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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

This paper presents a complete analysis of the positional errors of terrestrial laser scanning (TLS) data based on spherical statistics and 3D graphs. Spherical statistics are preferred because of the 3D vectorial nature of the spatial error. Error vectors have three metric elements (one module and two angles) that were analyzed by spherical statistics. A study case has been presented and discussed in detail. Errors were calculating using 53 check points (CP) and CP coordinates were measured by a digitizer with submillimetre accuracy. The positional accuracy was analyzed by both the conventional method (modular errors analysis) and the proposed method (angular errors analysis) by 3D graphics and numerical spherical statistics. Two packages in R programming language were performed to obtain graphics automatically. The results indicated that the proposed method is advantageous as it offers a more complete analysis of the positional accuracy, such as angular error component, uniformity of the vector distribution, error isotropy, and error, in addition the modular error component by linear statistics.

In the last decade, terrestrial laser scanning (TLS) systems have appeared on the market and found a firm place in geodetic metrology. When TLS laser scanners were introduced on the market, their performances were rather poor, having in general a measurement uncertainty in the range of centimeters. However, with the progressive improvement of technology and the consequent increase in the measurement precision, the potential range of purposes has been widened from some meters to hundreds of meters in forensics [

As a result of the absence of standards, the accuracy specifications given by laser scanner producers in their publications and pamphlets are not comparable [

The first suggestion for system calibrations, system tests and accuracy checks for TLS correspond to Lichti [

Reshetyuk [

The “technical” parameters representing the mechanical-optical stability such the geometry of the axes, eccentricity, and the addition constant were obtained for certain instruments [

Mechelke

Kersten

Lichti [

The International Organization for Standardization (ISO) was formed in 1947 as a non-governmental federation of standardization bodies from over 60 countries. The ISO is headquartered in Geneva, Switzerland. The Unites States is represented by the ANSI.

In 1984, ISO published the 1st edition of the “International vocabulary metrology _Basic and general concepts and associated terms (VIM)” [

It considers appropriate to review some terms:

▪ Precision: degree of closeness between independent measurement results obtained

▪ Accuracy: The closeness of agreement between a test result and the accepted reference value. The term accuracy, when applied to a set of test results, involves a combination of random components and a common systematic error or bias component.

▪ Uncertainty, parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand. The parameter may be, for example, a standard deviation, or the half-width of an interval having a stated level of confidence.

The most common descriptor in geosciences is the root mean square error (RMSE). The frequently used mean error (ME) and error standard deviation (S) are also given in accuracy tests for a more complete statistical description of errors. However, none of these descriptive statistics (RMSE, ME, S) reports more than a global summary statistic based on comparison with a limited sample of points and only from the perspective of analysing the modulus (vertical and horizontal are not considered) of such errors. The two angles of error can be found through statistical analysis of spherical data. This approach to error evaluation has been used in the earth sciences [

While several authors have contributed to providing accuracy evaluations of 3D laser scanning systems, 3D statistic analysis has been with available scanner data. In brief, the aim of this work is to present a novel proposal to analyse the positional accuracy in TLS data with a more complete analysis than currently available. Our proposal is characterised by some issues: the use of check points (CP) acquired by a technology with more accuracy (Proliner) and error analysis by means of spherical statistics.

In this proposal, an alternative way to analyse the error by means of spherical statistics is presented. The error of a control point “i” is defined as the value e_{i} = c_{i} − c_{j}, where c_{i} is the coordinate point measured and c_{j} the “real” or “true” coordinate, estimated by more precise methods.

The deviation between the true position (true data) and the corresponding point with TLS data is estimated as a vector. Each vector is defined by means of its modulus and two angles (colatitude and longitude—inclination and azimuthal), which allows us to analyse the errors in a 3D space or spherical coordinates, like the type of measured TLS data. Spherical coordinates, also called spherical polar coordinates, are a system of curvilinear coordinates that are natural for describing positions on a sphere (

The pairwise comparison of measured and reference coordinates allows the calculation of the ME, S, RMSE or similar statistics. RMSE is the square root of the average of the set of squared differences between the dataset coordinate values and coordinate values from an independent source of higher accuracy for identical points.

The statistical procedure proposed includes the basic statistic calculations (for modular and angular data) and the main tests for spherical distributions. The error analysis proposed in this paper consists of several parts. In the first part, the modular error component was analysed by linear statistics, similar to the conventional method. In the second part, the angular error components were analysed as well. In the third part, the most innovative part of the analysis, the graphical analysis 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 the distribution error data was done to complete the data analysis.

The error is a vector with three Cartesian components, one for each axis X, Y and Z, and denoted Δ

The basic statistics of the modulus (no angles are considered) are the sample mean, minimum and maximum values, standard deviation and root mean square error (RMSE).

▪ The sample mean (

▪ The range of errors is the difference between the largest (maximum) and the smallest (minimum) calculated error. It is a measure of the spread or the dispersion of the error observations.

There are several measures of dispersion, the most common being the standard deviation. These measures indicate to what degree the individual observations of a data set are dispersed or ‘spread out’ around their mean.

▪ The standard deviation (S or SD) is calculated by taking the square root of the variance. The sample variance is the sum of the squared deviations from their average divided by one less than the number of observations in the data set. It is a measure of the spread or dispersion of a set of data. In the measurements, high precision is associated with low dispersion:

▪ The root mean square error (RMSE), or standard error, is the square root of the average of the set of squared differences between dataset coordinate values and coordinate values from an independent source of higher accuracy for identical points. RMSE is a good measure of accuracy:

The RMSE is an expression equivalent to the SD in the absence of bias (

Both a magnitude and a direction must be specified for a vector quantity, in contrast to a scalar quantity, which can be quantified with just a number. In the same way that a vector has three Cartesian components, it can also be decomposed into polar components: modular distance, vertical angle and horizontal angle. The modular component was analysed in the previous step (Section 2.1).

There are different conventions for considering the angles. In this work, the following convention is proposed because it is the most appropriate to TLS and similar to geographical nomenclature (

▪ The vertical angle (

▪ The horizontal angle (

Angles are considered spherical data, so they are analysed as a point (vector) on a unit sphere [

By examining a sample of n spherical data (_{1}, _{2},…, _{n}), or (_{1}, _{1})…(_{n}, _{n}) (polar coordinates) with corresponding direction cosines, the resultant length of the data is R:

It can be observed that the following are the basic circular statistics for angles:

▪ Mean directions (

▪ Mean module (

Because we work with the unit vectors,

▪ Concentration parameter (

Calculation the estimation of

when

To n ≤ 16

▪ Circular standard deviation (

For more details [

Similar to the analysis of linear and circular data, while analysing spherical data, one should look for uniformity and normality (Fisher distribution) of distributions. The Fisher distribution is a symmetric unimodal distribution and can be considered as an analogue of the von Mises distribution in circular data; and normal distribution in linear data.

For spherical data, two tests are used:

▪ Rayleigh test: It is a uniformity test that detects a single modal direction in a sample of data. This test, developed by Lord Rayleigh in 1919, tests for uniformity against a unimodale alternate model, as assumed for the Fisher distribution. For n < 10, it compares the magnitude of the resultant vector, R, to a critical value. For n > 9, the test statistic, (3R)^{2}/n, is tested with the chi-squared distribution test with three degrees of freedom at the 95% confidence level. La hypothesis of uniformity is rejected if this value is too large.

▪ Beran/Giné test: In 1968, R. J. Beran devised a statistic, based on the angle between pairs of sample directions, for testing uniformity against alternate models that are not symmetric with respect to the centre of the sphere [

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

The equipment used in this work is listed below:

▪ TLS: a 3D long range TLS (TOF) Riegl LMS-Z390i, technical specification are provided in [

▪ 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 (

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 0.02-degree resolution (see

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:

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

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 CP-TLS 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.

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.

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 ^{2} + Δy^{2} + Δz^{2}] 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

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 was developed for this work. The results of the angular (vertical and horizontal) error components are summarized in

The spherical statistics for angular error are mean direction directions (

Although angular error components are summarized in

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,

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.

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 three-dimensional 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).

In the angular error analysis the results convey the angular parameters (mean directions, circular standard deviation and concentration) but on a global form (

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

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 (

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. TIN2008-03063, BIA2009-08012 and INCITE09304262 PR (sponsor and financial support acknowledgment goes here).

TLS Intrinsic Coordinates System.

The wall with the distribution of targets used in the case study.

The room with the targets in the case study.

3D spherical graphic of errors with all vectors in blue and the mean vector in red.

Data in circular diagram (2D).

Map of vectors in the ZX wall-plane, 2D circular graphics analysis for each point in the wall (made with VecSartGraph2D).

Parameters transformation used to align reference systems (Proliner to TLS).

9,057 | 1,703 | 1,055 | −1°, 5438 | −0°, 0133 | −1°, 5879 |

Basic statistic results for modulus analysis of errors.

Mean | −7.6 | −3.83 | −0.26 | 9.53 |

Min | −0.25 | 0.16 | 0.04 | 2.02 |

Max | −10.84 | −10.18 | −7.76 | 18.39 |

RMS | 3.75 | 3.24 | 3.08 | 3.23 |

Standard deviation | 0.51 | 0.44 | 0.42 | 0.44 |

Error analysis was calculated with a sample size 53 ICP.

Δx: X axis error; Δy: Y axis error; Δz: Z axis error; Δr: Radial error.

Spherical statistics results for angular error analysis.

Mean directions ( |
239.7°/−3.8° |

Circular Standard deviation ( |
±27.9° |

Mean module ( |
0.8 |

Concentration parameter ( |
6.7 |

Calculated with Spheristat 2.2.