Artículo patrocinado por Extraco, Misturas, Lógica, Enmacosa e Ingeniería InSitu, dentro del proyecto SITEGI, cofinanciado por el CDTI. (2012).
Article sponsored by Extraco, Misturas, Lógica, Enmacosa and Ingeniería Insitu inside the SITEGI project, cofinanced by the CDTI. (2012)
Contnúa de: Continued from: http://carreteraslaserescaner.blogspot.com/2014/10/erroranalysisofterrestriallaser.html
3. A Study Case
3.1. Experiment Description
In this study case, the experiment was performed over a set of 53 targets or check points (CP) distributed over a wall of 9.80 3.22 m as shown in Figure 2. The targets were circular shaped, retroreflective material, with a centred cross. The center of the targets was measured with TLS, 3,000 to 400 points per target, and other equipment of higher accuracy (Proliner).
Figure 2. The wall with the distribution of targets used in the case study. 
3.2. Materials The equipment used in this work is listed below:
TLS: a 3D long range TLS (TOF) Riegl LMSZ390i, technical specification are provided in [42]. This equipment measures distances in a range of 1.5 to 400 meters, with a nominal precision of ±6 mm at 50 m distance in normal illumination and reflectivity conditions. The vertical field of vision has amplitude of 80 degrees and 360 degrees in the horizontal plane. It has a minimum angular resolution of 0.2 degrees and a maximum of 0.002 degrees, and the rate of measurement of points oscillates between 8,000 and 11,000 points per second. This scanner is used in combination with a calibrated Nikon D200 camera incorporating a CCD sensor, DX format (10.2 megapixels in total).
The Proliner 5.7 system is especially designed for accurate measurement, gathered by locating a contact device, similar to a pen, on each point that defines the contours of the boat deck. This pen has a spherical tip that is joined to the machine by a cable. The 3D coordinates (X, Y, Z) of each point are stored in a memory device. As a result, all of the data can be exported to an ASCII file, where the 3D coordinates of all the points are included. Proliner is a machine that can be located in different positions: horizontal, vertical and tilted. The Proliner has a cable with a length of 5 meters, so the maximum distance that can be measured with the Proliner from a station position is 10 meters (in the absence of obstacles). Its precision in point coordinates, according to the manufacturer, is 0.3 mm.
Software package: All operations were performed using a variety of software such as RiSCAN PRO Software, Riegl© used for the recording and alignment of clouds of points. The calculation of the parameters that link the two reference systems of measurement equipment used (Proliner to Riegl LMS) was performed with Matlab 7.1. The specific Statistical calculations were made with the following software packages:
Statistical Package for the Social Sciences (SPSS): is a statistical analysis program, used for modular statistical analysis.
Spheristat v2.2: is a specific software for spherical statistics for angular statistical analysis.
VecStatGraph2D and VecStatGraph3D: are two packages in the R programming language (http://www.rproject.org), which is a language and environment for statistical computing and graphics and in addition available as Free Software under the terms of the Free Software Foundation's GNU General Public License in source code form. These two packages perform a 2D and 3D statistical analysis, both numerical and graphic, of a set of vectors and were developed for the Spherical graphics analysis proposed in this paper (http://gim.unex.es/VecStatGraphs2D, http://gim.unex.es/VecStatGraphs3D, respectively).
3.3. Data Collection, Preprocessing and Calculating Errors
The whole room was scanned with an angular resolution of 0.2 degrees. Detailed scans were performed for every target, 3,000 to 4,000 points per target. They were automatically detected on the 2D overview of the initial point cloud through a contrast algorithm that allowed discriminating the retroreflective material of targets from the intensity values of the others materials in the scene. Once detected and scanned, circles are approximated, and corresponding centers are estimated with 0.001 standard deviation. Then, a detailed scan of the scene is performed at a 0.02degree resolution (see Figure 3). Furthermore, the centre of each of the targets was also measured with Proliner through one station.
Figure 3. The room with the targets in the case study. 
Once the measurements were made with both equipments, data were needed to be aligned (or unified) in a common reference system in order to calculate the differences of coordinates obtained at each check point, and thus the error was obtained at each point. The alignment between the reference systems, Proliner and TLS, was performed using a rigid body transformation, more specifically, through 3D translation and three rotations in space. In this case, the translation was the origin point of reference of Proliner to the origin point of reference of TLS. The three cardanic rotations were made to coincide with the axes of both systems: ω in the X axis, φ in the Y axis and К on the Z axis. Therefore, the unification of the two coordinate systems requires knowledge of some parameters. To calculate the transformation parameters five points (1, 6, 35, 41, and 1,025) were measured by measuring both teams. The distribution of these points was selected as the most optimal (see Figure 2). The results of the parameters used for the alignment of coordinates are shown in Table 1.
Error is defined as the difference between a measure and the correct value. In this study case, the error in each point is the difference between the location of each CPTLS and its ―true‖ coordinate measured by Proliner technology. This error was calculated for each CP. Therefore, a 3D error vector was calculated for each point of each CP. Furthermore, each vector was defined in terms of its modulus and angles.
3.4. Vectorial Error Analysis The error analysis proposed of this paper was made in three parts. In the first part, the modular error component was calculated. This first part is similar to the conventional method based in the calculations of linear statistics. The second part was the directional error component analysis based in the calculations of spherical statistics. The third part was the most innovative part of the proposed analysis, which was developed by 2D and 3D graphics with two packages of the R programming language. In the last part, a study of the uniformity and normality of data distribution with errors data was achieved to complete data analysis.
3.4.1. Modular Error Analysis
Modular accuracy evaluation of TLS was calculated for a set of 53 CPs. The basic statistics of the modulus was calculated. The results of the modular error components are summarized in Table 2. The mean, minimum, maximum, standard deviation and root mean square error (RMSE) values along the X, Y and Z axis and denoted Δx, Δy and Δz, respectively were calculated. The Δr vector is equal to the square root of [Δx2 + Δy2 + Δz2] at each point, and Δr is the modular or radial statistic; however, the Δr basic modular statistics (mean, minimum, maximum, standard deviation and RMS) is not equal to the square root of [(statistic Δx)2 + (statistic Δy)2 + (statistic Δz)2].
We can observe, in Table 2, that the basic statistics of the Δr modular error were 9.53 mm of mean error, 2.02 mm of minimum error, 18.39 mm of maximum error, 3.23 mm of RMSE and ±0.44 mm of standard deviation. We can observe that the modular error result is reasonable according to the technical characteristics of the TLS. This is one of the main advantages of the modular error statistical analysis.
3.4.2. Angular Error Analysis
The next approach in the angular accuracy evaluation of error TLS was calculated for a set of CPs using spherical statistics. In this part of the study, specific software—Spheristat V2.2—for spherical statistics was used to calculate the spherical statistical parameters. Spheristat is a commercially available program that offers basic functionality for the analysis of spherical data. On the other hand, the graphic part was made by two packages implemented in the R programming language, which wasdeveloped for this work. The results of the angular (vertical and horizontal) error components are summarized in Table 3 and Figures 4, 5 and 6.
Figure 4. 3D spherical graphic of errors with all vectors in blue and the mean vector in red. 
Figure 5. Data in circular diagram (2D). (a) XZ wallplane. 
Figure 5. Data in circular diagram (2D). (b) XY groundplane 

Figure 6. Map of vectors in the ZX wallplane, 2D circular graphics analysis for each point in the wall (made with VecSartGraph2D). 
The spherical statistics for angular error are mean direction directions ( , ), circular standard deviation (υ), mean module ( ) of the all error vectors and the concentration parameter (κ). These statistics were explained in Section 2.2. The mean direction values were 249.7° of vertical angle and −3.8° of horizontal angle. The circular standard deviation value (υ) was 27.9°. On the other hand, a relatively high value of the mean module ( ) 0.8, shows that they were not in a uniform distribution.
As a last basic parameter for directions, the parameter (κ) is a measure of the concentration of data in a preferred orientation. The concentration parameter (κ) is 6.7. Therefore, if we consider that the value is not small, the data some show symmetrical distribution in relation to a preferred direction (see Figure 4).
3.4.3. Graphical Error Analysis
Although angular error components are summarized in Table 3, the angular analysis is not complete without Figures 4, 5 and 6, which depict some of these parameters.
Figure 4 shows some threedimensional representations of a sphere containing all error vectors. The error vectors radiate from the centre of the sphere and are represented by blue arrows. The mean vector was represented by the red arrow. This sphere allows a global view and a comparison of the modules and orientations of all vectors. Therefore, these 3D graphics are a complement to the data presented in Table 2 and Table 3 by providing a better and more complete analysis of the data. The 3D graphics of the Figure 4 were made with VecStatGraphs3D, a package in the R programming language.
Figure 5 shows twodimensional representations of results of the projections of the previous sphere, shown in Figure 4, for the three main planes: XZ wallplane (Figure 5(a)), XY groundplane (Figure 5(b)), and the plane perpendicular to these planes (Figure 5(c)). The 2D graphics of the Figure 5 were made with VecStatGraphs2D, another package in the R programming language, which was developed for circular data. Finally, as the last graphic analysis proposed, Figure 6 shows a twodimensional graphic that represent all of the errors in the XZ wallplane, as in Figure 5(a), but with each error vector with the corresponding correct position, also achieved with the VecStatGraphs2D software package. In this study case, although the error modules are small, we can see a trend in the angular errors (see Figures 4 and 5). On the other hand, the modular error values at the top and right of the wall have higher values than at the bottom and left of the wall (see Figure 6).
3.4.4. Uniformity and Distribution Error Analysis
The last step is to analyse the distribution of errors. Several tests were used to examine the uniformity and error distribution. For this part of the study, Spheristat software was used as well. SpheriStat also tests whether the sample is from a uniform distribution using Rayleigh‘s test of the magnitude of the resultant vector. This test compares the resultant vector, R, to a critical value at the 5% significance level. When R exceeds the critical value, the distribution cannot be considered to be uniform. In that case, SpheriStat compares the distribution to the Von Mises distribution, a circular equivalent to the Gaussian distribution. The result of Rayleigh‘s test is that the distribution has a preferred trend, with an R value of 88.9% and an R critical value (5% level) of 23.8%.
The concentration parameter, κ, measures the spread of the distribution (a lower κ for a wider spread), and the 95% confidence angle, derived from the standard error of the mean, gives the uncertainty in the resultant direction. The concentration parameter (κ) of Von Mises model is 6.7.
Therefore, if we consider that the value is not small, we can consider that the data show certain concentration. The last step is to analyse the distribution of errors by applying several tests. The Rayleigh test is a uniformity test that detects a single modal direction in a sample of data. The result of the test of sample uniformity, Rayleigh‘s test, is with a Rayleigh statistic of 115.97 better to a critical value of 7.81, so uniformity is rejected at the 95% confidence interval. The result of the test of sample uniformity, Beran/Giné test, is with a Beran/Giné statistic of 37.24 better to a critical value of 2.75, so uniformity is rejected too at the 95% confidence interval. 4. Discussion and Conclusions An improved methodology for the analysis of the vectorial errors of TLS data was presented in this paper. Although the method was applied to TLS data, it can also be applied to any threedimensional data. In this study, we argue that the real nature of the positional error is vectorial, and thus error vectors should be analysed. These error vectors have three metric elements (one module and two angles), and these magnitudes were used for a complete analysis of the positional error. In the case study presented as an example, 53 CP were measured by TLS with equipment to analyse and other equipment of higher accuracy (Proliner). We must take into account that the accuracy in the calculation of the errors was limited by the accuracy of the equipment used for this purpose. In the case study presented, the Proliner has ±0.3 mm accuracy compared to ±6 mm for TLS. Therefore, the calculation of errors was limited to ±0.3 mm of accuracy. We might highlight that this case is not designed to find the sources of errors in the TLS instruments; instead, it is intended to show the benefits of spherical graphics and statistical analysis. Results showed that the RMS modular error was 3.23 mm with ±0.44 mm of standard deviation. These values are reasonable if we know the technique characteristics of TLS. The first part of the analysis was performed as conventional analysis of the TLS error. The conventional analysis module provides information on the amount of error but not on the direction of error. This is analysed by means of the statistical analysis of the angles (Section 3.4.2). Figures 4, 5 and 6 show an important aspect of the error distribution that cannot be observed by analysing only the linear statistics: spatial error is clearly anisotropic. Error vectors from each CP show that TLS data are displaced to the east, but this displacement is not homogeneous because modular error values at the top and right of the wall have higher values than the bottom and left ones (Figure 6). The results show a preferential direction to the lower left corner. In the angular error analysis the results convey the angular parameters (mean directions, circular standard deviation and concentration) but on a global form (Table 3). In the graphical error analysis (Section 3.5.3) both parts (modular and angular) are performed together (Figures 4, 5 and 6). These graphs show the angular trend mentioned above.
We think that this trend could be due to the position of the measuring equipment Prolainer, which was placed on the ground at the bottom left (point 0,0,0 of Prolainer). In Figure 6, if we move away from this point we can see that the error modules increase. This may be due to the great influence of the accuracy on the coordinates measured of each point which depends on the distance to the equipment. Therefore, we believe that Prolainer equipment has limited use to analyse the accuracy of a TLS scanner because: (a) the accuracy is quite limited to calculate the standard deviation of error; (b) accuracy is highly dependent on distance. On the other hand, these results show the advantages of graphical error analysis using spherical statistics, which may reveal results that with conventional statistical analysis would be hidden. In the uniformity and distribution error analysis (Section 3.4.4.), the study case showed that in this dataset errors (study case with Riegl LMS z390i) are not spatially uniform but not necessarily for all TLS data. These local effects are interesting and can be detected with the proposed methodology; otherwise, they may be unnoticed if the error analysis is restricted to linear statistics and/or a limited set of checkpoints. Spherical statistics permit the analysis of a set of spatial properties—the angular error component, ―normality‖ or uniformity of vector distribution, and error isotropy and homogeneity—which cannot be taken into account in the error analysis based on traditional linear statistics, such as RMSE or standard deviation. These graphics complete the analysis of data error with a joint view of the sphere of errors from several perspectives (3D view and their respective projections in the three principal planes), composing a sufficient set of charts to analyse the results. Finally, one of the purposes of this paper, besides presenting the potential offered by these graphs, was the advantage of its development in R for the scientific community, which is available as Free Software under the terms of the Free Software Foundation's GNU General Public License in source code form (http://www.rproject.org/). In our future work, we propose to focus on the uncertainty of the measurements of TLS in terms of repeatability.
Acknowledgements
This work was financed by the Spanish Government (National Research Plan), the Galician Government (Galician Research Plan) and the European Union (FEDER founds) by means of the grants ref. TIN200803063, BIA200908012 and INCITE09304262 PR (sponsor and financial support acknowledgment goes here).
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