Automated As-Built Model Generation of Subway

sensors Article

Automated As-Built Model Generation of Subway Tunnels from Mobile LiDAR Data Mostafa Arastounia Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada; [email protected]; Tel.: +1-403-210-7140 Academic Editor: Changshan Wu Received: 10 August 2016; Accepted: 10 September 2016; Published: 13 September 2016

Abstract: This study proposes fully-automated methods for as-built model generation of subway tunnels employing mobile Light Detection and Ranging (LiDAR) data. The employed dataset is acquired by a Velodyne HDL 32E and covers 155 m of a subway tunnel containing six million points. First, the tunnel’s main axis and cross sections are extracted. Next, a preliminary model is created by fitting an ellipse to each extracted cross section. The model is refined by employing residual analysis and Baarda’s data snooping method to eliminate outliers. The final model is then generated by applying least squares adjustment to outlier-free data. The obtained results indicate that the tunnel’s main axis and 1551 cross sections at 0.1 m intervals are successfully extracted. Cross sections have an average semi-major axis of 7.8508 m with a standard deviation of 0.2 mm and semi-minor axis of 7.7509 m with a standard deviation of 0.1 mm. The average normal distance of points from the constructed model (average absolute error) is also 0.012 m. The developed algorithm is applicable to tunnels with any horizontal orientation and degree of curvature since it makes no assumptions, nor does it use any a priori knowledge regarding the tunnel’s curvature and horizontal orientation. Keywords: LiDAR; modeling; tunnel; laser scanning; mobile mapping

1. Introduction The safety of transportation tunnels (such as subway tunnels) is of great importance. Therefore, they need to be regularly monitored to ensure that they are neither deformed, nor displaced excessively. The tunnels are traditionally monitored by regular measurement of a few benchmarks, employing land surveying instruments such as total stations. The traditional technique in which very few benchmarks are recorded is quite time-consuming and costly. As a result, the redundancy of the acquired data is quite low implying that its reliability is also low. Furthermore, the traditional methods require physical presence of land surveyors in the tunnel, which interrupts the regular train schedule. Poor environmental conditions in tunnels can also negatively affect the quality of the collected data [1]. Mobile mapping technology, on the other hand, does not have any of the abovementioned shortcomings since mobile laser scanning (MLS) systems provide quite fast data collection with very high redundancy, which improves the reliability of the acquired data. Moreover, poor environmental conditions do not have a negative impact on MLS systems since they collect data automatically by using active Light Detection and Ranging (LiDAR) sensors. However, the processing of large-volume MLS data can be time-consuming and expensive. Automation of MLS data processing increases its pace and decreases the associated costs. As a result, deformation and displacement analyses can be carried out more frequently, which enhances the safety of tunnels. This study aims to demonstrate a workflow to automatically create as-built models of tunnels from MLS data. As-built models can be utilized for three salient applications: construction progress monitoring (for project management); deformation and displacement analysis (maintenance applications); and planning for future design enhancements. The construction progress can be Sensors 2016, 16, 1486; doi:10.3390/s16091486

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monitoring (for project management); deformation and displacement analysis (maintenance 2 of 20 applications); and planning for future design enhancements. The construction progress can be monitored by comparing the as-built model of a tunnel under construction with the associated as-designedbymodel. The deformation and displacement analyzed by comparing as-built monitored comparing the as-built model of a tunnel are under construction with the the associated models of amodel. tunnel The fromdeformation two (or more) epochs. an by as-built modelthe can be utilized for as-designed anddifferent displacement areFinally, analyzed comparing as-built models the planning of future design enhancements since it provides an accurate and precise survey of the of a tunnel from two (or more) different epochs. Finally, an as-built model can be utilized for the current state of a tunnel. The main focus of this work isan theaccurate generation of such survey as-builtofmodels. The planning of future design enhancements since it provides and precise the current following introduces thefocus typical structure of generation subway tunnels as the models. requiredThe background state of a tunnel. The main of this work is the of such as-built following knowledge. reviewsofthe most tunnels relevantasstudies in this domain, which is followed introduces theSection typical 2structure subway the required background knowledge. Sectionby 2 description of test datasetstudies used ininthis in Section 3. followed The methodology is explained in Section reviews the most relevant thisstudy domain, which is by description of test dataset used4. Section 5 discusses the3.achieved results andisthe conclusions are presented 6. the achieved in this study in Section The methodology explained in Section 4. Sectionin5 Section discusses Subway tunnels have a standard tubular physical shape. As is evident in Figure 1, the cross results and the conclusions are presented in Section 6. sections of subway two primary parts: theshape. upperAs part an in elliptic shape and the Subway tunnelstunnels have a have standard tubular physical is with evident Figure 1, the cross lower part constituting the railroad infrastructure. Though both bottom and upper parts deform sections of subway tunnels have two primary parts: the upper part with an elliptic shape and the underpart the terrain weightthe load, the bottom part will act more stiffly due to the of lower constituting railroad infrastructure. Though both bottom andlarger uppercross partssection deform the rail the most relevant willtotake place cross in thesection upper under theinfrastructure. terrain weightTherefore, load, the bottom part will actdeformations more stiffly due the larger (elliptic-shaped) part, since only thisthe area is part of thedeformations exterior structural support of the of the rail infrastructure. Therefore, most relevant will take place system in the upper tunnel, so only the upper elliptic-shaped of the is considered forsupport the construction ofthe the (elliptic-shaped) part, since only this areapart is part of tunnel the exterior structural system of as-built model. A crucial part of the developed algorithm is extraction of the tunnel cross sections tunnel, so only the upper elliptic-shaped part of the tunnel is considered for the construction of the since deformation and displacement analyses are performed on the basis of the cross sections [1]. as-built model. A crucial part of the developed algorithm is extraction of the tunnel cross sections Crossdeformation sections are and orthogonal to the analyses tunnel’s main axis and are identified their normal vector, since displacement are performed on the basis ofbythe cross sections [1]. semi-major axis, semi-minor axis, and the coordinates of their center. The tunnel main axis is defined Cross sections are orthogonal to the tunnel’s main axis and are identified by their normal vector, as a set of points lying on the center of cross sections. It is required to tunnel note that theaxis lower part of semi-major axis, semi-minor axis, and the coordinates of their center. The main is defined Figure 1 displays a sample railroad infrastructure, which may have different configurations in as a set of points lying on the center of cross sections. It is required to note that the lower part of different subway tunnels. However, these differences in configuration are not relevant to the current Figure 1 displays a sample railroad infrastructure, which may have different configurations in different study. tunnels. However, these differences in configuration are not relevant to the current study. subway

Semi-major axis (a)

Elliptic-shaped part of the tunnel

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infrastructure

Railroad

Center of the cross section

Semi-minor axis (b) Rail tracks

Figure 1. The cross sections of subway tunnels consist of two parts: the upper elliptic-shaped part; and Figure 1. The cross sections of subway tunnels consist of two parts: the upper elliptic-shaped part; the lower part constituting the railroad infrastructure. The arrows represent the weight of the above and the lower part constituting the railroad infrastructure. The arrows represent the weight of the terrain, which is loaded on the upper part of the tunnel cross sections. above terrain, which is loaded on the upper part of the tunnel cross sections.

2. Literature Review MLS systems have two primary units: an imaging unit (LiDAR sensor); and a navigation unit. LiDAR sensors are active sensors that collect data of visible sites and objects by laser range finding. They measure either the time of flight of each emitted laser pulse or the phase shift of the modulating envelope in order to derive the range. They are non-contact measurement instruments that produce a three-dimensional (3D) representation of objects called a point cloud i.e., a set of points

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with 3D coordinates in a global or national coordinate system. LiDAR sensors can be mounted on terrestrial/airborne and static/mobile platforms. On static terrestrial platforms, the surrounding environment is scanned by rotation of the instrument head around its vertical axis and deflection of the laser beam in a vertical plane using a rotating mirror. On mobile platforms, the LiDAR sensor is attached to a moving vehicle and acquires 3D geometrical information in two-dimensional (2D) profiles along the vehicle’s trajectory. Terrestrial LiDAR data typically provides higher accuracy and point sampling due to its shorter spatial offset from objects of interest, compared to airborne LiDAR data. Many researches have recently studied object extraction and modeling from LiDAR data in various environments. Dorninger and Pfeifer [2], Pu and Vosselman [3], and Kim and Habib [4] extracted and modeled buildings from LiDAR data. Jaakkola et al. [5] and Yan et al. [6] extracted and modeled road and road markings from mobile LiDAR data. Arastounia and Lichti [7] automatically recognized key components of electrical substations from terrestrial LiDAR data. Leslar et al. [8] employed mobile LiDAR to create asset inventory of railroad corridors. Oude Elberink and Khoshelham [9] and Arastounia [10] modeled railroad corridors exploiting mobile LiDAR data. The following reviews the studies that specifically focus on the extraction and modeling of tunnels from LiDAR data. Tsakiri et al. [11] and Wang et al. [12] discussed the potential of LiDAR technology for deformation analysis of various objects comprising tunnels. They also proposed some surface modeling approaches for deformation analysis. Gosliga et al. [13] analyzed the deformation of a tunnel by directly fitting a circular cylinder to a tunnel point cloud acquired by a terrestrial laser scanning (TLS) system. Their proposed algorithm can only be applied to straight tunnels with no curvature. Moreover, a grid is fitted to the data in this work and the deformation is analyzed at the grid cell level, which leads to less accurate results. Vezoˇcnik et al. [14] inspected the displacement of underground pipelines. LiDAR technology, global navigation satellite system (GNSS), and precise tacheometry are employed to compute the coordinates of some “representative” points on the pipelines. The pipeline displacement is then studied by monitoring the position of the representative points over a period of time. Qiu and Wu [1] employed an automatic electronic total station (ETS) for tunnel deformation analysis. Although using an ETS reduces the data collection time, the redundancy of data is still much less than that of MLS systems. Delaloye et al. [15] compared the traditional techniques (using total stations) with new techniques (employing LiDAR sensors) for deformation analysis of tunnels. They recommended employing LiDAR sensors since they provide fast data acquisition and full surface characterization and they do not need geo-referencing. Goncalves et al. [16] inspected the cross sectional deformation of a subway tunnel from MLS data by employing image processing techniques. Once the cross sections are extracted, they are converted into 2D images, which are further analyzed by image processing techniques. Han et al. [17] studied the deformation of a road tunnel in which the extraction of cross sections is carried out manually. The deformation is also analyzed by inspecting the displacement of some corresponding points from two different scans. They improved this work by eliminating the cross section extraction step in Han et al. [18]. Kang et al. [19] extracted the cross sections of a circular subway tunnel from TLS data. While the cross sections are successfully extracted, their proposed algorithm is limited to tunnels with circular cross sections. They also employed quadric parametric surface modeling, which makes the algorithm computationally intense and impartial for large datasets such as those obtained using MLS systems. Zai et al. [20] analyzed the deformation of a very small part of a road tunnel using TLS data. They fitted a grid to the tunnel and inspected the displacement of the cell centers from two different epochs. Walton et al. [21] employed the geometric genetic algorithm proposed by Ray and Srivastava [22] in tunnel deformation analysis. Nuttens et al. [23–25] employed a TLS system to inspect the deformation of a tunnel from immediately after placement until three months after construction; investigate a tunnel’s deformation under the influence of estuarine tides; and monitor the ovalization of a tunnel, respectively. The above-mentioned studies utilize two primary approaches for deformation analysis: recording benchmarks, or employing as-built models. The former studies the deformation of a very small portion of a tunnel while the latter portrays a thorough and precise picture of the current state of the tunnel so

construction; investigate a tunnel’s deformation under the influence of estuarine tides; and monitor the ovalization of a tunnel, respectively. The above-mentioned studies utilize two primary approaches for deformation analysis: recording benchmarks, or employing as-built models. The former studies the deformation of a very Sensors 16, 1486 of 20 small 2016, portion of a tunnel while the latter portrays a thorough and precise picture of the current4state of the tunnel so that the deformation of each and every part of the tunnel can be fully investigated. However, the above studies that pursued the modeling approach do not propose a comprehensive that the deformation of each and every part of the tunnel can be fully investigated. However, the above solution as they limit their scope to tunnels with a very specific shape and/or horizontal orientation studies that pursued the modeling approach do not propose a comprehensive solution as they limit or they process the data manually. This contribution aims to create the as-built model of subway their scope to tunnels with a very specific shape and/or horizontal orientation or they process the data tunnels in a fully-automated way with no assumptions or a priori knowledge regarding the manually. This contribution aims to create the as-built model of subway tunnels in a fully-automated curvature and horizontal orientation of the tunnel. Furthermore, MLS data is employed in this study way with no assumptions or a priori knowledge regarding the curvature and horizontal orientation due to its fast acquisition and very high redundancy. Only geometrical information (i.e., 3D of the tunnel. Furthermore, MLS data is employed in this study due to its fast acquisition and very coordinates of points) is utilized for the data processing and the data was processed in its original 3D high redundancy. Only geometrical information (i.e., 3D coordinates of points) is utilized for the data format to enhance the processing efficiency. processing and the data was processed in its original 3D format to enhance the processing efficiency. 3. Dataset 3. Dataset The dataset dataset was HDL 32E LiDAR sensor (Figure 2) that was The was collected collectedby byemploying employinga Velodyne a Velodyne HDL 32E LiDAR sensor (Figure 2) that placed on aon raila car scanned the subway tunnel when the train waswas in motion at 65atkm/h. The was placed rail and car and scanned the subway tunnel when the train in motion 65 km/h. specifications of the employed sensor are summarized in Table 1. The positional accuracy in Table The specifications of the employed sensor are summarized in Table 1. The positional accuracy in1 indicates the accuracy of the measurements made bymade sensor. 0.02 m accuracy Table 1 indicates the accuracy of the measurements byHerein, sensor. the Herein, thepositional 0.02 m positional implies aimplies 68% confidence interval (one (one standard deviation) on on either side measured accuracy a 68% confidence interval standard deviation) either sideofof the the measured coordinates [26]. coordinates [26].

Figure 2. Velodyne 32E LiDAR sensor, sensor, employed employed in in this this study. study. Table 1. Specifications Specifications of of the the Velodyne Velodyne32E 32ELiDAR LiDARsensor. sensor. Table 1. Parameters Parameters Dimensions Dimensionsrate Measurement Measurementrange rate Measurement Measurement range Positional accuracy Positional accuracy Angular resolution Angular resolution

Value Value 0.145 m × 0.086 m 0.145 m(points/s) × 0.086 m 700,000 700,000 (points/s) 100 m 100mm 0.02 0.02 m 1.33° ◦ 1.33

The dataset covers 155 m of a one-way subway tunnel containing more than six million points The 155 m ofThe a one-way subway containing moreorientation than six million (6,012,406dataset pointscovers to be precise). tunnel is along tunnel an arbitrary horizontal i.e., itpoints is not (6,012,406 points to be precise). The tunnel is along an arbitrary horizontal orientation i.e., it is not along any of cardinal directions. The tunnel has an approximate 7.4° vertical slope resulting in along any of m cardinal directions.between The tunnel has an approximate 7.4◦ vertical slope resulting in roughly a 20 height difference its two ends. The dataset contains geometrical information roughly a 20 m height difference betweeninformation its two ends.i.e., Thea dataset contains geometrical information (3D coordinates of points) and intensity quantized measure of the reflected laser (3D coordinates of points) and intensity information i.e., a quantized measure of the reflected beam power. However, only geometrical information is utilized for the processing in this worklaser and beam power. However, only geometrical information is utilized for the processing in this work and the the intensity information is utilized only to provide an appropriate visualization of the dataset. intensity information is utilized only to provide an appropriate visualization of the dataset. Figure 3a Figure 3a shows the entire dataset in an oblique view; Figure 3b portrays the front view of a cross shows entire dataset in an oblique view; Figure 3b portrays thetofront view of walls a cross section; section;the and Figure 3c indicates the equipment and cables attached the tunnel’s that make and Figure 3c indicates the equipment and cables attached to the tunnel’s walls that make the cross the cross sectional shape slightly different from a perfect ellipse. sectional shape slightly different from a perfect ellipse.

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(a)

Equipment and cables attached to tunnel’s walls

Equipment and cables attached to the tunnel’s wall

(b)

(c)

Figure Figure3.3.The Thedataset: dataset:(a) (a)The Theentire entiredataset datasetininan anoblique obliqueview; view;(b) (b)AAside sideview viewofofthe thetunnel tunnelcross cross section. The upper part is elliptic-shaped and the lower part constitutes the railroad infrastructure; section. The upper part is elliptic-shaped and the lower part constitutes the railroad infrastructure; (c) (c)The Theequipment equipmentattached attachedtotothe thetunnel’s tunnel’swall wallmaking makingcross crosssectional sectionalshape shapeslightly slightlydifferent differentfrom fromaa perfect ellipse. perfect ellipse.

The LiDAR sensor employed to acquire data is an oblique viewing sensor and the point spacing The LiDAR sensor employed to acquire data is an oblique viewing sensor and the point spacing is is consequently non-uniform in various parts of the tunnel. Table 2 presents an average point consequently non-uniform in various parts of the tunnel. Table 2 presents an average point sampling sampling on various parts of the tunnel. As is evident in this table, employing an oblique-viewing on various parts of the tunnel. As is evident in this table, employing an oblique-viewing LiDAR LiDAR sensor results in lower point sampling on the ceiling and the floor, compared to the tunnel’s sensor results in lower point sampling on the ceiling and the floor, compared to the tunnel’s side walls. side walls. In addition, no blind spots or occlusions by the support system of the employed LiDAR In addition, no blind spots or occlusions by the support system of the employed LiDAR sensor were sensor were observed in the dataset. observed in the dataset. Table Table2.2.Point PointSampling Samplingon onVarious VariousParts Partsofofthe theTunnel. Tunnel. Object Object Side wall SideCeiling wall Floor Ceiling (around rail tracks) Floor (around rail tracks)

Sampling (Points per/m2) 2 Sampling 1426 (Points per/m ) 509 1426 652 509 652

4. Methodology 4. Methodology The proposed methodology consists of four parts. First, the tunnel’s main axis is automatically identified. Second, the points belonging aretunnel’s extracted and next, the as-built The proposed methodology consiststoofeach four cross parts. section First, the main axis is automatically model is generated and refined. Finally, the constructed is evaluated. 4 shows a identified. Second, the points belonging to each cross section model are extracted and next,Figure the as-built model flowchart for the methodology. is generated anddeveloped refined. Finally, the constructed model is evaluated. Figure 4 shows a flowchart for the developed methodology.

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Figure 4. Flowchart of the developed methodology. methodology.

4.1. Identification of the the Tunnel’s Tunnel’sMain MainAxis Axis 4.1. Identification of The tunnel’s tunnel’smain mainaxis axisisisfound foundby byconnecting connecting points center of the cross sections. First, The points onon thethe center of the cross sections. First, a 3Da 3D plane is fitted to 3D each 3Dspherical local spherical neighborhood with of a radius of one meter using a plane is fitted to each local neighborhood with a radius one meter using a parametric parametric Least Squares adjustment [27]. The general equation of a 3D plane is: Least Squares adjustment [27]. The general equation of a 3D plane is: N x x  N y y  N z z  D  0  N x x  N x y  N z z  1  0  N x x  N x y  N z z  1 (1) Nx x + Ny y + Nz z − D = 0 ⇒ Nx0 x + Ny0 y + Nz0 z − 1 = 0 ⇒ Nx0 x + Ny0 y + Nz0 z = 1 (1) Ny N N where: N x0  Nx x , N y0  Ny and N z0  Nz z where: Nx = DD , Ny = DD and Nz = DD → directions Nx, , Ny, , Nz, represent , representcomponents componentsof of the the plane’s plane’s normal normal vector vector (( N)) along along the the cardinal cardinal directions i.e., → = ( , , )
and D is the normal distance of the plane from the coordinate system center. i.e., N = Nx , Ny , Nz and D is the normal distance of the plane from the coordinate system center. The general equation of parametric least squares adjustment and its closed-form solution are The general equation of parametric least squares adjustment and its closed-form solution are presented presented in Equations (2) and (3), respectively [27]: in Equations (2) and (3), respectively [27]:

Anu X u 1  Yn 1

An×u Xu×1 = Yn×1 T

−11

T

( AT PA PA PY XX = ( A ) ) AAT PY

(2) (2) (3) (3)

where A, A, X thethe number of where X and and Y Y are are the thedesign, design,unknowns unknownsand andconstants constantsmatrices, matrices,respectively. respectively.n is n is number observations (number of data points, in this case) and u is the number of unknowns (number of of observations (number of data points, in this case) and u is the number of unknowns (number of plane parameters, parameters, in in this this case). case). P P denotes denotes the the weight weight matrix matrix of of observations, observations, which which is is calculated calculated by by plane multiplication of the a priori variance factor ( ) and the co-factor matrix of the observations ( ) as follows:

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multiplication of the a priori variance factor (σ02 ) and the co-factor matrix of the observations (Ql ) as follows: 1 P = σ02 Q− (4) l The a priori variance factor is set to one (σ02 = 1) and the elements of the co-factor matrix (σPi ) are equal to the positional accuracy of the employed sensor, which is provided by the sensor manufacturer. Herein, the positional accuracy of the utilized LiDAR sensor is 0.02 m (σPi = 0.02 m) that implies a 68% confidence interval (one standard deviation) on either side of the measured coordinates [26], so the co-factor matrix of the observations and weight matrix are as follows:      Ql =   

σP21 0 0 0 0

0 σP22 0 0 0

0 0 0 0 . 0 0 . 0 0

0 0 0 0 2 σPn

      

⇒

1 P = σ02 Q− l

n×n

σ02 σP2

 1   0   =  0   0  0

0

0

0

σ02 σP2

0

0

0 0

. 0

0 .

0

0

0

2

0



  0     0   0  2 

σ0 2 σPn

(5)

n×n

Considering the equation of a 3D plane in Equation (1), the design, unknowns, and constants matrices are:     x1 y1 z1 1    x   1  Nx0  2 y2 z2       0    A= . , X =  Ny  and Y =  .  (6) . .       .  .  . .  Nz0 3×1 x n y n z n n ×3 1 n ×1 Thus, the normal vectors of all local spherical neighborhoods are calculated by employing the least squares adjustment (Equation (3)) and matrices in Equations (5) and (6). Only horizontal and vertical normal vectors go through the center of cross sections. This is evident in Figure 5 that indicates a sample cross section with an arbitrary horizontal orientation. The Z component of perfectly horizontal and vertical unit vectors is zero and one, respectively. However, due to measurement inaccuracies, herein, a normal vector is considered horizontal if it satisfies the condition in Equation (7) and is considered vertical if it meets the condition in Equation (8) in which Vz represents the Z component →
of the unit normal vector i.e., V = Vx , Vy , Vz . Moreover, if multiple normal vectors are found to meet the conditions in Equations (7) and (8), only the normal vector with the smallest/largest VZ is considered horizontal/vertical, respectively:

|Vz | ≤ 0.01

(7)

|Vz | ≥ 0.99

(8)

That being said, points with horizontal normal vectors that are within one meter 3D Euclidean distance of one another are clustered. This results in two segments (one segment on either side of the tunnel) and each segment contains points with a horizontal normal vector. These two segments are indicated in Figure 6 as one segment in blue and another segment in red on opposite walls of the tunnel (let us call them “blue and red segments”). One should note that Figure 6 illustrates a sample tunnel and the proposed methodology is applicable to tunnels with any degree of curvature since no assumption is made regarding the tunnel’s curvature in the developed algorithm. The one-meter threshold utilized for the neighborhood size of plane fitting and clustering in this section is selected based on the data point sampling, tunnel’s dimensions, and specifications of the employed LiDAR sensor. The neighborhood needs to be large enough to provide reliable information regarding the object

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sensor.The Theneighborhood neighborhoodneeds needstotobe belarge largeenough enoughtotoprovide providereliable reliableinformation informationregarding regardingthe the sensor. object of interest. It is also required to include only points belonging to the object of interest and object of interest. It is also required to include only belonging points belonging to theofobject of and interest and of interest. It is also required to include only points to the object interest exclude excludepoints pointsbelonging belongingtotoother otherobjects. objects.Herein, Herein,the theemployed employeddataset datasetcontains containsmore morethan thanfive five exclude points belonging to other objects. Herein, the employed dataset contains more than five hundred hundred points per square meter (Table 2) implying that a spherical neighborhood with a radius hundred square meter (Table 2) that implying that aneighborhood spherical neighborhood with radius ofof points perpoints squareper meter (Table 2) implying a spherical with a radius ofaone meter one meter provides reliable information regarding the normal vector of the local best-fit plane. one meterreliable provides reliable information regarding the normal of the plane. local best-fit plane. provides information regarding the normal vector of the vector local best-fit Furthermore, Furthermore,there thereare areno noother otherexternal externalobjects objectsininone-meter one-meterneighborhood neighborhoodofofthe thetunnel’s tunnel’swalls walls Furthermore, there are no other external objects in one-meter neighborhood of the tunnel’s walls suggesting that the suggesting that the considered spherical neighborhoods (with a radius of one meter) only include suggesting spherical that the considered spherical (with aonly radius of one meter) only include considered neighborhoods (with neighborhoods a radius of one meter) include points belonging to the pointsbelonging belongingtotothe thetunnel’s tunnel’swalls. walls.One Oneshould shouldnote notethat thatthe thepresence presenceofofpoints pointsbelonging belongingtoto points tunnel’s walls. One should note that the presence of points belonging to equipment on the tunnel’s equipmenton onthe thetunnel’s tunnel’swalls walls(Figures (Figures3b 3band andc)c)ininthe theconsidered consideredneighborhoods neighborhoodsisisinevitable inevitable equipment walls (Figure 3b,c) in the considered neighborhoods is inevitable regardless of the neighborhood size regardless of the neighborhood size since such equipment are attached to the walls. These points regardless of the neighborhood since such equipment are be attached to and the walls. These points since such equipment are attachedsize to the walls. These points will identified excluded by further willbe beidentified identifiedand andexcluded excludedby byfurther furtheranalysis analysisininthe thefollowing followingsteps steps(Section (Section4.3.2) 4.3.2)though. though. will analysis in the following steps (Section 4.3.2) though. Verticalnormal normalvector vector Vertical

ZZ

XX

YY

Horizontal Horizontal normalvectors vectors normal

Figure The normal vector ofofthe local best-fit planes the The grey planes Figure5. Thenormal normalvector vectorof thelocal localbest-fit best-fitplanes planeson onvarious variousparts partsof thetunnel. tunnel.The Thegrey greyplanes planes Figure 5.5.The the on various parts ofofthe tunnel. represent the local best-fit planes and the arrows show their normal vectors. Only horizontal and represent the local best-fit planes and the arrows show their normal vectors. Only horizontal and represent the local best-fit planes and the arrows show their normal vectors. Only horizontal and vertical normal vectors go through the cross section’s center (black point). verticalnormal normalvectors vectorsgo gothrough throughthe thecross crosssection’s section’scenter center(black (blackpoint). point). vertical

Figure6. Identificationofofpoints pointson onthe thetunnel’s tunnel’smain mainaxis axis(black (blackpoints) points)by byfinding findingthe themid-point mid-pointofof Figure Figure 6.6.Identification two closest points from opposite walls of the tunnel (blue and red points). The arrows represent the two points from from opposite The arrows arrows represent two closest closest points opposite walls walls of of the the tunnel tunnel (blue (blue and and red red points). points). The represent the the normalvector vectorof thelocal localbest-fit best-fitplanes. planes. normal normal vector ofofthe the local best-fit planes.

Next,aaapoint point from the blue segment randomly chosen and itsclosest closest point from thered red Next, point from blue segment isisrandomly chosen its point the Next, from thethe blue segment is randomly chosen and itsand closest point from thefrom red segment segment on the opposite wall is found. Enforcing the minimum distance between two points from segment on the opposite wall is found. Enforcing the minimum distance two between two points from on the opposite wall is found. Enforcing the minimum distance between points from opposite opposite walls ensures that they lie on the “exact” opposite sides of the tunnel implying that their opposite walls ensures that they lie on the “exact” opposite sides of the tunnel implying that their walls ensures that they lie on the “exact” opposite sides of the tunnel implying that their mid-point mid-point is located on the center of the cross section, so the center of the query cross section mid-point is located on of thethe center the cross section, the section query cross sectionby isis is located on the center crossofsection, so the centersoofthe thecenter queryofcross is obtained obtained by calculating the midpoint of two points from the “exact” opposite sides of the tunnel. obtained bythe calculating midpoint twothe points from the “exact” sidesThis of the tunnel. calculating midpointthe of two points of from “exact” opposite sides opposite of the tunnel. process is This process is pursued for all points of the blue and red segments and consequently the center all This process is pursued points thesegments blue andand red consequently segments andthe consequently center ofofall pursued for all points offor theall blue andof red center of allthe cross sections cross sections are identified. The tunnel’s main axis is then extracted by connecting the identified cross sections The are tunnel’s identified. Theaxis tunnel’s axis by is then extracted connecting theofidentified are identified. main is thenmain extracted connecting the by identified centers the cross centersofofthe thecross crosssections. sections.Figure Figure66shows showsthree threesample samplepoints pointson onthe thetunnel’s tunnel’smain mainaxis axis(black (black centers

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sections. Sensors 2016,Figure 16, 14866 shows three sample points on the tunnel’s main axis (black points). Afterwards, 9 of 20 the normal vector of each cross section is computed. To this end, the vector connecting two closest points). Afterwards, the normal vector → of each cross section is computed. To this end, the vector points from opposite sides of the tunnel (V) and the vector connecting the center of cross section to the → connecting two closest points from opposite sides of the tunnel ( ) and the vector connecting the topmost part of the cross section (W) are calculated. The topmost point is found by seeking for the center of cross section to the topmost part of the cross section ( ) are calculated. The topmost point point with a vertical normal vector (satisfying the condition in Equation (8)) with the shortest spatial is found by seeking for the point with a vertical normal vector (satisfying condition in → the → offset from thewith cross center. Asoffset indicated Figure vectors V and lie on theinplane that Equation (8)) thesection shortest spatial fromin the cross 7, section center. AsWindicated Figure 7, → goes through the cross section so their cross product defines the cross section’s normal vector ( N): vectors and lie on the plane that goes through the cross section so their cross product defines the cross section’s normal vector ( ): → → → N = V × W (9) (9) N  V W

Figure The calculation calculation of of aa sample sample cross cross section’s section’s normal normal vector vector (Section (Section 4.1). The dashed dashed rectangle rectangle Figure 7. 7. The 4.1). The represents the plane fitted to the cross section in Section 4.2. represents the plane fitted to the cross section in Section 4.2.

4.2. Extraction of of the the Tunnel Tunnel Cross 4.2. Extraction Cross Sections Sections The The cross cross sections sections are are extracted extracted by by fitting fitting 3D 3D planes planes orthogonal orthogonal to to the the tunnel’s tunnel’s main main axis axis and and identifying points within a certain distance from the fitted planes. The plane fitting is carried identifying points within a certain distance from the fitted planes. The plane fitting is carried out out → 0, and 0) and normal vector of cross the cross sections that 3D coordinates coordinatesofofthe thecenter center using 3D (x0(x , y0,0 ,yand z0 )zand normal vector ( N) (of )the sections that were were calculated in the previous Afterwards, points within 0.05 m normal distance calculated in the previous section. section. Afterwards, points within 0.05 m normal distance from (eitherfrom side (either side of) theare fitted plane are considered to the cross section As a of) the fitted plane considered as belonging toas thebelonging cross section under study. As aunder result, study. the tunnel’s result,sections the tunnel’s sections 0.1 m intervals are extracted. This (0.05 m) is equal of to cross at 0.1 cross m intervals areatextracted. This threshold (0.05 m) is threshold equal to half of the interval half ofsection the interval of cross points within mfitted “on either of the fitted cross extraction sincesection pointsextraction within 0.05since m “on either side” 0.05 of the planeside” are considered as plane are to considered as belonging to the crossThat section inspection. Thatis being since belonging the cross section under inspection. beingunder said, since modeling carriedsaid, out at the modeling is carried out at the possible cross section level, the shortest possible interval of cross section cross section level, the shortest interval of cross section extraction corresponds to the most extraction corresponds theshortest most precise modeling shortest possible precise modeling results.toThe possible interval results. of cross The section extraction (thatinterval is twice of as cross large section extraction (that is twicetoas as the query threshold) needs to be selectedofwith respect to as the query threshold) needs belarge selected with respect to the positional accuracy the employed the positional of theaccuracy employed sensor. Herein, the positional the confidence employed sensor. Herein,accuracy the positional of the employed sensor (σ = 0.02 m)accuracy implies aof95% sensor (σof= 2σ 0.02(0.04 m) implies a 95% side confidence interval ofcoordinates. 2σ (0.04 m) on either of the measured interval m) on either of the measured Thus, theside 0.05-m threshold is coordinates. Thus, the 0.05-m threshold is selected to be slightly larger than 2σ (0.04 m). The formula selected to be slightly larger than 2σ (0.04 m). The formula utilized to calculate the normal distance utilized to calculate normal i) between a queryinpoint and the planexin (D a querythe point anddistance the fitted(D plane in indicated Equation (10)fitted in which 0 , yindicated 0 , and z0 i ) between in Equation (10)coordinates in which x0of , ythe 0, and z0 denote 3D coordinates of the cross section and xi, yofi, denote the 3D cross sectionthe center and xi , yi , and zi denote the 3Dcenter coordinates and zi denote 3D coordinates query In Equation (10), ,of the and the a query point. the In Equation (10), Nx ,ofNyaand Nz ,point. represent the components cross, represent section’s unit components of the cross section’s unit normal vector: normal vector: Di = Nx ( xi − x0 ) + Ny (yi − y0 ) + Nz (zi − z0 ) (10) Di  N x ( xi  x0 )  N y ( yi  y0 )  N z ( zi  z0 ) (10)

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4.3. As-Built Model Generation 4.3.1. Modeling by Least Squares Adjustment The ellipse and circle are two common cross sectional shapes of subway tunnels. Considering that a circle is a specific form of an ellipse (with equal semi-major and semi-minor axes), an ellipse is fitted to the points belonging to each extracted cross section using least squares adjustment. Least squares fitting algorithms generally fall into two primary categories: algebraic methods and geometric methods. Algebraic methods are more common due to their simplicity, linear nature, and high computational efficiency. Geometric methods provide a more robust fitting approach by solving a non-linear problem. Though geometric approaches are computationally more expensive, they provide a more reliable solution in the presence of outliers in data [22]. This is important since Least Squares adjustment is quite sensitive to outliers implying that if outliers are not removed, they will have a significant negative impact on the generated model. In subway tunnel point clouds, outliers are introduced by the ever-present equipment attached to the tunnel’s walls (Figure 3b) and random irregularities in the tunnels’ cross sectional shape (Figure 3c), which make the tunnels’ cross sectional shape slightly different from a perfect ellipse. Such outliers are easily identifiable due to their large deviation from the actual data points. Therefore, herein, an algebraic (linear) least squares fitting method is employed, which is followed by sequential application of residual analysis and Baarda’s data snooping method [28] to refine the modeling parameters (Section 4.3.2). This results in a precise modeling as well as a computationally efficient algorithm since a linear least squares adjustment is applied to the outlier-free data. Fitting an ellipse to a cross section with an arbitrary orientation requires the general form of quadric surfaces (Equation (11)) and application of complicated constraints [29–32], which makes the algorithm computationally very intense. However, if the normal vector of the cross section is along one of the cardinal directions, a simple ellipse model can be employed rather than the general form of quadric surfaces. Employing an ellipse model significantly improves the computational efficiency since the number of unknowns decreases from ten parameters (coefficients of Equation (11)) to two parameters (semi-major and semi-minor axes of the ellipse). Thus, each cross section is automatically →

rotated so that its normal vector ( N) is aligned along a cardinal direction (in this case Y axis) in the dataset’s coordinate system. The rotation is carried out by two sequential rotations: one rotation around Z axis; and one rotation around X axis, as in Equation (12). The rotation is automated since the →

rotation angles (θ z and θ x ) are computed using the normal vector ( N) of cross sections, which were calculated in the previous section. The rotation is executed as in Equation (12) in which Rx and Rz denote the rotation matrices around X and Z axis, respectively. Pi denotes 3D coordinates of a sample point and Pi0 represents the respective rotated coordinates: Ax2 + By2 + Cz2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 

  xi0 1  0    yi  =  0 zi0 0 | {z } | Pi0

 0 0 cos(θz ) cos(θz )  cos(θ x ) cos(θ x )  cos(θz ) cos(θz ) cos(θ x ) cos(θ x ) 0 0 {z }| {z Rx

Rz

where: θz = arctan( θ x = arctan( q

Nx ) Ny Nz

Nx2 + Ny2

(11)

 0 xi  0   yi 1 zi } | {z Pi

  

(12)

}

(13)

)

(14)

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Afterwards, an ellipse is fitted to the upper part of each and every cross section using the least squares adjustment. The upper part is defined as points (belonging to the queried cross section) whose height is equal to or higher than height of the cross section center. The equation of an ellipse orthogonal to Y axis is shown in Equation (15) and the associated design, unknowns, and constants matrices are presented in Equation (16) in which a denotes the semi-major axis of the fitted ellipse; b represents the semi-minor axis of the fitted ellipse; xi and zi are the coordinates of data points; and n is the total number of data points belonging to the queried cross section: 1

    A=  

z21 z22 . . z2n

1

c= 2 and d= 2 z2 x2 + 2 = 1 a → b cz2 + dx2 = 1 2 a b   x12  ! x22    c   , X= and Y =  .    d 2×1  .  2 x n n ×2

(15) 1 1 . . 1

      

(16) n ×1

The semi-major axis (a) and semi-minor axis (b) of all cross sections are then calculated by applying Least Squares adjustment (Equation (3)) to the matrices in Equation (16). Next, the preliminary model is generated by calculating the coordinates of its points using Equation (17) and the calculated parameters i.e., 3D coordinates of the cross section center (x0 , y0 and z0 ) and the associated axes (a and b).  ◦ ◦   zi = z0 + acos(θi ) 1 ≤ θi ≤ 360 ( z − z0 )2 ( x − x0 )2 (17) + = 1 → xi = x0 + bsin(θi ) 1◦ ≤ θi ≤ 360◦  a2 b2  y = y 0 i It is required to note that the rotation angles of each cross section are specifically computed using the components of the respective normal vector (Equations (13) and (14)). This implies that the modeling is carried out at the cross section level and makes no assumption regarding the tunnel’s curvature and that is why the developed algorithm is applicable to tunnels with any degree of curvature. 4.3.2. Model Refinement Once the preliminary model is constructed, the residuals are computed using Equation (18) and the 90th percentile residual of each cross section is calculated as the cut-off threshold. Then, points whose residual is larger than this threshold are identified as outliers and are removed: V = Y − AX

(18)

Afterwards, Baarda’s data snooping method is used to remove the remaining outliers based on standardized residuals, which are computed as follows: nVi =

|V | pi σ0 × QV (i, i )

(19)

where Vi and nVi are the residual and standardized residual of ith point and QV (i,i) represents the element on the ith row and ith column of the co-factor matrix of residuals (QV ). The co-factor matrix of residuals (QV ) is calculated in the following equation: QV = Ql − A( A T PA)

−1

AT

(20)

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According to Baarda’s method, points whose standardized residual is larger than four are considered as outliers and are not considered for further processing. Therefore, outliers including points belonging to the tunnel’s equipment are identified and eliminated by residual analysis and Baarda’s method. Finally, the least squares adjustment is applied to the outlier-free data and the final as-built model is generated. Once the modeling is accomplished, the extracted cross sections and their associated models are rotated back to their original orientation so that the original shape, curvature, and orientation of the tunnel are preserved in the constructed as-built model. Moreover, the dataset’s coordinate system does not make any difference in this particular section since this section only calculates the dimensions of each cross section i.e. its semi-major and semi-minor axes. This is carried out by: aligning each cross section along a cardinal direction (Y axis); calculating its dimensions (by ellipse fitting and model refinement); and rotating it back to its original orientation. The rotation of cross sections is applied only to enhance the computational efficiency since it reduces the number of modeling unknowns (coefficients) from ten to two unknowns. Since a circle is a specific form of an ellipse (with equal semi-major and semi-minor axes), the developed algorithm is applicable to transportation tunnels with either elliptic or circular cross sections. 4.4. Evaluation of the Generated Model 4.4.1. Dimensions of the Extracted Cross Sections The dimensions of cross sections can be fully investigated by calculating their area and eccentricity, which indicates the deviation of the tunnel’s cross sectional shape from being circular. Thus, once the semi-major and semi-minor axes and their standard deviations are obtained, the area (A) and eccentricity (e) of cross sections along with their standard deviations (σA and σe ) are calculated using Equations (21)–(24). The standard deviations of axes are calculated by Equations (25)–(27): πab 2

A= r

1−

e=

(21)

b2 a2

(22)

πab Law of Error Propagation −−−−−−−−−−−−−→ 2 − q ∂f 2 2 ∂f 2 2 ( ∂a ) σa + ( ∂b ) σb → σA = π2 (b2 σa2 ) + ( a2 σb2 )

A = f ( a, b) =

2 = σA

q

2

Law of Error Propagation

∂g 2

(23)

∂g 2

1 − ba2 −−−−−−−−−−−−−−→σe2 = ( ∂a ) σa2 + ( ∂b ) σb2 → q q 2 2 2 2 σe = ( ab3 e σa ) + ( ab2 e σb ) = ab2 e ( ba2 σa2 ) + σb2

e = g( a, b) =

(24)

4.4.2. Precision of the Estimated Parameters Once modeling was accomplished, the standard deviation of the semi-major and semi-minor axes is calculated using the following formulas. The standard deviation of the axes are quite important since they indicate how precisely the axes are computed: σc2 σdc

σcd σd2

!

_2

= σ 0 ( A T PA)

−1

(25)

1 1 Law of Error Propagation df 2 c = 2 → a = f (c) = √ −−−−−−−−−−−−−−→σa2 = ( ) σc2 → σa = dc a c 2

d=

1 1 Law of Error Propagation dg → b = g(d) = √ −−−−−−−−−−−−−−→σb2 = ( ) σd2 → σb = dd b2 d

r

s

σc2 4c3

(26)

σd2 4d3

(27)

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where: _2 σ0

=

V T PV DF

(28)

In the above equations, DF represents the degree of freedom and σˆ 02 denotes the a posteriori variance factor. DF is calculated as the subtraction of unknowns from the number of equations. The number of unknowns of all cross sections is two (semi-major and semi-minor axes) and the number of equations is the number of data points belonging to each cross section. Thus, DF of each cross section is different from others and is computed as the number of data points belonging to each cross section minus 2. The average number of cross section points is 2090 (for the dataset employed), which corresponds to an average DF of 2088. Furthermore, the normal distance of points from the model (i.e., absolute error) is computed as another measure to evaluate the constructed as-built model. The normal distance of points from the model is calculated as in Equation (29) in which xi , yi and zi denote the coordinates of a query point and xm , ym and zm represent the coordinates of the closest point on the model from the query point: Dist =

q

( x i − x m )2 + ( y i − y m )2 + ( z i − z m )2

(29)

5. Results and Discussion 5.1. Results of Cross Section Extraction 1551 cross sections at 0.1 m intervals were successfully extracted from the dataset described in Section 3. Figure 8 shows the tunnel’s centerline and extracted cross sections at one-meter and ten-meter intervals. As was mentioned in Section 1, the cross section centers (shown as white points) constitute the tunnel’s main axis. It should be noted that the extraction and modeling of cross sections are performed at 0.1 m intervals. However, cross sections at such short interval (0.1 m) are so dense that they cannot be conveniently visualized and differentiated from one another. Thus, Figure 8 depicts the extracted cross sections at larger intervals so that they can be conveniently seen. The spatial variations and histograms of the semi-major axis, semi-minor axis, area, and eccentricity of 1551 extracted cross sections are demonstrated in Figures 9 and 10, respectively. The achieved results show that cross sections have an average semi-major axis of 7.8508 m with a standard deviation of 0.2 mm and an average semi-minor axis of 7.7509 m with a standard deviation of 0.1 mm. Their average area and eccentricity are also 95.5838 m2 with a standard deviation of 0.0003 m2 and 0.17606 with a standard deviation of 0.00002, respectively. Such parameters of a subway tunnel should ideally be constant for all cross sections meaning that the dimensions of cross sections do not vary throughout the tunnel. Although there are some fluctuations in Figure 9, the vast majority of the presented measures have a small deviation from their mean value. This is also evident in Figure 10 that depicts a great portion of semi-major axes are between 7.8 m and 7.9 m and the semi-minor axes predominantly range from 7.7 m to 7.8 m. This indicates that the dimensions of cross sections are quite consistent, considering the positional accuracy of the employed LiDAR sensor (0.02 m). The uniform dimensions of cross sections is confirmed by Figure 9c,d and Figure 10c,d where the area and eccentricity of a large number of cross sections are centered around 95.6 m2 and 0.15 (unit-less), respectively. However, a few cross sections with inconsistent dimensions exist, which can be due to one or both of the following two reasons. First, the tunnel’s as-built and as-designed plans might not be fully-compatible suggesting that the tunnel is not constructed exactly according to the designed plans. Second, the tunnel may be experiencing deformation. The former can be investigated by comparing the as-built models with as-designed plans and the latter can be inspected by comparing as-built models of a tunnel from two different epochs. However, neither of them fits into the scope of this study since this work is primarily aimed at automated construction of such as-built models. Table 3 summarizes the average dimensions of cross sections including the mean and standard deviation of axes, area, and eccentricity. The magnitude of the semi-major and semi-minor axes of

Dist  ( xi  xm )  ( yi  ym )  ( zi  zm ) 5. Results and Discussion Sensors 2016, 16, 5.1. Results of 1486 Cross

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1551 cross sections at 0.1 m intervals were successfully extracted from the dataset described in Section 3. Figure shows tunnel’s and extracted crosscross sections at one-meter and each cross section 8are quitethe close to one centerline another, implying an elliptic sectional shape that is ten-meter intervals. As was in by Section 1.1, theeccentricity cross section as white close to a circular shape. Thismentioned is confirmed the average thatcenters is close (shown to zero (0.17606). points) constituteregarding the tunnel’s axis. It of should be noted the extraction and modeling of cross The information the main dimensions a tunnel can bethat employed in many applications such as sections are performed at 0.1 manalysis. intervals. However, cross sectionsofatstandard such short intervalof(0.1 m)axes are displacement and deformation Moreover, the calculation deviation both soofdense they cannot conveniently visualized and one differentiated Thus, is greatthat importance whenbe deformation occurs only along of the axesfrom sinceone suchanother. deformations Figure 8 depicts the extracted cross sections at larger intervals so that they can be conveniently seen. cannot be reflected by the eccentricity.

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(a)

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(b)

(c) Figure 8. The sections: (a) cross Figure 8. The extracted extracted centerline centerline and and cross cross sections: (a) The The tunnel’s tunnel’s centerline centerline (center (center of of cross sections) indicated in white color; color; (b) (b) The The extracted extracted cross cross sections sections at at one-meter one-meter intervals; intervals; (c) The sections) is is indicated in white (c) The extracted cross sections at ten-meter intervals. extracted cross sections at ten-meter intervals.

The spatial variations and histograms of the semi-major axis, semi-minor axis, area, and eccentricity of 1551 extracted cross sections are demonstrated in Figures 9 and 10, respectively. The achieved results show that cross sections have an average semi-major axis of 7.8508 m with a standard deviation of 0.2 mm and an average semi-minor axis of 7.7509 m with a standard deviation of 0.1 mm. Their average area and eccentricity are also 95.5838 m2 with a standard deviation of 0.0003 m2 and 0.17606 with a standard deviation of 0.00002, respectively. Such parameters of a subway tunnel should ideally be constant for all cross sections meaning that the dimensions of cross sections do not vary throughout the tunnel. Although there are some fluctuations in Figure 9, the vast majority of the presented measures have a small deviation from their mean value. This is also

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Semi-major Semi-major axis axis (m) (m)

Semi-minor Semi-minor axis axis (m) (m)

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Cross sections’ number

Cross sections’ number (b)

Area Area (m (m22))

Eccentricity Eccentricity

(a)

Cross sections’ number

Cross sections’ number

(c)

(d)

Number Number of of cross cross sections sections

Number Number of of cross cross sections sections

Figure 9. The dimensions of the extracted cross sections: (a) Semi-major axis; (b) Semi-minor axis; Figure 9. The dimensions of the extracted cross sections: (a) Semi-major axis; (b) Semi-minor axis; (c) Area; (d) Eccentricity of the extracted cross sections. (c) Area; (d) Eccentricity of the extracted cross sections.

Semi-major axis (m)

Semi-minor axis (m) (b)

Number Number of of cross cross sections sections

Number Number of of cross cross sections sections

(a)

Area (m22) (c)

Eccentricity (d)

Figure10. 10. Histogram of of thethe cross sections’ dimensions: (a) The (a) histogram of semi-major axis, (b) Theaxis; Figure Histogram cross sections’ dimensions: The histogram of semi-major histogram of semi-minor axis; (c) The histogram of area; (d) The histogram of eccentricity. The (b) The histogram of semi-minor axis; (c) The histogram of area; (d) The histogram of eccentricity. histogram bin width of the axes, area, and eccentricity is 0.01 m, 0.1 m, and 0.01 (unit-less), The histogram bin width of the axes, area, and eccentricity is 0.01 m, 0.1 m, and 0.01 (unit-less), respectively. The bin widths are selected with respect to the dataset point sampling, the tunnel’s respectively. The bin widths are selected with respect to the dataset point sampling, the tunnel’s dimensions and the employed LiDAR sensor specifications. dimensions and the employed LiDAR sensor specifications.

deviation of axes, area, and eccentricity. The magnitude of the semi-major and semi-minor axes of each cross section are quite close to one another, implying an elliptic cross sectional shape that is close to a circular shape. This is confirmed by the average eccentricity that is close to zero (0.17606). The information regarding the dimensions of a tunnel can be employed in many applications such as displacement and deformation analysis. Moreover, the calculation of standard deviation of both axes Sensors 2016, 16, 1486 16 of 20 is of great importance when deformation occurs only along one of the axes since such deformations cannot be reflected by the eccentricity. Table 3. Average Dimensions of the Extracted Cross Sections. Table 3. Average Dimensions of the Extracted Cross Sections. Statistics Mean Standard Deviation Statistics Mean Standard Deviation Semi-major axis (m)(m) 7.8508 0.0002 Semi-major axis 7.8508 0.0002 Semi-minor axis (m) 7.7509 0.0001 Semi-minor axis (m) 7.7509 0.0001 95.5838 0.0003 Area (m2 ) 2 95.5838 0.0003 Area (m ) Eccentricity 0.17606 0.00002 Eccentricity 0.17606 0.00002

5.2. The Generated Generated As-Built 5.2. The As-Built Model Model Figure Figure 11 11 demonstrates demonstrates the the front front and and oblique oblique views views of of the the generated generated model model at at 0.1 0.1 m m intervals. intervals.

(a)

(b)

Figure 11. Two Two different differentviews viewsofofthe theconstructed constructedmodel: model:(a)(a) front view; oblique view of AA front view; (b)(b) an an oblique view of the the model. model.

The standard standard deviation deviation of of the the semi-major semi-major and and semi-minor semi-minor axes axes indicates indicates how how precisely precisely they they are are The estimated. For section are are estimated estimated as as 7.8485 7.8485 m m and and 7.7364 7.7364 m m with with aa estimated. For instance, instance, the the axes axes of of 700th 700th cross cross section standard deviation of 0.1 mm for both axes. The standard deviations of all computed axes are standard deviation of 0.1 mm for both axes. The standard deviations of all computed axes are indicated indicated in Figure 12. The average standard deviation of semi-major axes and semi-minor axes are in Figure 12. The average standard deviation of semi-major axes and semi-minor axes are 0.2 mm 0.2 mm and 0.1 mm, respectively. As isin evident 12, the majority of the standard and 0.1 mm, respectively. As is evident Figurein 12,Figure the majority of the standard deviationsdeviations are quite are quite small (smaller than 0.5 mm) suggesting that the generated model is quite However, small (smaller than 0.5 mm) suggesting that the generated model is quite precise.precise. However, some some sections cross sections (indicated with arrows in Figure 12) slightly have slightly standard deviations. cross (indicated with arrows in Figure 12) have higherhigher standard deviations. Such Such inconsistencies are introduced by non-flat surfaces on some parts of the tunnel, which inconsistencies are introduced by non-flat surfaces on some parts of the tunnel, which are displayedare in displayed Figure 13. Moreover, the point sampling ofmeters the firstoften is less than Figure 13. in Moreover, the point sampling of the first ten themeters tunnelof is the lesstunnel than the tunnel’s the tunnel’s average sampling and leadstandard to higherdeviations standard deviations forhundred the first hundred cross average sampling and this lead tothis higher for the first cross sections sections (illustrated by dashed grey brackets in Figure 12). The lower sampling of the first ten meters (illustrated by dashed grey brackets in Figure 12). The lower sampling of the first ten meters of the of the can tunnel can be visualized 14 that the pointofsampling of all cross sections. tunnel be visualized in Figurein 14Figure that depicts thedepicts point sampling all cross sections. Furthermore, a gradual increaseofinpoints the number pointsofalong the length of the tunnel can 14. be aFurthermore, gradual increase in the number along theoflength the tunnel can be observed in Figure observed in Figure 14. The author believes this can be the result of one or both of the following The author believes this can be the result of one or both of the following reasons: reasons: 1. Speed reduction of the train carrying the LiDAR sensor 1. Speed reduction of the train carrying the LiDAR sensor 2. Change in in the the setting setting of of the the employed employed LiDAR LiDAR sensor sensor during during data data collection collection 2. Change However, the the author authorwas wasnot notable abletoto verify these assumptions since he was not involved in verify these assumptions since he was not involved in data data acquisition. acquisition. Figure 15 shows the histogram of the normal distance of points from the as-built model (absolute errors). Recall that the positional accuracy of the employed LiDAR sensor (σ = 0.02 m) implies a 95% confidence interval of 2σ (0.04 m) on either side of the measured coordinates. This is confirmed by Figure 15 that indicates 95% of points lie within 0.04 m of the generated model. The average normal distance of all points from the model (i.e., average absolute error) is 0.012 m.

Standard Standard deviation deviation of of semi-major axis (mm) semi-major axis (mm)

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Lessprecise precise Less

Largerresiduals residualsintroduced introduced Larger

modellingdue duetotopoor poor modelling

bynon-flat non-flatsurfaces surfaceson onthe the by

pointsampling sampling point

tunnel’swalls walls tunnel’s

Crosssections’ sections’number number Cross (a) (a)

Standard Standard deviation deviation of of semi-minor axis (mm) semi-minor axis (mm)

Lessprecise precise Less modellingdue duetotopoor poor modelling pointsampling sampling point

Largerresiduals residuals Larger introducedby bynon-flat non-flat introduced surfaceson onthe the surfaces tunnel’swalls walls tunnel’s

Crosssections’ sections’number number Cross (b) (b) Figure 12. 12.The Thestandard standard deviation the cross cross sections’ sections’ axes: (a) The The standard standard deviations deviations Figure The standard deviation ofofcross the axes: ofof Figure 12. deviation of the sections’ axes: (a) The (a) standard deviations of semi-major semi-major axis; (b)The The standardof deviations semi-minoraxis. axis. semi-major axis; (b) standard deviations ofofsemi-minor axis; (b) The standard deviations semi-minor axis.

Non-flatsurfaces surfaceson on Non-flat thetunnel’s tunnel’swalls walls the

Non-flat Non-flat surfaceson onthe the surfaces tunnel’s’swalls walls tunnel

Figure 13. Non-flat surfaces on the tunnel’s walls. Figure13. 13.Non-flat Non-flatsurfaces surfaceson onthe thetunnel’s tunnel’swalls. walls. Figure

Number of points Number of points

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Cross sections’ number

Figure 14. The point sampling of cross sections. The sampling of the first ten meters (corresponding to the first hundred cross sections) is lower than that of the other cross sections.

Figure 15 shows the histogram of the normal distance of points from the as-built model sections’ (absolute errors). Recall that the positionalCross accuracy of number the employed LiDAR sensor (σ = 0.02 m) implies a 95% confidence interval of 2σ (0.04 m) on either side of the measured coordinates. This is Figure14. 14.The Thepoint pointsampling sampling of of cross cross sections. sections. The meters Figure The sampling sampling of of the thefirst firstten meters(corresponding (correspondingto confirmed byhundred Figure 15 that indicates 95%than of points lie other within 0.04 m ten of the generated model. The the first cross sections) is lower that of the cross sections. to the first hundred cross sections) is lower than that of the other cross sections. average normal distance of all points from the model (i.e., average absolute error) is 0.012 m.

Number of points Number of points

Figure 15 shows the histogram of the normal distance of points from the as-built model (absolute errors). Recall that the positional accuracy of the employed LiDAR sensor (σ = 0.02 m) implies a 95% confidence interval of 2σ (0.04 m) on either side of the measured coordinates. This is confirmed by Figure 15 that indicates 95% of points lie within 0.04 m of the generated model. The average normal distance of all points from the model (i.e., average absolute error) is 0.012 m.

Normal distance (m)

Figure Figure15. 15.The Thehistogram histogramofofnormal normaldistance distanceofofall allpoints pointsfrom fromthe theconstructed constructedmodel. model.

6.6.Conclusions Conclusions This novel fully-automated fully-automatedmethodology methodologyfor foras-built as-builtmodel modelgeneration generation Thiscontribution contribution presents presents aa novel of Normal distance (m) of subwaytunnels tunnelsemploying employing3D 3DMLS MLSdata. data.The Thedataset dataset used used in in this this work work covers subway covers 155 155 m mof ofaaone-way one-way subway containing more than sixsix million points, are by aby Velodyne HDLHDL 32E subwaytunnel tunnel containing more than million points, which are a Velodyne Figure 15. The histogram of normal distance of allwhich points fromacquired theacquired constructed model. device. Although the intensity information is alsois captured, only only geometrical information (3D 32E device. Although the intensity information also captured, geometrical information coordinates of points) is employed forfor data processing. 6. Conclusions (3D coordinates of points) is employed data processing.The Thedeveloped developedalgorithm algorithmautomatically automatically identifies the main which isisthen for ofof cross sections. Next, a identifies thetunnel’s tunnel’spresents mainaxis, axis, which thenutilized utilizedmethodology forthe theextraction extraction cross sections. Next, This contribution a novel fully-automated for as-built model generation preliminary model is is generated byby fitting an ellipse totoeach cross section. The preliminary preliminary model generated fitting eachextracted extracted cross section. The ofa subway tunnels employing 3D MLS data. an Theellipse dataset used in this work covers 155 m ofpreliminary a one-way model is refined by using residual analysis and Baarda’s data snooping method to remove outliers. model is refined by using residual analysis and Baarda’s data snooping method to remove outliers. subway tunnel containing more than six million points, which are acquired by a Velodyne HDL 32E The final model is then generated by applying the least squares adjustment to the outlier-free data. The final model is then generated by applying the least squares adjustment to the outlier-free device. Although the intensity information is also captured, only geometrical information data. (3D The results indicate that tunnel’s main 1551 cross at intervals Theobtained obtained thatthe thefor tunnel’s mainaxis axisand and 1551 crosssections sections at0.1 0.1m m intervalsare are coordinates ofresults points)indicate is employed data processing. The developed algorithm automatically successfully extracted. Cross sections have an average semi-major axis of 7.8508 m with a standard successfully Cross sections an average axis of of 7.8508 with a standard identifies the extracted. tunnel’s main axis, whichhave is then utilized semi-major for the extraction crossmsections. Next, a deviation of 0.2 mm and an average semi-minor axis of 7.7509 m m with with aa standard standard deviation deviationof of0.1 mm. deviation of 0.2 mm and an average semi-minor axis of 7.7509 preliminary model is generated by fitting an ellipse to each extracted cross section. The preliminary 0.1 The modelingshow results that the dimensions of cross are quitesuggesting consistent Themm. modeling thatshow the dimensions of crossdata sections are sections quite consistent model is refinedresults by using residual analysis and Baarda’s snooping method to remove outliers.a suggesting a minimal variation in the tunnel’s cross sectional shape and dimensions. The minimal variation in thegenerated tunnel’s cross sectionalthe shape dimensions. The average normal average distance The final model is then by applying leastand squares adjustment to the outlier-free data. normal distance of the all points from model the constructed model (average absolute error) is also 0.012since m. of all points from constructed (average absolute error) is also 0.012 m. Moreover, The obtained results indicate that the tunnel’s main axis and 1551 cross sections at 0.1 m intervals are Moreover, since subway tunnels are either ellipticor circular-shaped and a circle is a specific form subway tunnels are either ellipticor circular-shaped a circle isaxis a specific form an ellipse (with successfully extracted. Cross sections have an averageand semi-major of 7.8508 mofwith a standard of an ellipse (with and equal semi-majoraxes), and semi-minor axes), the developed algorithm is applicable to equal semi-major semi-minor the developed algorithm is applicable to tunnels with either deviation of 0.2 mm and an average semi-minor axis of 7.7509 m with a standard deviation of tunnels with either cross elliptic or circular sections. The proposed methodology istoalso applicable to elliptic circular Thecross proposed methodology also applicable tunnels with any 0.1 mm.orThe modeling sections. results show that the dimensions ofiscross sections are quite consistent horizontal aorientation and degree of curvature sincesectional it makesshape no assumptions, nor does use any suggesting minimal variation in the tunnel’s cross and dimensions. Theitaverage priori knowledge regarding the curvature and horizontal orientation of the tunnel. normal distance of all points from the constructed model (average absolute error) is also 0.012 m. This since studysubway can be tunnels pursuedareineither threeellipticaspects:or construction progress monitoring (in project Moreover, circular-shaped and a circle is a specific form displacement and deformation analysis and for the ofmanagement); an ellipse (with equal semi-major and semi-minor axes), (maintenance the developedapplications); algorithm is applicable to planning of future Thesections. construction progress can be monitored by comparing tunnels with either design ellipticenhancement. or circular cross The proposed methodology is also applicable the to

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as-built and as-designed models of a tunnel. The displacement and deformation can be analyzed by: scanning the tunnel in two or more epochs; generating the tunnel’s as-built model in each epoch using the developed algorithm; and comparing the dimensions of extracted cross sections (for deformation analysis) and 3D coordinates of cross sections center (for displacement analysis). Deformation and displacement analyses require the data from different epochs to be either in the same local reference system or in a national/global reference system. Finally, a tunnel’s as-built model can be used for the planning of future design enhancement since it provides a very precise and accurate survey of its current state. Acknowledgments: USLASER is highly acknowledged for data collection. Conflicts of Interest: The author declares no conflict of interest.

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