments using different sensors
like the Microsoft Kinect and Intel Realsense, and are processed into point cloud fragments by
fusing 50 consecutive depth frames using TSDF volumetric fusion [10]. The dataset contains 46 scenes for training, scenes for validation and scenes for testing. The original 3DMatch 44
only considers point cloud pairs with > 30% overlap. In addition to this benchmark (3DMatch),
21

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(a) Input
(b) Ground-truth
(c) MAC
(d) SC²-PCR
(e) Ours
3DMatch
3DMatch
3DMatch
3DLoMatch
3DLoMatch
3DLoMatch
Figure 12: Qualitative registration results on the 3DMatch and 3DLoMatch datasets.
we follow 17 to include point cloud pairs with overlaps between 10% and 30% to form another benchmark (3DLoMatch).
Table 7: Raw data used in the 3DMatch dataset and their licenses.
Datasets
License
SUN3D (36
CC BY-NC-SA 4.0
7-Scenes
28
Non-commercial use only
RGB-D Scenes
21
(License not stated)
Shalysis-by-Synthee
31
CC BY-NC-SA 4.0
BundleFusion
CC 4.0
Halbel et al.
[6
CC-NNN-- 4.0
22

---

Input
Ours
Input
Ours
(a) With the FCGF descriptor
(b) With the FPFH descriptor
Figure 13: Visualizations of registration results on the KITTI dataset.
23

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Reviews:
[{"review_id": "kPvnoLPFnd", "reviewer": "Reviewer_9KUM", "summary": "The paper presents an inlier identification algorithm for point cloud registration by reformulating the conventional registration problem as an alignment error ℓ0-minimization problem. This approach focuses on enhancing the accuracy of point cloud registration under conditions with high outlier ratios and noise. The proposed method involves a novel two-stage decoupling strategy that first separates the alignment error into rotation and translation fitting errors, and then employs null-space matrices to decouple inlier identification from the estimation of these errors. Bayesian theory is applied to solve the decoupled ℓ0-minimization problems, identifying correspondences with the smallest errors as inliers. Extensive experiments on the KITTI, 3DMatch, and 3DLoMatch datasets demonstrate that the proposed algorithm is effective in various scenarios, both indoor and outdoor, compared to traditional and learning-based methods.", "questions": "Q1. Runtime Performance. Can this method achieve real-time performance? What is the exact running time for this method?", "limitations": "L1. Failure Cases. It could be helpful to understand the method better if the author can provide some failure cases analysis.\n\nL2. Scalability Considerations. Exploring the scalability of the algorithm and its suitability for real-time applications would enhance its practical value. Providing insights into potential optimizations for faster execution could also be beneficial. Can the authors comment on limitations in this regard?", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. Registration as Alignment with Decoupling. The paper introduces a novel approach to point cloud registration by reformulating the problem as an alignment error ℓ0-minimization problem and applying a two-stage decoupling strategy, offering a fresh perspective on tackling the issue of outliers in point cloud registration. \n\nS2. Method Details and Experimentation. The proposed algorithm is meticulously described and tested across multiple datasets, including KITTI, 3DMatch, and 3DLoMatch. The comprehensive experimental results demonstrate the robustness and effectiveness of the algorithm under various challenging conditions, including high outlier ratios and noise.", "weaknesses": "W1. Efficiency Considerations Unclear. While the paper demonstrates the robustness and accuracy of the proposed method, it does not extensively address the computational complexity and efficiency of the algorithm quantitatively. E.g. in Fig.7 and Fig.4, it is hard to distinguish the flat curves, it would be better if the author can show the running time as numbers in the paper for better comparison or change the axis scaling.\n\nW2. Parameter Ablation Missing. The sensitivity of the algorithm to various parameters, such as the number of local sets and the thresholds used for inlier identification is not thoroughly investigated. Understanding the impact of these parameters on the performance could provide valuable insights for practical implementation. \n\nW3. Questionable Performance on Small Scale Data. This paper conducts experiments on large scale scene-level datasets, which contains somehow distinctive features, however, it might fail when handling some small-scale object-level scens which may contain less distinctive feature, or some ambiguous scene. Can it be anticipated how the model would work in these cases / is the experimental evidence for it?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer 9KUM,\n\nThanks for your insightful comments. We have carefully responded to every comment raised by the reviewer and will make corresponding revisions to the manuscript.\n\n__W1-Efficiency Considerations__\n\nAs the reviewer suggests, we provide the figure that changes the axis scaling (Fig. 1 of the rebuttal PDF). As the number of correspondences increases, GORE's running time rises significantly while ours increases slightly. While our inference speed is comparable to and slightly slower than RANSAC and MAC, our algorithm improves registration accuracy by 90% and 80% over these methods (Fig. 7 of the submitted manuscript). The results demonstrate that our method achieves higher accuracy while being competitive in time efficiency. We will update this figure as the efficiency subfigure in both Fig. 4 and Fig. 7.\n\n**W2-Parameter Ablation**\n\nWe conduct ablation studies on the KITTI dataset with the FCGF descriptor to evaluate the sensitivity of our algorithm to various parameters.\n\nFirstly, we ablate the number of local sets $N_1$ and the number of correspondences $N_2$ in each local set. As shown in tables below, our method is insensitive to $N_1$ and $N_2$, consistently achieving high registration rates (RR) and low errors (RE and TE).\n\n#### Table 1. Ablation on the number of local sets\n|$N_1$|RR(%)|RE(°)|TE(cm)\n|-|-|-|-\n|20|98.02|0.33|20.68\n|25|98.02|0.32|20.62\n|30|98.20|0.32|20.73\n|35|98.02|0.33|20.60\n|\n\n#### Table 2. Ablation on the number of correspondences\n|$N_2$|RR(%)|RE(°)|TE(cm)\n|-|-|-|-\n|5|98.02|0.33|20.72\n|10|98.02|0.33|20.69\n|15|98.02|0.33|20.68\n|20|98.20|0.32|20.73\n|\n\nThen, we evaluate the impact of the rotation estimation threshold $K_R$ and translation estimation threshold $K_t$. As shown in Fig. 3 of the rebuttal PDF, the curves of registration metrics (RR, RE, and TE) remain stable when $K_R$ and $K_t$ increase, indicating the insensitivity of our method to these parameters.\n\nIn summary, our algorithm is parameter-insensitive and practical in real-world scenes. We will discuss these results in the revised manuscript.\n\n**W3-Performance on Small Scale Data** \n\nThe experiments in Section 4.2 of the manuscript can demonstrate our performance on small-scale object-level scenes with less distinctive features. These experiments are conducted on the object-level Bunny dataset with varying noise, outlier ratios, and number of correspondences. Under high noise or high outlier ratios, the geometric features of Bunny data are corrupted, consistent with challenges in scenes with less distinctive features. As shown in Figures 2-4 of the submitted manuscript, our algorithm achieves accuracy with strong robustness to high levels of noise and outlier ratios. Therefore, our method can perform effectively even when features are not obvious or scenes are ambiguous.\n\n**Q1-Runtime Performance** \n\nTo address this concern, we compare the running time of our method with state-of-the-art methods on two real-world datasets. The table below provides the comparison of both accuracy and efficiency on KITTI and 3DMatch. Our method achieves average running times of 0.54 seconds on KITTI and 0.36 seconds on 3DMatch. The results demonstrate that our method achieves high registration accuracy (RR, RE, and TE) while being competitive in time efficiency. Besides, results on the Bunny dataset show that as the number of correspondences increases, the speed of our method remains relatively stable (Fig. 1 of the rebuttal PDF).\n\nExperiments demonstrate that existing methods struggle to achieve real-time performance. However, our method achieves a balance between high accuracy, robustness, and comparable efficiency, making it highly competitive in real-time applications. Furthermore, the speed of our algorithm can be improved through some engineering techniques such as parallel computing. We will discuss these results in the Experiment section.\n\n#### Table 3. Comparison on KITTI with the FPFH descriptor and 3DMatch with the 3DSmoothNet descriptor\n|Method｜|RR(%)|RE(°)|TE(cm)|Time(s)｜|RR(%)|RE(°)|TE(cm)|Time(s)\n|-|-|-|-|-|-|-|-|-\n|||**KITTI**|**Dataset**|||**3DMatch**|**Dataset**\n|FGR|5.23|0.86|43.84|3.88|73.26|2.51|7.45|0.89\n|RANSAC|74.41|1.55|30.20|5.43|92.30|2.59|7.91|2.86\n|TEASER++|91.17|1.03|17.98|0.07|92.05|2.23|6.62|0.03 \n|SC²-PCR|99.46|0.35|7.87|0.31|94.82|1.76|5.98|0.12\n|MAC|97.66|0.41|8.61|3.29 |94.57|2.21|6.52|5.54\n|DGR|77.12|1.64|33.10|2.29|90.57|2.35|7.24|1.53\n|PointDSC|98.92|0.38|8.35|0.45|93.65|2.17|6.75|0.10\n|VBReg|98.92|0.45|8.41|0.24|37.09|6.15|15.65|0.20\n|**Ours**|99.56|0.34|7.85|0.54|95.07|1.75|5.97|0.36\n|\n\n__L1-Failure Cases__ \n\nThe visualization of failure cases is provided in Fig. 4 of the rebuttal PDF. We observe that when there are repeated patterns (e.g., similar chairs appearing in different locations) or textureless structures (e.g., walls, floors), failure cases may occur due to the feature matching ambiguity. These remain challenging problems in point cloud registration and have not yet been effectively addressed. Potential solutions include improving feature extraction or applying point cloud completion based on the scene context. We will discuss these failure cases in the revised manuscript.\n\n__L2-Scalability Considerations__\n\nExploring the scalability of our algorithm and its suitability for real-time applications is important for practical deployment. Existing algorithms struggle to achieve both fast speed and high accuracy. Experiments show that our algorithm can achieve high accuracy and robustness while being competitive in efficiency, which highlights its potential in real-time applications. The speed of our method can be further improved through some techniques such as parallel computing and C++ implementation. For scalability, the proposed two-stage decoupling strategy is a key step for inlier identification and can be flexibly combined with other methods to improve accuracy. We will discuss potential optimizations and scalability of our algorithm as suggested.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gl2WUQFyce", "reviewer": "Reviewer_Z5SY", "summary": "This study presents a framework for robust inlier identification in multi-model fitting tasks, which is a significant problem in computer vision. The proposed method and its evaluation show promising results.", "questions": "How sensitive is your method to the choice of parameters in the initial inlier identification and the fine-tuning stages? It would be helpful to see an analysis or discussion on parameter sensitivity and its impact on performance.", "limitations": "-Discuss the scope within which the proposed method is expected to perform well. Are there specific types of data or noise levels where the method might not be as effective? Providing this context will help readers understand the boundaries of the method's applicability.\n-The current comparative analysis in the paper is somewhat limited and does not include enough recent and advanced baseline methods. For a more thorough evaluation, it is recommended to compare the proposed method with a wider array of baseline algorithms. Specifically, the inclusion of EGST (EGST: Enhanced geometric structure transformer for point cloud registration), EMFPCR (Evolutionary multiform optimization with two-stage bidirectional knowledge transfer strategy for point cloud registration) and EMTR-SSC (Evolutionary Multitasking with Solution Space Cutting for Point Cloud Registration) would provide a more comprehensive benchmark.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Originality:\n-Novel Framework: The paper introduces a new framework for robust inlier identification in multi-model fitting tasks, which addresses a significant problem in computer vision. This two-stage decoupling strategy (TDS) is a unique approach that separates the initial inlier identification and fine-tuning phases, enhancing robustness and accuracy.\n-Innovative Approach: The combination of graph model construction and optimization algorithms represents a creative integration of existing techniques, providing a fresh perspective on solving the inlier identification problem. This method stands out by effectively reducing the impact of noise and outliers, which is a persistent challenge in this domain.\nQuality\n-Thorough Evaluation: The paper conducts comprehensive experiments on multiple public datasets, including KITTI, 3DMatch, and 3DLoMatch, ensuring the proposed method's performance is rigorously tested across diverse scenarios.\n-Multiple Metrics: The use of various evaluation metrics such as Rotation Error (RE), Translation Error (TE), and Registration Rate (RR) provides a well-rounded assessment of the method’s effectiveness and robustness.\nClarity\n-Structured Presentation: The paper is well-organized with clear sections delineating the problem, methodology, experimental setup, and results. Each section builds logically on the previous one, facilitating understanding.\n-Detailed Results: The presentation of experimental results is clear, with well-labeled tables and figures that effectively communicate the performance of the proposed method compared to baseline techniques.\nSignificance\n-Addressing a Key Challenge: The problem of robust inlier identification in the presence of high noise and outliers is crucial for various computer vision tasks. The proposed method’s ability to handle such challenging conditions has significant implications for improving the reliability of multi-model fitting applications.\n-Broader Applicability: The methodology's potential for application across different domains (e.g., 3D reconstruction, object recognition, and scene understanding) underscores its broad significance. By improving inlier identification, the paper contributes to advancing the state-of-the-art in these areas.", "weaknesses": "-The abstract should succinctly summarize the problem, methodology, and key results. Currently, it lacks clarity in explaining the significance of the proposed method and its comparative advantage over existing techniques.\n-While the paper outlines the methodology, some parts lack detailed explanations. Specifically, the algorithmic steps should be elaborated with clear justifications for each step. Including pseudo-code or flowcharts could improve understanding.\n-The paper should clearly state any assumptions made during the development of the methodology and discuss potential limitations. This transparency helps in assessing the validity and applicability of the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer Z5SY,\n\nThanks for your precious time and efforts in reviewing our work. We have thoroughly addressed the concerns raised by the reviewer and will revise the paper accordingly.\n\n__W1-Clarify significance and advantage in abstract__\n\nAs suggested by the reviewer, we will revise the abstract to clarify the significance and advantages of our method. In point cloud registration, correspondences are prone to outliers due to inaccurate feature matching, which significantly reduces registration accuracy and highlights the need for precise inlier identification. Despite many efforts, most existing methods still struggle with accurately identifying inliers. Our method achieves high accuracy even in the presence of high noise and outlier ratios while being competitive in time efficiency, making it practical for real-world applications.\n\n__W2-Pseudo-code for some parts__\n\nThere are two key parts in our algorithm: inlier identification via $\\ell_0$-minimization and two-stage decoupling strategy. In Appendix A.3 of the submitted manuscript, we have provided the pseudo-code for two-stage decoupling strategy. We will add the pseudo-code for inlier identification via $\\ell_0$-minimization, as shown in Fig. 2 of the rebuttal PDF. In addition, we will elaborate the steps of our algorithm in more detail in Method section, making it easier for readers to understand.\n\n__W3-Assumptions of the algorithm__\n\nThere are three assumptions in our algorithm:\n\n1. Only inliers can be fitted by the same rigid transformation while outliers are transformed by different rigid transformations. \n\n2. The noise is assumed to follow the Gaussian distribution. This allows us to apply Bayes Theorem in the two-stage decoupling strategy to solve fitting errors. \n\n3. The transformation that can be fitted by the maximum number of inliers is considered the best estimation. It is used to select the final transformation from a series of transformation hypotheses estimated from local sets. \n\nThese are general assumptions in point cloud registration and widely used in existing methods. While they may not perfectly reflect real-world conditions and can lead to some reduction in accuracy, they are generally sufficient for most applications. We will clarify these assumptions and discuss their potential limitations in the revised manuscript.\n\n**Q1-Sensitivity to parameters**\n\nWe conduct ablation studies on the KITTI dataset with the FCGF descriptor to evaluate the sensitivity of our algorithm to various parameters.\n\nFirstly, we ablate the number of local sets $N_1$ and the number of correspondences $N_2$ in each local set. As shown in tables below, our method is insensitive to $N_1$ and $N_2$, achieving high registration rates (RR) and low errors (RE and TE).\n\n#### Table 1. Ablation on the number of local sets\n|$N_1$|RR(%)|RE(°)|TE(cm)\n|-|-|-|-\n|20|98.02|0.33|20.68\n|25|98.02|0.32|20.62\n|30|98.20|0.32|20.73\n|35|98.02|0.33|20.60\n|\n\n#### Table 2. Ablation on the number of correspondences\n|$N_2$|RR(%)|RE(°)|TE(cm)\n|-|-|-|-\n|5|98.02|0.33|20.72\n|10|98.02|0.33|20.69\n|15|98.02|0.33|20.68\n|20|98.20|0.32|20.73\n|\n\nThen, we evaluate the impact of rotation estimation threshold $K_R$ and translation estimation threshold $K_t$. As shown in Fig. 3 of the rebuttal PDF, the curves of registration metrics (RR, RE, and TE) remain stable when $K_R$ and $K_t$ increase, indicating the insensitivity of our method to these parameters.\n\nTherefore, our method is parameter-insensitive and can be reliable in practical applications. We will analyze these results in the revised manuscript.\n\n__L1-Scope of method__\n\nTo discuss the scope of our method, we provide the visualization of some failure cases in Fig. 3 of the rebuttal PDF. When there are repeated patterns (e.g., similar chairs appearing in different locations) or textureless structures (e.g., walls, floors), our method may fail due to the feature matching ambiguity. These remain challenging problems in point cloud registration and have not yet been effectively addressed. Potential solutions include improving feature extraction or applying point cloud completion based on the scene context. \n\nRegarding the impact of noise, we have thoroughly discussed this in Section 4.2. Our experiments demonstrate that our algorithm is highly robust to noise, achieving accurate registration even as the noise standard deviation increases to 0.1. We will discuss these results in the revised manuscript.\n\n__L2-Comparison with provided methods__\n\nFor a more comprehensive evaluation, we have thoroughly reviewed the literature provided by the reviewer and compared our method with these works where possible. EGST is a learning-based method, while EMFPCR and EMTR-SSC are optimization-based methods. To ensure a fair comparison with EGST, we re-evaluate our method under the same dataset settings and metrics as EGST. The comparison results on KITTI and 3DMatch are shown in the table below. The results of EGST reported in the table follow its published paper. Our method shows better performance in rotation error and comparable results in translation error.\n\n#### Table 3: Comparison results with EGST\n|Method｜|Error(R)|Error(t)｜|Error(R)|Error(t)\n|-|-|-|-|-\n||**KITTI**|**Dataset**|**3DMatch**|**Dataset**\n|EGST|0.0168|0.0018|0.2086|0.0087\n|Ours|0.0059|0.0078|0.0305|0.0059\n|\n\nDue to the use of different datasets and lack of publicly available codes, direct comparisons with EMFPCR and EMTR-SSC are difficult. We have contacted the authors via email to request the code but have not yet received a response. We will conduct the comparison if we receive the code. Additionally, we will discuss these two methods in the Related Work section to provide further insights. EMFPCR proposes a multiform optimization approach through evolutionary multitasking, while EMTR-SSC introduces a novel evolving registration algorithm via evolutionary multi-task optimization.\n\nThese studies will be discussed and compared in the revised manuscript as suggested.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C7bFzJQEa2", "reviewer": "Reviewer_TzsD", "summary": "The paper addresses the problem of point cloud registration, namely the challenge of outliers that can negatively affect registration quality.  The proposed algorithm first constructs a compatibility graph from which local sets of possible correspondences are extracted. The method further involves decoupling the alignment error into rotation and translation fitting errors and then using null-space matrices to decouple inlier identification from the rotation and translation estimation. The method demonstrates robustness under large outlier ratios and noise, achieving state-of-the-art performance on multiple datasets. \n\nOverall, the paper introduces an innovative approach to a longstanding problem in Computer Vision, and I am willing to raise my score if my comments and questions are addressed.", "questions": "* What is meant by Bayesian Theory? Do you mean Bayes Theorem? Properly introducing the concept to the reader would be beneficial. \n* Line 123: What is K2? This seems to be an undefined variable.", "limitations": "Limitations are discussed in Appendix A.8", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method seems novel and effective. While I am not very familiar with the literature on the topic, the l0 optimization scheme, rotation and translation decoupling, and solving the whole system in an analytical way contain sufficient novelty. The experiments show that the proposed method is more robust to outliers and noise in the point clouds than previously proposed methods.\n\n2. I appreciate the thorough empirical validation of the approach. By studying the method's robustness on simpler datasets and comparing real-world datasets like KITTI, the paper makes a compelling argument for its strength. \n\n3. Overall, the paper is well-motivated and written, with some small errors that can be corrected to increase its readability.", "weaknesses": "1. The proposed method involves complex mathematical formulations and a multi-stage process that seems computationally intensive, either in speed or memory. The paper would benefit from a more thorough discussion of the computational complexity and efficiency of the proposed algorithm, especially in terms of how time complexity affects the practicability of the proposed method in real-world scenarios.\n\n2. The evaluation of the real-world datasets (Tables 1, 2, 3) shows that all analyzed methods fall tightly together, suggesting that point cloud registration performance is saturating. There seem to be multiple methods that perform almost similarly. This would suggest that on real-world data, one does not need such high levels of robustness to noise or outliers as suggested in the paper's motivation. I suggest the paper restate additional advantages of the method (such as time complexity or practicality in real-world applications) in the experiment section by adding time complexity as a metric to these tables.  \n\n3. Figure 1 is quite small. I suggest increasing the size of these figures so that the reader can grasp the concept of the approach more easily.\n\n4. I would exchange Figure 5 for a more representative figure showing the proposed methods' qualitative results. It isn't easy to see the proposed method's advantage qualitatively. Maybe adding a case where other methods fail could highlight the proposed method's strengths. Additionally, it would be better to show the ground truth alignment in the same pose as the different baselines. \n\nMinor points:\n* I suggest mentioning the dataset used in the captions of Figures 2-4.\n* Line 22: overlapped -> overlapping\n* Lines 25-27: This is a hard-to-understand sentence; maybe rewrite it.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer TzsD,\n\nThanks for the positive assessment of our contributions. We have thoroughly addressed the concerns raised by the reviewer and provided further clarification on our study. \n\n__W1-Time complexity and practicality__\n\nThe complex mathematical formulas in the manuscript are intended to detail the derivation of our algorithm. These formulas are mainly used to derive the explicit solutions of optimization problems (in Eqs. (12) and (16)). In practice, we directly calculate explicit solutions for fitting errors without complex derivation. Therefore, our speed and GPU memory are comparable to existing methods. For a clearer comparison, we evaluate accuracy and efficiency on real-world KITTI and 3DMatch datasets, as shown in Table 1 below.\n\n#### Table 1. Comparison on KITTI with the FPFH descriptor and 3DMatch with the 3DSmoothNet descriptor\n|Method｜|RR(%)|RE(°)|TE(cm)|Time(s)｜|RR(%)|RE(°)|TE(cm)|Time(s)\n|-|-|-|-|-|-|-|-|-\n|||**KITTI**|**Dataset**|||**3DMatch**|**Dataset**\n|FGR|5.23|0.86|43.84|3.88|73.26|2.51|7.45|0.89\n|RANSAC|74.41|1.55|30.20|5.43|92.30|2.59|7.91|2.86\n|TEASER++|91.17|1.03|17.98|0.07|92.05|2.23|6.62|0.03 \n|SC²-PCR|99.46|0.35|7.87|0.31|94.82|1.76|5.98|0.12\n|MAC|97.66|0.41|8.61|3.29 |94.57|2.21|6.52|5.54\n|DGR|77.12|1.64|33.10|2.29|90.57|2.35|7.24|1.53\n|PointDSC|98.92|0.38|8.35|0.45|93.65|2.17|6.75|0.10\n|VBReg|98.92|0.45|8.41|0.24|37.09|6.15|15.65|0.20\n|**Ours**|99.56|0.34|7.85|0.54|95.07|1.75|5.97|0.36\n|\n\nOur method achieves the highest registration accuracy (RR, RE, and TE) while being competitive in time efficiency. Accuracy and robustness are crucial to real-world applications. As illustrated in [1], they have always been the most important and widely recognized metrics in point cloud registration. As discussed in [2], while inference speed and GPU memory can be improved through engineering techniques, accuracy and robustness are entirely limited by the algorithm in real-world applications. Our method achieves a balance between high accuracy, robust performance, and comparable efficiency, making it competitive for practical applications.\n\n[1] Bai X, Luo Z, Zhou L, et al. Pointdsc: Robust point cloud registration using deep spatial consistency[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021: 15859-15869.\n\n[2] Shuvo M M H, Islam S K, Cheng J, et al. Efficient acceleration of deep learning inference on resource-constrained edge devices: A review[J]. Proceedings of the IEEE, 2022, 111(1): 42-91.\n\n__W2-Robustness and efficiency in real-world applications__\n\nFor real-world applications, improving the robustness of algorithms to noise and outliers is important and valuable. Experiments in Section 4.2 demonstrate our high robustness and accuracy even under high noise and outlier ratios. Compared to MAC, our algorithm improves accuracy by more than 80% at a noise level of 0.05. These experiments are consistent with real-world challenging scenarios. As discussed in [3], in autonomous driving, point clouds obtained by LiDAR and Radar contain significant noise, requiring strong robustness and accuracy for registration methods. The above experiments and analysis demonstrate our algorithm's practicality in real-world applications.\n\nThe performance saturation in KITTl and 3DMatch datasets can be attributed to distinctive features of those simpler scenes, which makes it easier for existing methods to achieve high registration accuracy. Nevertheless, our algorithm achieves state-of-the-art robustness and accuracy in both challenging and simple scenarios.\n\nAs suggested by the reviewer, we add a comparison of time complexity on KITTI and 3DMatch in Table 1. In response to W1, our algorithm achieves the highest registration accuracy while being comparable in efficiency. Thus, our method is highly competitive in both robustness and efficiency. We will restate these advantages in the revised manuscript.\n\n[3] Li Y, Ma L, Zhong Z, et al. Deep learning for lidar point clouds in autonomous driving: A review[J]. IEEE Transactions on Neural Networks and Learning Systems, 2020, 32(8): 3412-3432.\n\n__W3-Size of Fig. 1__\n\nWe will increase the size of Fig. 1 to enhance readability.\n\n__W4-Qualitative results__\n\nFigure 5 of the submitted manuscript is presented to visualize the effectiveness of inlier identification. The number of green lines (inliers) identified by our method is much more than red lines (outliers). Besides, we have provided qualitative comparisons in Appendix A.9 to show the better alignment of our algorithm.\n\nTo further highlight our method's strengths, we will add a new comparison figure in the main text as suggested. This figure,  shown in Fig. 5 of the rebuttal PDF, qualitatively compares our approach to state-of-the-art methods and ground truth. The first and second rows are comparisons on 3DMatch and 3DLoMatch. Methods such as MAC may fail in scenes with ambiguous features or limited overlap. Ours still achieves satisfactory alignment and is close to the ground truth.\n\n__Minor points 1-3__\n\nWe will mention the Bunny dataset in captions of Figures 2-4, revise overlapped to overlapping, and rewrite the sentence in Lines 25-27 for clearer expression.\n\n__Q1-Bayesian Theory__\n\nThe Bayesian Theory mentioned in our paper refers to using the Bayes Theorem [4] and Maximum A Posteriori (MAP) to solve the fitting error optimization. Taking rotation estimation for example, considering i.i.d. Gaussian distribution noise, we can deduce the posterior distribution of the rotation fitting error according to the Bayes Theorem. Then the unconstrained optimization problems are constructed via MAP for solving fitting errors. For clarity, we will revise Bayesian Theory into Bayes Theorem and explain the concept in more detail.\n\n[4] Jaynes E T. Probability Theory: The Logic of Science[M]. Cambridge University Press, 2003.\n\n__Q2-Definition of K2__\n\nWe will change K2 to N2 defined in Line 122, referring to the correspondence number in each local set.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear Reviewers 9KUM, Z5SY, and TzsD,

Thank you for your constructive comments and positive assessment of the manuscript. We have thoroughly addressed the concerns raised. Additional experiments have been conducted and more detailed analyses have been provided. We will also revise the manuscript carefully to improve its readability. Hope our responses meet your expectations.

For convenience, we have summarized the main responses to each reviewer:

1. Response to Reviewer 9KUM: We have revised the efficiency comparison figure and provided the exact running time comparisons. Experiments have been conducted to show the insensitivity of our method to various parameters. We have analyzed the algorithm's performance on small-scale data. Failure case analysis and algorithm scalability discussion have also been provided.

2. Response to Reviewer Z5SY: The significance and advantages of our approach will be clarified in the abstract. We have provided the pseudo-code and assumptions of our algorithm to enhance understanding. Experiments have been conducted to show the insensitivity of our method to various parameters. We have analyzed the scope of our method and compared it with the provided methods.

3. Response to Reviewer TzsD: We have provided the time complexity comparisons and demonstrated the practicality of our method in real-world applications. A qualitative comparison figure has been provided. We have introduced the concept of Bayesian Theory used in our method. Some inappropriate expressions raised by the reviewer will be revised.

In the rebuttal to each reviewer, we have carefully addressed the specific comments one by one. Five additional figures have been included in the submitted author rebuttal PDF. 

Thanks again for your precious time and valuable comments.

Meta-review: Overall, all the reviewers are interested in the topics, have good opinions towards the proposed method and the experimental results, and have positive rating towards and tend to accept the paper. Most concerns about computational complexity, missing ablation studies, further validation over small scale objects, and missing clarities from the reviewers have been well addressed by the authors with more discussions, clarifications and experimental results.
Decision: Accept (poster)