# A Robust Rigid Registration Framework of 3D Indoor Scene Point Clouds Based on RGB-D Information

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## Abstract

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## 1. Introduction

- We present a point normal estimation method by coupling total variation with second-order variation. The method is capable of effectively removing noise while keeping sharp geometric features and smooth transition regions simultaneously.
- We present a robust correspondence points extraction method, based on a descriptor (TexGeo) encoding both texture and geometry information. With the help of the TexGeo descriptor, the proposed method is robust when handling low-quality point clouds.
- We design a point-to-plane registration method based on a nonconvex regularizer. The method can automatically ignore the influence of those false correspondences and produce an exact rigid transformation between a pair of noisy point clouds.
- We verify the robustness of our approach on a variety of low-quality RGB-D point clouds. Intensive experiments demonstrate that our approach outperforms the selected state-of-the-art methods visually and numerically.

## 2. Related Work

**Point normal estimation.**As an important signal indicating the direction field of the scanned surface, the point normal field has been widely applied for constructing 3D point descriptors, such as the FPFH [7]. Note that the 3D point descriptor is fundamental to correspondence extraction. However, it is challenging to robustly estimate point normals, since the captured point clouds are inevitably corrupted by noise and outliers. To address this issue, researchers have proposed many valuable methods. Here, we only review remarkable ones related to our work. Avron et al. [32] have applied ${\ell}_{1}$ regularization to recover the point normal field. In order to preserve sharp edges and corners, Sun et al. [33] have derived a sparsity-based method that uses ${\ell}_{0}$ minimization for effectively processing point clouds whose underlying surfaces are piecewise constant. These two methods can keep sharp geometric features while removing noise effectively. However, both of them inevitably suffer from serious staircase artifacts in smooth transition regions [34,35,36,37,38,39,40]. For alleviating these artifacts, Liu et al. [41] have recently introduced a point cloud denoising framework, which presents an anisotropic second-order regularizer to remove noise and preserve sharp geometric features as well as smooth transition regions.

**Correspondence extraction.**Correspondence extraction consists in matching points to determine a coarse alignment. Existing methods are designed based on either point features or structure information to construct descriptors. Gelfand et al. [8] have identified features and extracted correspondences using a novel integral volume descriptor. Similar to [7], Zhou et al. [23] have utilized the FPFH descriptor to match points efficiently. In order to register RGB-D point clouds, the authors of [9,14] have applied texture information for extracting correspondences. Though the above methods can extract correspondences effectively, they are easily disturbed by large noise. Different from the above methods, the methods based on structure information need to construct some meaningful structures. For example, Aiger et al. [15] have matched points by comparing approximately congruent coplanar four-point sets selected from a pair of point clouds. Their approach, called 4PCS, can robustly register point clouds without any assumption about their initial poses. However, it costs a lot of time when handling large-scale point clouds, because it performs the RANSAC random iteration process [42,43]. As a result, to improve the efficiency of 4PCS, Mellado et al. [44] have derived Super4PCS. Moreover, by using structural information including planes and lines as well as their interrelationship, Chen et al. [16] have matched points for pairs of point clouds whose overlap ratios are small. Zhang et al. [17] have introduced a registration framework that computes correspondences by using middle-level structural features.

**Point clouds alignment.**Point clouds alignment estimates the rigid transformation for registering a pair of point clouds given the extracted correspondences. To this end, the ICP algorithm iteratively minimizes the sum of the ${\ell}_{2}$ distance (i.e., point-to-point distance or point-to-plane distance) between correspondence points [4,18]. Though the ICP algorithm is simple, it is not only sensitive to noise and outliers, but also computationally expensive. To overcome these limitations, Chetverikov et al. [21] have introduced a trimmed ICP algorithm, which can robustly register incomplete point clouds with noise. By utilizing a branch-and-bound scheme, Yang et al. [25] have presented the Go-ICP, which is a global algorithm for point cloud registration. Bouaziz et al. [22] have presented the sparse ICP algorithm that formulates the registration problem as an ${\ell}_{p}$ minimization problem. Though their method can limit the effect of noise and outliers on the aligned results by adjusting the value of parameters p, it is time-consuming to solve the nonconvex optimization problem. To conquer the issue, Mavridis et al. [45] have improved the sparse ICP for more efficiently solving the nonconvex optimization problem. Wu et al. [24] have eliminated the interference of outliers and noise by using the maximum correntropy criterion. Zhou et al. [23] have introduced a robust global approach by utilizing a scaled Geman–McClure function, which can automatically reduce wrong correspondences. To improve the convergence of the ICP algorithm, Rusinkiewicz [46] have presented a symmetric objective function. Furthermore, to speed up the ICP algorithm, Pavlov et al. [19] have proposed AA-ICP, a novel modification of the ICP algorithm based on Anderson acceleration, which substantially reduces the number of iterations with a negligible cost. However, when the ground-truth rotation is close to a gimbal lock [47], the AA-ICP method cannot produce the desired result. To alleviate this issue, Zhang et al. [20] have recently proposed a fast and robust variant of the ICP algorithm using Welsch’s function. Moreover, an Anderson-accelerated majorization-minimization algorithm has been proposed to solve their problem.

## 3. Methodology

#### 3.1. Point Normal Estimation

**Rough point normal computation.**We can easily compute rough point normals from the corresponding depth image. Formally, for each point ${\mathbf{p}}_{i,j}$, we compute its rough normal, $\widehat{\mathbf{N}}$, as

**Point normal filtering**. To filter out the noise of the rough point normals, we present a sparsity-inspired global optimization method, whose goal is to find the smoothed point normals that best fit the input rough normals and the given sparsity constraints. We first design a point normal filtering model by using the total variation model and a second-order operator. Then, we present an iterative algorithm to minimize the point normal filtering model.

**Effectiveness of our point normal estimation**. To test the effectiveness of our point normal filter, we ran it on indoor point clouds, and compared it to state-of-the-art methods including RIMLS [53], MRPCA [54], and L0P [33]. Figure 3 shows the comparison results. Note that, for better visualization, we adopt the strategy presented in [41] for updating point positions after filtering point normals. Apparently, all these tested methods can effectively filter out noise. However, RIMLS blurs sharp geometric features (e.g., edges and corners), though it recovers smooth transition regions well (see Figure 3c). On the contrary, MRPCA and L0 can preserve sharp geometric features. Nevertheless, L0 inevitably causes staircase artifacts in smooth transition regions (see Figure 3d), and MRPCA tends to over-sharpen smooth features (see Figure 3c). Unlike the above methods, our normal filtering method can simultaneously keep sharp geometric features and smooth transition regions (see Figure 3e). The comparisons in Figure 3 demonstrate that, our approach outperforms the other ones in handling indoor point clouds, especially those containing sharp features and smooth regions.

Algorithm 1: The iterative algorithm for minimizing problem (3). |

#### 3.2. Correspondence Extraction

#### 3.3. Point Clouds Alignment

Algorithm 2: Robust rigid transformation computation. |

## 4. Experimental Results

#### 4.1. Qualitative Comparison

#### 4.2. Quantitative Comparison

#### 4.3. Ablation Study

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$\mathcal{L}(i,j)$ | The auxiliary line connecting point ${\mathbf{p}}_{i,j}$ with some midpoint |

$len(\xb7)$ | The length of $(\xb7)$ |

$disk(\xb7)$ | The area of $(\xb7)$ |

${D}^{1}$ | The first-order operator |

${D}^{2}$ | The second-order operator |

RIMLS | Robust implicit moving least squares |

MRPCA | Moving robust principal components analysis |

L0P | Denoising point sets via ${\ell}_{0}$ minimization |

PCL | A point cloud library implementation of Rusu et al. [7] |

S4PCS | Super 4pcs fast global point cloud registration via smart indexing |

GICP | Go-ICP: a globally optimal solution to 3D ICP point-set registration |

GICPT | A trimming variant of GICP |

FGR | Fast global registration |

SymICP | A symmetric objective function for ICP |

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**Figure 1.**The pipeline of our method consisting of point normal estimation, correspondence extraction, and point clouds alignment.

**Figure 2.**(

**a**) Auxiliary geometric elements of point ${p}_{i,j}$, including neighbor points (colored in blue) and the midpoints of two neighbor points (colored in red), and auxiliary edges (colored in red). (

**b**) Demonstration of second-order variation defined over the line l.

**Figure 4.**The example RGB images of six indoor scene point clouds named Lr1, Lr2, Lr3, Of1, Of2, and Teddy, respectively.

**Figure 5.**Registration results of Lr1 ($\sigma =0.2\%\overline{d}$) which have small rotation and translation between them. (

**a**) Input. (

**b**) PCL. (

**c**) GICP. (

**d**) GICPT. (

**e**) S4PCS. (

**f**) FGR. (

**g**) Our approach.

**Figure 6.**Registration results of Lr2 ($\sigma =0.4\%\overline{d}$) with both comparatively large translation and rotation between them. (

**a**) Input. (

**b**) PCL. (

**c**) GICP. (

**d**) GICPT. (

**e**) S4PCS. (

**f**) FGR. (

**g**) Our approach.

**Figure 7.**Registration results of indoor scene point clouds Lr3 ($\sigma =0.4\%\overline{d}$) with rich geometric features and large textureless regions. (

**a**) Input. (

**b**) PCL. (

**c**) GICP. (

**d**) GICPT. (

**e**) S4PCS. (

**f**) FGR. (

**g**) Our approach.

**Figure 8.**Registration results of indoor scene point clouds Of1 ($\sigma =0.2\%\overline{d}$) with both large translation and rotation between two input point clouds. (

**a**) Input. (

**b**) PCL. (

**c**) GICP. (

**d**) GICPT. (

**e**) S4PCS. (

**f**) FGR. (

**g**) Our approach.

**Figure 9.**Registration results of indoor scene point clouds Of2 ($\sigma =0.4\%\overline{d}$) with few geometric features and rich textures. (

**a**) Input. (

**b**) PCL. (

**c**) GICP. (

**d**) GICPT. (

**e**) S4PCS. (

**f**) FGR. (

**g**) Our approach.

**Figure 10.**Registration results of point clouds Teddy, which are corrupted by real noise. (

**a**) Input. (

**b**) PCL. (

**c**) GICP. (

**d**) GICPT. (

**e**) S4PCS. (

**f**) FGR. (

**g**) Our approach.

**Figure 11.**Comparison of registration results produced by SymICP and our approach. (

**a**) input; (

**b**) registration result produced by SymICP; (

**c**) registration result produced by our approach.

**Figure 12.**Ablation of point normal estimation. From left to right: (

**a**) input; (

**b**) registration result produced using noisy point normals; (

**c**) registration result produced using estimated point normals.

**Figure 13.**Ablation of correspondence extraction. (

**a**) input; (

**b**) registration result produced using the FPFH descriptor; (

**c**) registration result produced using the proposed TexGeo descriptor.

**Figure 14.**Ablation of rigid transformation computation scheme. (

**a**) input; (

**b**) registration result produced using the ICP algorithm; (

**c**) registration result produced using our approach.

Point Clouds | $\mathbf{RMSE}(\times {10}^{-3})$ | |||||
---|---|---|---|---|---|---|

PCL | GICP | GICPT | S4PCS | FGR | Our Approach | |

Lr1 | 2.800 | 4.791 | 2.658 | 4.275 | 2.614 | 2.526 |

Lr2 | 3.067 | 4.747 | 3.691 | 3.566 | 2.917 | 2.613 |

Lr3 | 4.485 | 5.509 | 3.352 | 3.492 | 3.614 | 3.202 |

Of1 | 3.006 | 2.715 | 3.049 | 2.218 | 2.070 | 1.869 |

Of2 | 5.013 | 4.584 | 5.547 | 5.043 | 4.283 | 3.935 |

Teddy | 5.893 | 6.356 | 5.767 | 6.545 | 5.710 | 5.661 |

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## Share and Cite

**MDPI and ACS Style**

Zhong, S.; Guo, M.; Lv, R.; Chen, J.; Xie, Z.; Liu, Z.
A Robust Rigid Registration Framework of 3D Indoor Scene Point Clouds Based on RGB-D Information. *Remote Sens.* **2021**, *13*, 4755.
https://doi.org/10.3390/rs13234755

**AMA Style**

Zhong S, Guo M, Lv R, Chen J, Xie Z, Liu Z.
A Robust Rigid Registration Framework of 3D Indoor Scene Point Clouds Based on RGB-D Information. *Remote Sensing*. 2021; 13(23):4755.
https://doi.org/10.3390/rs13234755

**Chicago/Turabian Style**

Zhong, Saishang, Mingqiang Guo, Ruina Lv, Jianguo Chen, Zhong Xie, and Zheng Liu.
2021. "A Robust Rigid Registration Framework of 3D Indoor Scene Point Clouds Based on RGB-D Information" *Remote Sensing* 13, no. 23: 4755.
https://doi.org/10.3390/rs13234755