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SAC-COT（Sample Consensus by Sampling Compatibility Triangles）是一种用于三维点云配准的稳健六自由度（6-DOF）位姿估计算法。其核心思想是通过在图中对兼容性三角形（Compatibility Triangle, COT）进行采样，以此来实现样本一致性（Sample Consensus）估计，并解决三维点云配准问题。该算法主要解决的是从特征对应关系中进行位姿估计的问题，尤其针对那些初始对应集中存在大量异常值的情况。SAC-COT算法主要由以下几个部分组成： 1. 特征对应关系的图模型表示：SAC-COT首先将对应关系集表示为一个图模型，其中的节点代表兼容的特征对应关系。通过分析这些对应关系，可发现图中能够连接这些兼容对应关系的节点。 2. 引导式的三采样方法（Guided three-point sampling approach）：此为SAC-COT算法的核心创新，它利用了一种新颖的对应关系采样表示，即兼容性三角形（COT）。通过对COT进行排序和采样，算法能够在迭代的早期阶段生成正确的假设（hypothesis）。 3. 假设生成与最大化共识：算法将通过那些由三元环形成的COTs进行排序和采样，最终根据产生最大共识的COT来确定SAC-COT的输出假设。 4. 三维点云与配准：三维点云配准是将一个三维点云（源点云）与另一个三维点云（目标点云）对齐的过程。该技术广泛用于计算机视觉、机器人学和其他三维数据处理领域。 5. 特征对应关系与一致性：特征对应关系是指在源点云和目标点云之间找到的匹配特征点对。一致性是指这些特征点对之间保持某种一致性的程度。一致性分数（如c(ps,pt)）通常用来评估对应关系的可信度。 6. 四元组变换矩阵：SAC-COT算法中可能会涉及的是一种四元组变换矩阵，它是一个4×4的刚体变换矩阵，能够表达三维空间中从一个坐标系到另一个坐标系的变换，包含了3×3的旋转矩阵和3×1的平移向量。 7. 算法的鲁棒性与对比：SAC-COT通过实验验证，在多个数据集上的广泛实验以及与现有先进技术的全面比较显示，该算法具有以下两个显著优点：一是能够通过少量迭代实现准确的配准；二是即使在面对高斯噪声、数据降采样、存在空洞、杂物、部分重叠、输入对应关系的尺度变化以及数据模态变化等不利条件下，SAC-COT的性能仍然表现出极高的鲁棒性。 8. 研究支持：文章指出研究受到了中国国家自然科学基金（National Natural Science Foundation of China, NFSC）部分资助。通过对SAC-COT算法的深入分析，可以看出它为三维点云配准领域提供了一种简单有效的方法，能够解决传统方法在面对大量噪声和异常值时难以处理的问题。SAC-COT的提出不仅丰富了三维点云配准的理论，也为实际应用中的稳健配准提供了强有力的工具。

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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1

SAC-COT: Sample Consensus by Sampling

Compatibility Triangles in Graphs for 3-D

Point Cloud Registration

Jiaqi Yang , Zhiqiang Huang, Siwen Quan , Zhaoshuai Qi, and Yanning Zhang , Senior Member, IEEE

Abstract— Six-degree-of-freedom (6-DOF) pose estimation

from feature correspondences remains a popular and robust

approach for 3-D registration. However, heavy outliers that

existed in the initial correspondence set pose a great challenge

to this problem. This article presents a simple yet effective

estimator called SAmple Consensus by sampling COmpatibility

Triangles in graphs (SAC-COT) for robust 6-DOF pose estimation

and 3-D registration. The key novelty is a guided three-point

sampling approach. It is based on a novel correspondence sample

representation, i.e., COmpatibility Triangle (COT). We ﬁrst

model the correspondence set as a graph with nodes connecting

compatible correspondences. Then, by ranking and sampling

COTs formed by ternary loops, we show that correct hypotheses

can be generated in early iteration stage. Finally, the hypothesis

generated by the COT yielding to the maximum consensus is the

output of SAC-COT. Extensive experiments on six data sets and

extensive comparisons with the state-of-the-art estimators con-

ﬁrm that: 1) SAC-COT can achieve accurate registrations with

a few iterations and 2) SAC-COT outperforms all competitors

and is ultrarobust when confronted with Gaussian noise, data

decimation, holes, clutter, partial overlap, varying scales of input

correspondences, and data modality variation.

Index Terms— 3-D point cloud, 3-D registration, feature cor-

respondences, sample consensus.

NOMENCLATURE

Source point cloud.

Target point cloud.

p Generic keypoint in point cloud P.

c(p

, p

) Generic feature correspondence.

s(c

, c

) Compatibility score between c

and c

T 4 × 4 rigid transformation matrix.

R 3 × 3 rotation matrix.

t 3 × 1 translation vector.

Manuscript received November 17, 2020; revised January 26, 2021;

accepted February 3, 2021. This work was supported in part by the National

Natural Science Foundation of China (NFSC) under Grant 62002295, Grant

62006025, and Grant U19B2037; and in part by the Ningbo Natural Sci-

ence Foundation Project under Grant 202003N4058. (Corresponding author:

Siwen Quan.)

Jiaqi Yang, Zhaoshuai Qi, and Yanning Zhang are with the National

Engineering Laboratory for Integrated Aero-Space-Ground-Ocean Big Data

Application Technology, School of Computer Science, Northwestern Poly-

technical University, Xi’an 710129, China (e-mail: jqyang@nwpu.edu.cn;

zhaoshuaiqi1206@163.com; ynzhang@nwpu.edu.cn).

Zhiqiang Huang is with the School of Software, Northwestern Polytechnical

University, Xi’an 710129, China (e-mail: zhiqianghuang@mail.nwpu.edu.cn).

Siwen Quan is with the School of Electronic and Control Engineering,

Chang’an University, Xi’an 710064, China (e-mail: siwenquan@chd.edu.cn).

Color versions of one or more ﬁgures in this article are available at

https://doi.org/10.1109/TGRS.2021.3058552.

Digital Object Identiﬁer 10.1109/TGRS.2021.3058552

I. INTRODUCTION

EGISTRATION of 3-D point clouds is essential in a

number of 3-D vision and has been applied to various

areas, e.g., remote sensing [1]–[3], 3-D reconstruction [4],

3-D objection recognition [5], and odometry [6]. The goal

is to recover the relative six-degree-of-freedom (6-DOF) pose

between the source point cloud and the target point cloud,

which is further employed to align two point clouds.

Establishing point-to-point feature correspondences between

two point clouds remains a popular and robust approach for

3-D registration [7]–[10]. However, this problem is still very

challenging due to heavy outliers contained in the initial

correspondence set generated by matching local geometric

descriptors. Speciﬁcally, three main factors can result in out-

liers.

1) Keypoint Localization Error: The repeatability of most

existing 3-D keypoint detectors is still limited [11], and

thus, the corresponding keypoints may miss each other.

2) Limited Distinctiveness of Loca l Descriptors: The 3-D

local descriptors are devised to represent the geometric

information around each keypoint, while most existing

3-D local descriptors still suffer from limited distinctive-

ness on real-world data sets [12]–[14].

3) Data Nuisances: Raw point clouds usually suffer from

many data nuisances such as noise, data resolution vari-

ation, holes, clutter, and partial overlap, which greatly

improve the difﬁculty of consistent correspondence

calculation.

The problem of estimating a 6-DOF pose from

correspondences with outliers has been studied for decades,

and a number of methods have been proposed, such as random

sample consensus (RANSAC)-based [7], [15]–[17], voting-

based [18], [19], and clustering-based [20] ones. RANSAC

and its variants, due to their strong reliability and high

efﬁciency, are arguably de facto choices in most scenarios [8].

RANSAC methods iteratively sample correspondences to

generate hypotheses and serve the hypothesis yielding the

maximum consensus as the ﬁnal pose result. Clearly, the

sampling process in the RANSAC pipeline is critical for

at least two reasons. First, only correct correspondences

can generate correct hypotheses, and high-quality samples

indicate noiseless hypotheses as well as accurate 6-DOF

poses. Second, the iteration can stop early if with high-quality

samples, indicating quick convergence. Existing sampling

techniques for RANSAC can be categorized into three classes:

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2 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING

three-point-based, two-point-based, and one-point-based.

Three-point-based ones purely leverage keypoint location

information and sample three correspondences per iteration

to generate hypothesis; two-point- and one-point-based

ones, respectively, consider normals and local reference

frames (LRFs) in addition to keypoint locations, and thus,

less correspondences are required to form a sample. Because

three-point-based sampling approaches have a theoretical

time complexity of O(n

), most studies pay more attention

to two-point- and one-point-based methods and develop

guiding strategies for methods in the two categories [1],

[21] to achieve higher efﬁciency. However, because normals

and LRFs are very sensitive to common nuisances such as

noise and occlusion, two-point- and one-point-based methods,

even with guided sampling techniques, are not reliable in

the presence of common nuisances (as will be veriﬁed

in Section IV). At present, it is still very challenging for

6-DOF pose estimators to achieve a good balance in terms of

accuracy, robustness, and efﬁciency.

A. Motivation and Contributions

To achieve accurate, robust, and fast 6-DOF pose estimation

from noisy correspondences, we propose a robust estimator

called SAmple Consensus by sampling COmptibility Triangles

in graphs (SAC-COT). Unlike previous methods, we focus

on a three-point-based sampling technique due to its stability

and develop a robust approach to guide three-point sampling,

making SAC-COT accurate and robust while even being more

efﬁcient than several two-point- and one-point-based methods.

SAC-COT ﬁrst models the correspondence set as a graph with

edges connecting compatible correspondences. Then, guided

three-point sampling is performed in the graph by ranking

Compatibility Triangles (COTs) formed by ternary loops.

Finally, the 6-DOF pose hypothesis generated by the COT

yielding the maximum consensus remains the output of SAC-

COT. Here, the concept “compatibility,” in the context of 3-D

feature correspondences, indicates that two correspondences

are geometrically consistent, which is usually reﬂected by

enforcing robust geometric constraints [8], [22]. Experiments

have been carried out on four data sets with different applica-

tion scenarios, nuisances, and data modalities. Comparisons

with several state-of-the-art estimators further conﬁrm the

outstanding performance of SAC-COT. To summarize, this

article presents two main contributions.

1) We propose a guided approach for three-point corre-

spondence sampling to achieve convincing 6-DOF pose

hypotheses. It is based on a novel correspondence sam-

ple representation, i.e., COT. COT, on the one hand,

inherits the merit of distance constraints of being robust

to common nuisances and, on the other hand, greatly

alleviates their ambiguity problem. By ranking and

sampling COTs, we show that correct hypotheses can

be generated in the early iteration stage. Moreover, the

proposed sampling technique is general, i.e., it can boost

the registration performance of other three-point-based

RANSAC methods.

2) We propose an accurate, robust, and fast 6-DOF

pose estimator, i.e., SAC-COT, for 3-D registration.

SAC-COT manages to generate reasonable hypotheses

with a few iterations due to its smart sampling strategy.

Comparative experiments on six data sets with different

application scenarios, nuisances, and data modalities

demonstrate that SAC-COT is efﬁcient, accurate, and

resilient to Gaussian noise, data decimation, holes, clut-

ter, partial overlap, varying scales of input correspon-

dences, and data modality variation.

B. Novelty Illustration With Respect to Related Methods

To highlight the novelty of our method, we compare our

method against related methods to illustrate its distinctions.

1) Guided Sampling Methods: Critical to guided sam-

pling is the rule of deﬁning/mining good samples.

Although some guided sampling methods have been

proposed, e.g., compatibility-guided sampling consensus

(CG-SAC) [1], our guiding rule is different with them

from at least three aspects: 1) the guiding targets are

different, i.e., SAC-COT guides correspondence triplets

while CG-SAC guides correspondence pairs; 2) the

guiding rule of SAC-COT is based on properties of a

graph, while others simply employ correspondence cues;

and 3) SAC-COT employs a simple distance constraint

to mine good samples, while CG-SAC additionally con-

siders other constraints that are more time-consuming

and could even degrade the robustness of the estimator.

2) Compatibility-Based Methods: Two compatibility-based

methods, i.e., CG-SAC [1] and compatibility features

(CFs) [23], also leverage the compatibility cue of

3-D feature correspondences. The novelty of SAC-COT

against CG-SAC has been previously stated (CG-SAC is

also experimentally compared with SAC-COT). Regard-

ing CF, we note that CF is proposed for 3-D corre-

spondence grouping rather than 6-DOF pose estimation.

In addition, CF still uses a pairwise correspondence

constraint, whereas SAC-COT proposes a triplet one to

form COTs and introduces a guiding rule for COTs to

achieve fast and accurate 6-DOF pose estimation.

3) Geometric-Constraint-Based Methods: It is common to

leverage geometric constraints to assist the 3-D registra-

tion [1], [24]–[26]. Compared with existing methods, the

proposed COT representation is grounded on the corre-

spondence compatibility cue and extracted from a com-

patibility graph rather than greedy spatial search [24],

merely employs a simple distance constraint while deliv-

ering high accuracy, without the need of trying and

combining different geometric constraints, and is a cor-

respondence triplet representation, and SAC-COT ranks

triplets rather than correspondences or correspondence

pairs [1], [26].

C. Article Organization

The rest of this article is structured as follows. Section II

brieﬂy reviews the estimators in the RANSAC family.

Section III introduces the technique details of our SAC-COT

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YANG et al.: SAC-COT FOR 3-D POINT CLOUD REGISTRATION 3

method. Section IV presents the deployed experiments to

validate the effectiveness of our method. Finally, Section V

concludes this article and presents future research directions.

II. R

ELATED WORK

This section ﬁrst gives a brief review on 6-DOF pose estima-

tors, geometrics constraints, and deep learning techniques for

3-D registration. Then, since SAC-COT is a guided sampling-

based estimator, existing guided sampling approaches in the

RANSAC family is recapped.

A. 6-DOF Pose Estimators for 3-D Registration

For the problem of estimating a 6-DOF pose from cor-

respondences with outliers in the context of 3-D regis-

tration, RANSAC [15] and its variants remain the main-

stream solution. The classical RANSAC exhibits two main

limitations when faced with heavy outliers: low time efﬁ-

ciency and limited accuracy. To overcome the two limita-

tions, Rusu et al. [10] presented a sample consensus-based

initial alignment (SAC-IA) method, which samples corre-

spondences spread out on the point cloud and leverages the

Huber penalty for hypothesis evaluation. To achieve more

reliable hypothesis evaluation, Yang et al. [7] proposed a point

cloud distance metric to assess pose hypotheses and intro-

duced the optimal sample consensus (OSAC) method. Both

SAC-IA and OSAC are three-point-based sampling methods

that exhibit a theoretical computational complexity of O(n

Some methods that sample fewer correspondences per itera-

tion have also been proposed. For instance, two-point-based

sample consensus with global constraint (2SAC-GC) [21] and

compatibility-guided sample consensus (CG-SAC) [1] addi-

tionally consider the normal information of keypoints during

the sampling process; globally constrained one-point-based

sample consensus (GC1SAC) [17] and one-point RANSAC

(1P-RANSAC) [16] additionally employ the LRF cue and only

sample one correspondence per iteration. Unfortunately, these

RANSAC methods still fail to achieve a good balance in terms

of accuracy, speed, and robustness to common nuisances (as

will be veriﬁed in Section IV).

In addition to RANSAC methods, Guo et al. [20] proposed a

clustering-based method that ﬁrst generates pose hypotheses in

the rotation and translation spaces and then locates the cluster

centers in both spaces; Tombari and Di Stefano [18] and Buch

et al. [19] proposed two voting-based methods that serve pose

hypotheses as voters and perform voting in a Hough space. All

these methods are dependent on the normals or LRFs, which

have been demonstrated to be instable in the presence of noise,

partial overlap, and occlusions [13], [27], [28].

B. Geometric Constraints for 3-D Registration

Point-, plane-, and correspondence-level geometric con-

straints have been proposed to assist 3-D registration. For

point-level constraints, Aiger et al. [29] enforced coplanar

constraints to four-point sets in point clouds to achieve reg-

istration; Drost et al. [30] presented a point pair voting-

based method for 3-D registration, where distance and angle

constraints are employed to describe point pair features. For

plane-level constraints, Xu et al. [31] introduced a voxel-

based four-plane congruent set (V4PCS) method for pair-

wise coarse registration, which ﬁrst extracts the planes for

point clouds and then leverages the plane-based geometric

constraints formed by four planes to achieve registration;

Chen et al. [32] presented a plane-based descriptor using dis-

tance and angle properties derived from planes for 3-D regis-

tration. For correspondence-level geometric constraints, most

of the existing methods focus on pairwise constraints. They

typically include the L

distance constraint [22], [33], normal

deviation angle constraint [21], [34], and LRF constraint [35].

Nonetheless, these pairwise constraints either suffer from

ambiguities or lack robustness to common nuisances. The

readers can refer to a related survey [36] for terrestrial laser

scanner point cloud registration for more details on geometrics

constraints proposed for 3-D registration.

SAC-COT leverages the correspondence-level geometric

constraints for registration, while it proposes a novel triplet

constraint that is validated to be far less ambiguous and more

robust than existing pairwise constraints.

C. Deep Learning for 3-D Registration

In addition to geometric-only methods, a few deep learning-

based point cloud registration methods have been pro-

posed recently. We review some typical examples here.

Aoki et al. [37] proposed the PointNetLK network for 3-D

registration in an iterative manner, where PointNet [38]

was employed for descriptor extraction combined with

a Lucas/Kanade-like optimization algorithm. Wang and

Solomon [39] presented deep closest point (DCP) that incor-

porates a point cloud embedding network, an attention-based

module to approximate combinatorial matching, and a differ-

entiable singular value decomposition (SVD) layer for 6-DOF

pose estimation. PRNet [40], as an extension to DCP, speciﬁ-

cally focuses on partial-to-partial registration, which contains

a keypoint detection module and is able to establish partial

correspondences.

Deep learning techniques have demonstrated a great poten-

tial for 3-D registration, while usually, adequate labeled train-

ing data should be prepared. This may not be feasible in many

practical scenarios, and as such, we still focus geometric-only

methods.

D. Guided Sampling Approaches for RANSAC

As the key novelty of SAC-COT is its sampling method,

we herein review existing guided sampling methods for

RANSAC. In the 2-D domain, guided sampling approaches

preferentially sample correspondences with high conﬁdence

scores [26], [41]. By contrast, guided sampling approaches

for RANSAC in the 3-D domain, as will be discussed in the

following, are more diverse.

For one-point-based sampling methods [16], [17], because

they have a linear time complexity, guided or random sampling

ways generate the same outcome as long as all correspon-

dences have been traversed. However, these one-point-based

sampling methods typically guide the sampling process based

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