﻿ 基于点云去噪的球形标靶中心拟合研究
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 大地测量与地球动力学  2019, Vol. 39 Issue (8): 849-855  DOI: 10.14075/j.jgg.2019.08.016

### 引用本文

YU Teng, LI Mingfeng, HU Wusheng, et al. Research on Spherical Target Center Location Based on Three-Dimensional Laser Scanning Point Cloud after De-Noising[J]. Journal of Geodesy and Geodynamics, 2019, 39(8): 849-855.

### Foundation support

National Natural Science Foundation of China, No. 41274009, 41574022; Science and Technology Project of Suqian, No. Z2018102.

### 第一作者简介

YU Teng, senior experimentalist, majors in precise engineering survey and integrated of surveying instruments, E-mail:164002786@qq.com.

### 文章历史

1. 宿迁学院建筑工程学院，江苏省宿迁市黄河南路399号，223800;
2. 南京工业大学测绘科学与技术学院，南京市浦珠南路30号，211800;
3. 东南大学交通学院，南京市进香河路35号，210096

1 球形标靶特性与球心拟合 1.1 球形标靶的特征

1.2 球形标靶配准基本原理

 $\left[ {\begin{array}{*{20}{c}} {x_m^2}\\ {y_m^2}\\ {z_m^2} \end{array}} \right] = \lambda \left[ {\begin{array}{*{20}{c}} {{r_{00}}}&{{r_{01}}}&{{r_{02}}}\\ {{r_{10}}}&{{r_{11}}}&{{r_{12}}}\\ {{r_{20}}}&{{r_{21}}}&{{r_{22}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {x_m^1}\\ {y_m^1}\\ {z_m^1} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{t_x}}\\ {{t_y}}\\ {{t_z}} \end{array}} \right]$ (1)
1.3 球形标靶球心拟合原理

 $\begin{array}{*{20}{c}} {{e_i}\left( {{x_0},{y_0},{z_0},r} \right) = {{\left( {{x_i} - {x_0}} \right)}^2} + }\\ {{{\left( {{y_i} - {y_0}} \right)}^2} + {{\left( {{z_i} - {z_0}} \right)}^2} - {r^2}} \end{array}$ (2)

 $E\left( {{x_0},{y_0},{z_0},r} \right) = \sum\limits_{i = 1}^N {{e_i}{{\left( {{x_0},{y_0},{z_0},r} \right)}^2}}$ (3)

Ex0y0z0r的函数。令E分别对x0y0z0r求偏导数，令其值为0，即可求出x0y0z0r

 $\begin{array}{l} \left\{ \begin{array}{l} \frac{{\partial E}}{{\partial {x_0}}} = 0\\ \frac{{\partial E}}{{\partial {y_0}}} = 0\\ \frac{{\partial E}}{{\partial {z_0}}} = 0\\ \frac{{\partial E}}{{\partial r}} = 0 \end{array} \right. \to \left\{ \begin{array}{l} \sum\limits_{i = 1}^N {{e_i}\frac{{\partial {e_i}}}{{\partial {x_0}}}} = 0\\ \sum\limits_{i = 1}^N {{e_i}\frac{{\partial {e_i}}}{{\partial {y_0}}}} = 0\\ \sum\limits_{i = 1}^N {{e_i}\frac{{\partial {e_i}}}{{\partial {z_0}}}} = 0\\ \sum\limits_{i = 1}^N {{e_i}\frac{{\partial {e_i}}}{{\partial r}}} = 0 \end{array} \right. \to \\ \left\{ \begin{array}{l} \sum\limits_{i = 1}^N {{e_i}\left( {{x_i} - {x_0}} \right)} = 0\\ \sum\limits_{i = 1}^N {{e_i}\left( {{y_i} - {y_0}} \right)} = 0\\ \sum\limits_{i = 1}^N {{e_i}\left( {{z_i} - {z_0}} \right)} = 0\\ \sum\limits_{i = 1}^N {{e_i}r} = 0 \end{array} \right. \to \left\{ \begin{array}{l} \sum\limits_{i = 1}^N {{e_i}{x_i}} = 0\\ \sum\limits_{i = 1}^N {{e_i}{y_i}} = 0\\ \sum\limits_{i = 1}^N {{e_i}{z_i}} = 0\\ \sum\limits_{i = 1}^N {{e_i}} = 0 \end{array} \right. \end{array}$ (4)

 $\left\{ \begin{array}{l} \bar x = \frac{1}{N}\sum\limits_{i = 1}^N {{x_i}} ,\bar y = \frac{1}{N}\sum\limits_{i = 1}^N {{y_i}} ,\bar z = \frac{1}{N}\sum\limits_{i = 1}^N {{z_i}} \\ \overline {xy} = \frac{1}{N}\sum\limits_{i = 1}^N {{x_i}{y_i}} ,\overline {xz} = \frac{1}{N}\sum\limits_{i = 1}^N {{x_i}{z_i}} ,\\ \;\;\;\;\overline {yz} = \frac{1}{N}\sum\limits_{i = 1}^N {{y_i}{z_i}} \\ \overline {{x^2}} = \frac{1}{N}\sum\limits_{i = 1}^N {x_i^2} ,\overline {{y^2}} = \frac{1}{N}\sum\limits_{i = 1}^N {y_i^2} ,\\ \;\;\;\;\overline {{z^2}} = \frac{1}{N}\sum\limits_{i = 1}^N {z_i^2} \\ \overline {{x^2}y} = \frac{1}{N}\sum\limits_{i = 1}^N {x_i^2{y_i}} ,\overline {{x^2}z} = \frac{1}{N}\sum\limits_{i = 1}^N {x_i^2{z_i}} ,\\ \;\;\;\;\overline {x{y^2}} = \frac{1}{N}\sum\limits_{i = 1}^N {{x_i}y_i^2} \\ \overline {{y^2}z} = \frac{1}{N}\sum\limits_{i = 1}^N {y_i^2{z_i}} ,\overline {x{z^2}} = \frac{1}{N}\sum\limits_{i = 1}^N {{x_i}z_i^2} ,\\ \;\;\;\;\overline {y{z^2}} = \frac{1}{N}\sum\limits_{i = 1}^N {{y_i}z_i^2} \\ \overline {{x^3}} = \frac{1}{N}\sum\limits_{i = 1}^N {x_i^3} ,\overline {{y^3}} = \frac{1}{N}\sum\limits_{i = 1}^N {y_i^3} ,\overline {{z^3}} = \frac{1}{N}\sum\limits_{i = 1}^N {z_i^3} \end{array} \right.$ (5)

 $\left\{ \begin{array}{l} \frac{{\overline {{x^3}} }}{{\bar x}} - 2{x_0}\frac{{\overline {{x^2}} }}{{\bar x}} + x_0^2 + \frac{{\overline {x{y^2}} }}{{\bar x}} + 2{y_0}\frac{{\overline {xy} }}{{\bar x}} + \\ \;\;\;\;y_0^2 + \frac{{\overline {x{z^2}} }}{{\bar x}} - 2{z_0}\frac{{\overline {xz} }}{{\bar x}} + z_0^2 = {r^2}\\ \frac{{\overline {{x^2}y} }}{{\bar y}} - 2{x_0}\frac{{\overline {xy} }}{{\bar y}} + x_0^2 + \frac{{\overline {{y^3}} }}{{\bar y}} - 2{y_0}\frac{{\overline {{y^2}} }}{{\bar y}} + \\ \;\;\;\;y_0^2 + \frac{{\overline {{z^2}y} }}{{\bar y}} - 2{z_0}\frac{{\overline {zy} }}{{\bar y}} + z_0^2 = {r^2}\\ \frac{{\overline {{x^2}z} }}{{\bar z}} - 2{x_0}\frac{{\overline {xz} }}{{\bar z}} + x_0^2 + \frac{{\overline {{y^2}z} }}{{\bar z}} - 2{y_0}\frac{{\overline {yz} }}{{\bar z}} + \\ \;\;\;\;y_0^2 + \frac{{\overline {{z^3}} }}{{\bar z}} - 2{z_0}\frac{{\overline {yz} }}{{\bar z}} + z_0^2 = {r^2}\\ \overline {{x^2}} - 2{x_0}\bar x + x_0^2 + \overline {{y^2}} - 2{y_0}\bar y + y_0^2 + \\ \;\;\;\;\overline {{z^2}} - 2{z_0}\bar z + z_0^2 = {r^2} \end{array} \right.$ (6)

 $\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\overline {{x^2}} - {{\bar x}^2}}&{\overline {xy} - \bar x\bar y}&{\overline {xz} - \bar x\bar z}\\ {\overline {xy} - \bar x\bar y}&{\overline {{y^2}} - {{\bar y}^2}}&{\overline {yz} - \bar y\bar z}\\ {\overline {xz} - \bar x\bar z}&{\overline {yz} - \bar y\bar z}&{\overline {{z^2}} - {{\bar z}^2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_0}}\\ {{y_0}}\\ {{z_0}} \end{array}} \right] = \frac{1}{2} \cdot \\ \left[ \begin{array}{l} \left( {\overline {{x^3}} - \bar x\overline {{x^2}} } \right) + \left( {\overline {x{y^2}} - \bar x\overline {{y^2}} } \right) + \left( {\overline {x{z^2}} - \bar x\overline {{z^2}} } \right)\\ \left( {\overline {{x^2}y} - \bar y\overline {{x^2}} } \right) + \left( {\overline {{y^3}} - \bar y\overline {{y^2}} } \right) + \left( {\overline {y{z^2}} - \bar y\overline {{z^2}} } \right)\\ \left( {\overline {{x^2}z} - \bar z\overline {{x^2}} } \right) + \left( {\overline {z{y^2}} - \bar z\overline {{y^2}} } \right) + \left( {\overline {{z^3}} - \bar z\overline {{z^2}} } \right) \end{array} \right] \end{array}$ (7)

2 离散小波阈值去噪基本原理

 $\begin{array}{*{20}{c}} {{D_f}\left( {j,k} \right) = \left\langle {f,{\psi _{j,k}}} \right\rangle = }\\ {{{\left| a \right|}^{ - \frac{j}{2}}}\int {\psi \left( {a_0^{ - j} - k{b_0}} \right)f\left( x \right){\rm{d}}x} } \end{array}$ (8)

 $f\left( x \right) = {A_{j - 1}}f\left( x \right) = {A_j}f\left( x \right) + {D_j}f\left( x \right)$ (9)

 ${\mathit{\boldsymbol{C}}_{j + 1}} = \mathit{\boldsymbol{H}}{\mathit{\boldsymbol{C}}_j},j = 1,2, \cdots ,J$ (10)
 ${\mathit{\boldsymbol{D}}_{j + 1}} = \mathit{\boldsymbol{G}}{\mathit{\boldsymbol{C}}_j},j = 1,2, \cdots ,J$ (11)

 ${\mathit{\boldsymbol{\tilde C}}_j} = \mathit{\boldsymbol{\bar H}}{\mathit{\boldsymbol{\tilde C}}_{j + 1}} + \mathit{\boldsymbol{\bar G}}{\mathit{\boldsymbol{\tilde D}}_{j + 1}},j = J,J - 1, \cdots ,1$ (12)

 $\hat f\left( x \right) = {A_J}\hat f\left( x \right) = \sum\limits_{k \in Z} {{{\mathit{\boldsymbol{\tilde C}}}_{J,k}}{\varphi _{J,k}}\left( x \right)}$ (13)

3 实例分析

3.1 实验概况

 图 1 三维激光扫描布置与实景图 Fig. 1 3D laser scanning resettlement and real view

3.2 小波阈值法去噪过程 3.2.1 适用于点云去噪的小波基与分解层数的确定

 图 2 不同小波基函数去噪效果对比图 Fig. 2 Contrast diagram of denoising effect of different small wave base functions
3.2.2 去噪过程与精度分析

 图 3 T2标靶球点云去噪过程 Fig. 3 T2 target ball point cloud de-noising process map

 图 4 T3标靶球点云去噪过程 Fig. 4 T3 target ball point cloud de-noising process map

3.3 去噪后数据的球心拟合

 图 5 去噪前后球形标靶球面对比 Fig. 5 Contrast diagram of spherical target spherical surface before and after de-noising

 图 6 球形标靶去噪前后配准点云对比 Fig. 6 Contrast map of registration point cloud before and after de-noising of spherical target

4 结语

1) 针对标靶点云的特性，分析不同小波基的数学特点，对比不同小波基去噪效果后认为，db-3小波基更适合此类数据的去噪。在用小波阈值去噪方法对球形标靶点云进行去噪处理后发现，去噪后的球形标靶点云曲面光滑，噪声能量减弱，对于实验标靶球的大小而言，扫描点至球心距离与球半径差值Δdi的最大值减少约1.7 mm，均值减少约0.8 mm。

2) 基于去噪后的球形标靶点云拟合出精确的球心坐标，以此坐标作为精密配准的关联点，配准后的点云影像较未经标靶去噪配准的点云影像更为清晰，重叠度减弱。将楼房中的清晰窗角点作为检查点进行检核，以高精度全站仪采集得到的坐标作为真值，并与拼接后的楼房点云检查点坐标进行比较可发现，对于同一点的拼接距离平均缩短5 mm左右。基于ASIFT算法对影像重叠度进行判别和评估后认为，扫描物体整体点云配准精度提升20%左右。

3) 必须重视用于坐标转换的地面三维激光扫描标靶研究，其扫描质量和数据纠偏质量直接影响最终拼接的点云精度。用小波阈值去噪方法对球形标靶点云去噪是有效的，去噪后的数据可以更精确地拟合出同名特征点的中心坐标，是提高点云配准精度的有效手段。

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Research on Spherical Target Center Location Based on Three-Dimensional Laser Scanning Point Cloud after De-Noising
YU Teng1     LI Mingfeng2     HU Wusheng3     SUN Xiaorong1
1. School of Civil Engineering and Architecture, Suqian College, 399 South-Huanghe Road, Suqian 223800, China;
2. School of Geomatics Science and Technology, Nanjing Technology University, 30 South-Puzhu Road, Nanjing 211800, China;
3. School of Transportation, Southeast University, 35 Jinxianghe Road, Nanjing 210096, China
Abstract: When the spherical target is used as the feature point of the same name for point cloud data registration, if there is a large amount of noise in the vicinity of the target with interference or scanning target point cloud, it will have a great influence on the point cloud registration quality. In view of the current situation that the point cloud registration has neglected target self-scanning noise, the characteristics of spherical target are analyzed, the applicability of the wavelet threshold denoising method is discussed, and the method of selecting the wavelet base function is tested. A wavelet threshold denoising method of the spherical target point cloud discrete noise is proposed. Experimental results show that the denoising of the target's own point cloud cannot be neglected; the result of case analysis shows that the method can filter the rough noise near the sphere more effectively, the fitting accuracy of the center position of a single spherical target is increased by about 0.8 mm, as compared with the point cloud stitching result of the spherical target without denoising. The co-ordinate stitching distance error of the scan feature check point is reduced by about 5 mm, and the registration accuracy of the point cloud is increased by about 20%. It is an efficient preprocessing method for point cloud registration data, and can provide reference for the application of related engineering.
Key words: spherical target; three-dimensional laser scanning; point cloud; wavelet de-noising; sphere centre