Terrestrial laser scanning (TLS) technology is one of the most efficient and accurate tools for 3D measurement which can reveal surfacebased characteristics of objects with the aid of computer vision and programming. Thus, it plays an increasingly important role in deformation monitoring and analysis. Automatic data extraction and high efficiency and accuracy modeling from scattered point clouds are challenging issues during the TLS data processing. This paper presents a data extraction method considering the partial and statistical distribution of the point clouds scanned, called the windowneighborhood method. Based on the point clouds extracted, 3D modeling of the boundary of an arched structure was carried out. The ideal modeling strategy should be fast, accurate, and less complex regarding its application to large amounts of data. The paper discusses the accuracy of fittings in four cases between whole curve, segmentation, polynomial, and Bspline. A similar number of parameters was set for polynomial and Bspline because the number of unknown parameters is essential for the accuracy of the fittings. The uncertainties of the scanned raw point clouds and the modeling are discussed. This process is considered a prerequisite step for 3D deformation analysis with TLS.
Structural health monitoring is of significant importance for the historical and architectural structures for safety reasons [
Morphologyoriented analysis with TLS is a new direction for deformation analysis [
This paper presents efficient extraction and discusses accurate modeling of the boundary of an arched structure, which are prerequisite steps for deformation analysis. Data extraction is carried out by the windowneighborhood method, which is closely related to the partial and statistical distribution of the scanned point clouds. In this paper, the mathematical functions of polynomial and Bspline are adopted to model the data. A polynomial has a deficiency in describing local features, but its advantages are efficiency and simplicity, while Bspline is exactly the opposite. Four cases are discussed, including a localized segmentation, to find an efficient and accurate modeling solution for the arched structure measured. Since the number of parameters has an influence on the fitting results, the numbers of parameters of polynomial and Bspline models are proximate to emphasize the model function.
Terrestrial laser scanning (TLS), which can capture up to millions of points per second and with a linear accuracy in the millimeter range, is one of the most efficient tools to measure 3D objects and structures. Traditional methods for monitoring, for example, total station, inclinometer, accelerometer, and leveling are generally pointbased measurements. Some lowcost sensors require preembedding or contact [
One important task for the presteps of modeling is to detect or recognize features from the 3D data. Reference [
It is common to fit the data collected with geometric forms or mathematical functions [
Curve fitting is the process of constructing a curve or mathematical function with the best approximation to data points [
Polynomial and Bspline are commonly used forms to fit a curve. Reference [
This paper focuses on the statistical analysis of spatial multidimensional point clouds data, which is the basis for the deformation analysis of an arched structure. The work flow of this paper is presented in Figure
Work flow of data extraction and fitting analysis.
The archshaped curve is then extracted to investigate the deformation behavior where the window selection method is adopted. The latter includes four steps: window definition, distribution analysis, threshold determination, and boundary extraction.
Using the resulting extracted data, point clouds fitting is performed for four kinds of combinations, which are explained later in Table
Different methods of the fitting for whole curve and segmentation.
Case 
Case 
Case 
Case 


Functions  P  B  P  B 
Points scope  Whole curve  Whole curve  Segmentation  Segmentation 
An experiment was carried out to investigate the deformation behavior of an arched structure using TLS point clouds. The main components are the arched structure specimen, Z + F Imager 5006, load equipment, and two supports, which are shown in Figure
Plot of experiment.
In this paper, the archshaped part of the object is vital for structural monitoring, because it bears the main load and shows significant deformation. The side surface of the object is taken into account for more precise deformation analysis. However, the archshaped object is occluded with some other objects, such as steel Ibeams, and needs to be separated for more accurate analysis.
The main concern of the deformation analysis lies in the edge of the arched structure, with the motivation that the arched edge defines the border of the arched surfaces with significant deformation, and is well connected to photogrammetry, since it can also be detected from digital images. From this point of view, the edge of the arched structure should be extracted and fitted with a high accuracy taken into consideration. Point clouds are preprocessed to pick up the side surface of the arched structure, which is extracted, and the result is shown in Figure
Point clouds of side surface of the arch.
As the laser beam rotates and scans with a uniform angular speed, footprints of laser beams have a theoretically closetoeven distribution considering a general surface. Suppose that each point of the surface carries a square selection window with a predefined size, which gives a reference to the neighboring points, there are mostly two cases. The first one is in the middle part of the surface, where those main points have almost the same number of neighbors, and such main points occupy the most population of all the point clouds. The second is in the boundary, with approximately half of the number of neighbors compared to the first case. A schematic diagram with a simple example is shown in Figure
Diagrammatic sketch of the edge point picking.
The neighboring numbers of the boundary points identified can only be within a certain range rather than an exact value, due to the inclination of the boundary line and the scattered distribution of points in the boundary area, as shown in Figure
Different cases of boundary and neighboring points.
According to the schematic diagram in Figures
The size of the selection window has an influence on the boundary extraction. If the size is too small, the noisy point will be mistaken for a boundary point, and vice versa; if the size is too large, some boundary points will be lost. The thickness of the arched structure is 10 cm.
The threshold
Distribution of neighboring points.
Because the value of
Edges extracted from MATLAB.
The extracted edges contain not only the archedshape curves, but also shadows of the occlusions. The boundary point clouds are imported to CloudCompare® for the purpose of separation of the archedshape curves. The openaccess software CloudCompare is an independent open source project and a free software for the processing of point clouds, which provides a set of basic tools to process 3D point clouds [
The extracted point clouds are approximated by both polynomial curve and Bspline. The 3D coordinates
The fitted curve of the polynomial and Bspline is presented in Figure
Comparison of polynomial and Bspline curves.
Epoch 3
Epoch 9
Due to the displacement of
The statistical errors of the fittings are considered to get a deeper insight into the fitting effects on the spatially scattered points. The polynomial approximation of the whole extracted line is defined as the original case/case 1 for the later comparison; the other cases are listed in Table
The uncertainties of the polynomial curve fitted for epoch 3 (E3) and 9 (E9) are presented in Table
Uncertainties of two forms of curves with unit mm.





P methods  
E3  

1.41  ⋯  ⋯ 

0.62  0.28  ⋯ 

−0.18 

0.31 
SD  37.49  16.70  17.51 
E9  

0.61  ⋯  ⋯ 

0.28  0.13  ⋯ 



0.17 
SD  24.75  11.26  12.98 


B methods  
E3  


⋯  ⋯ 



⋯ 




SD  2.84  1.34 

E9  


⋯  ⋯ 



⋯ 




SD  2.60  1.20 

The number of unknown parameters is considered to compare the Bspline and polynomial for spatial point clouds more impartially. The uncertainties of the Bspline for epochs 3 and 9 are presented in Table
According to Table
Occlusion of the laser beam in the experiment is unavoidable, because of the shelter of the loads equipment, which will cause fragmentation of the point clouds and then increase the uncertainties during the parametric estimation of the approximation. A segmentation fitting is employed for this perspective, which is presented in Figure
Segmentation fitting analysis of Epoch 9.
The segmentation approximations are described in Figure
The uncertainties of polynomial lines for each epoch are listed in Table
The standard deviations in the
Uncertainties during segmentation with unit mm.





P methods  
E3  


⋯  ⋯ 



⋯ 




SD  3.92  2.31 

E9  


⋯  ⋯ 



⋯ 




SD  3.61  2.23 



B methods  
E3  


⋯  ⋯ 



⋯ 




SD  0.89  0.66 

E9  


⋯  ⋯ 



⋯ 




SD  1.00  0.71 

Improvement analysis.
Ratio (%)  Epoch  X  Y  Z 

Improvement by Bspline approximation  
Whole curve  E3 



E9 




Segmentation  E3  77.07  71.25  66.24 
E9  72.22  67.87  60.52  


Improvement by segmentation  
P methods  E3 



E9 




B methods  E3  68.51  50.66  53.43 
E9  61.54  41.10  49.24 
Similarly, the segmentation curve is fitted by the Bspline function. The uncertainties of the approximation are shown in Table
Based on the uncertainties analysis in Tables
According to the analysis, the optimization of segmentation is shown in Table
It is also revealed that curve fitting results have a close relation with the accuracy and reliability of the scanned point clouds. The TLS employed is Imager Z + F 5006, whose scanning precision parameters are listed in Table
Technical parameters of employed TLS.
Measurement range  0.4–79 m 
Scan speed  up to 1016727 points/sec 
Range noise at 10 m  
Reflectivity 10% (black)  1.2 mm rms 
Reflectivity 20% (dark grey)  0.7 mm rms 
Reflectivity 100% (white)  0.4 mm rms 
Distance resolution  0.1 mm 
Linear error (50 m)  1 mm 
This paper focuses on the comparative analysis of the Bspline and polynomial approximation based on the extraction of multidimensional data which are collected by TLS during deformation analysis. An innovative window selection method is adopted to efficiently extract the edge data of the arch structure, where the partial and statistical distribution of the scanned point clouds are considered. An optimal extraction is chosen from the aspect of noise and blank of the extracted point clouds.
The Bspline and polynomial approximation are presented and analyzed with four cases, where conclusions can be drawn as follows:
For the experiment in this paper, the uncertainty ranking of the four cases is segmentation with B method (less than 1 mm), whole curve with B method and segmentation with P method (12 mm) and whole curve with P method (10–20 mm).
It is verified experimentally, through comparing the standard deviations of the two fitting methods, that the Bspline has a more satisfactory fitting precision, where the standard deviation of the whole edge curve of the Bspline fitting is about 90% better than the polynomial approximation.
However, in the case of segmented fitting, the accuracy improvement of the Bspline is not as much as the polynomial approximation. It is revealed that the Bspline has an advantage of better accuracy in complex situations, but it is also implied that the polynomial approximation can greatly improve the fitting accuracy by segmentation, simplifying the complex case.
It is indicated that, in the approximation of curves during deformation monitoring with TLS measurement, segmentation method can be adopted, where efficiency of polynomial and Bspline could be combined to construct a fast and accurate 3D model.
The authors declare no conflicts of interest.
The publication of this article was funded by the Open Access Fund of the Leibniz Universität Hannover. The authors also would like to acknowledge the support of Natural Science Foundation of Jiangsu, China (no. BK20160558) and Six Talent Peaks Project in Jiangsu Province (no. 2015JZ009). The authors wish to acknowledge the support of all the colleagues in Geodetic Institute of Leibniz University Hanover for their valid information and help.