Measurement 51 (2014) 259–264

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Measurement

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Point cloud comparison under uncertainty. Application to beambridge measurement with terrestrial laser scanning

http://dx.doi.org/10.1016/j.measurement.2014.02.0130263-2241/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +34 985458027; fax: +34 985458188.E-mail address: ordonezcelestino@uniovi.es (C. Ordóñez).

Francisco de Asís López a, Celestino Ordóñez a,⇑, Javier Roca-Pardiñas b, Silverio García-Cortés a

a Department of Mining Explotation and Prospecting, University of Oviedo, 33600 Oviedo, Spainb Department of Statistics and Operations Research, University of Vigo, 36208 Vigo, Spain

a r t i c l e i n f o

Article history:Received 5 November 2013Received in revised form 14 January 2014Accepted 4 February 2014Available online 18 February 2014

Keywords:BootstrapKernel smoothingLaser scannerPoint cloud

a b s t r a c t

In this paper a methodology to compare two point clouds of an object, obtained withdifferent equipment or in different conditions, is proposed. First, the point clouds areregistered to the same reference system using an iterative algorithm that performs a rigidbody transformation. Then, the standard deviation of the measurements is estimated, inorder to evaluate the uncertainty in the measurements. Afterwards, two surfaces areadjusted to each of the point clouds by means of a kernel smoothing technique andcompared. Finally, the effect of uncertainty in the point coordinates is considered by meansof a bootstrap analysis.

The methodology was used to compare two point clouds of a beam bridge measuredusing two different types of terrestrial laser scanner (time-of-flight and phase-shift basedsystems). According to the results obtained, some statistically significant differences existbetween both point clouds.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

There are many practical situations where technicianshave to make decisions about the most appropriate equip-ment for a particular task. For instance, they have tochoose between different terrestrial laser scanners (TLS)available on the market. Each of these devices has itsown technical characteristics, accuracy and precision ofthe point coordinates being two of the most importantfor the technicians. Usually the cost of the measuringequipment increases with accuracy and precision.

Both characteristics, accuracy and precision, are givenby the manufacturer for an isolated point. However, terres-trial laser scanners are not designed to measure isolatedpoints but to construct surface models of the measuredobjects [1]. In fact, these devices are able to measure

millions of points in few minutes. By adjusting surfacesto point clouds, factors such as the point density becomesrelevant for the accuracy of the models [2–4].

Accuracy and precision of TLS measurements influencethe quality of the surface model adjusted to a point cloudrepresenting a real object. Consequently, point cloud com-parison can be used to detect differences between TLS sys-tems. Point cloud comparison is also one of the toolsimplemented in many point cloud processing software[5,6] given its practical use in inspection work [7]. Changedetection is another common application of point cloudcomparison [8,9].

Although point cloud comparison can be performeddirectly [10], many approaches for point cloud comparisonstart adjusting surfaces to at least one of the two pointclouds and transforming the initial irregular meshes to acommon regular mesh [11]. Surface matching is thenaccomplished. The ICP (iterative closest point) algorithmis one of the basic algorithms for surface matching [12].Once the two point clouds are registered in the same

260 F. de Asís López et al. / Measurement 51 (2014) 259–264

coordinate system, the differences between the two pointclouds are determined by calculating the vectors betweencorresponding nodes of the two meshes.

Usually, software for point cloud comparison providesgraphics of displacement vectors and statistics concerningthe relative position of two point clouds. This informationis useful to estimate the magnitude and location of thedistance vectors between the point clouds. However, thecomparison is normally done assuming that there is nouncertainty in their position. Instead, in this paper weconsider a point cloud just as a realization of a stochasticprocess. Accordingly, we have developed a methodologyto compare point clouds that provides, besides the pointto point distances, an estimation of the uncertainty onthese distances and information regarding the statisticalsignificance of the differences between the point clouds.

2. Methodology

2.1. Point cloud registration

Many types of sensors such as rotating laser scannersand stereo and 3D cameras with all their variants, produce3D point clouds from different stations and with differentangular orientations which need to be registered (spatiallyaligned) in a common coordinate system. This registrationprocess is formulated mathematically through a rigid bodytransformation whose six parameters (one translation andthree angles in 3D space) are calculated by the iterativeclosest point (ICP) algorithm.

The ICP algorithm is one of the methods commonlyused for 3D shape alignment. It is used for real-world mod-el construction, robot navigation, inspection and reverseengineering among other applications. Since its introduc-tion [13,12] the ICP algorithm has derived in many differ-ent variants [14]. However, the main concept remainsstable and can be stated as the iterative search of therigid body transformation between two different pointclouds which minimizes an error metric through repeatedgeneration of pairs of corresponding points on the clouds.

The process starts with an initial estimation of the sixdegrees of freedom for the rigid body transformation andthe selection of points in both clouds. A matching betweencorresponding pairs is established after applying the actual3D transformation to the original point cloud and weightedappropriately. Some of the pairs are then rejected underdifferent criteria and an error metric value is calculatedfor the remaining pairs in a process which tries to mini-mize the total error.

Some formulations of the ICP algorithm use the point topoint error metric [12], others use the point to plane metricwhich tries to minimize the sum of the squared distancesbetween each source point and the tangent plane at its cor-responding point in the destination point cloud [15]:

M ¼ Tðtx; ty; tzÞRða;b; cÞ ð1Þ

Mopt ¼ argminX

i

ððMsi � diÞÞ2

where M represents a rigid body transformation matrix(translation T and rotation R), si is a source point vector,

di is a destination point vector and ni is the tangent planenormal in the destination point. If the initial estimationis reasonable, the overlap between the point clouds is suf-ficient and the process reaches to a local minimum withthe best transformation between point clouds.

2.2. Surface estimation

Once the point clouds have been registered in the samecoordinate system, a surface is adjusted to each of them bymeans of a nonparametric estimation method. Let us con-sider ðX;Y ; ZÞ the spatial co-ordinates of each point on theobject surface and assume that the third co-ordinate Z canbe obtained from ðX;YÞ using an unknown functionmðX;YÞ, which represents a kind of smooth surface, so thatZ ¼ mðX;YÞ. Given that m is not known, we need to esti-mate this function using the point cloud X�i ;Y

�i ; Z

�i

� �for

i ¼ 1; . . . ;n. Each of these points can be understood as ameasure of the real point ðXi;Yi; ZiÞ on the object surface,so that

X�i ;Y�i ; Z

�i

� �¼ ðXi;Yi; ZiÞ þ eX

i ; eYi ; e

Zi

� �ð2Þ

where eXi ; eY

i ; eZi

� �represents the measurement error on the

ith point.For any given point ðx0; y0Þ, a smoothed version of the

principal surfaces [16] is proposed to obtain an estimationof mðx0; y0Þ. These estimators are based on the fact thatsurface ðx; y;mðx; yÞÞ can be approximated by a plane:

mðx; yÞ � aþ bxþ cyþ d ¼ 0

in values ðx; yÞ near ðx0; y0Þ. Thus, the normal vector of theplane ða; b; cÞ can be obtained as the smallest component ofa local principal component analysis. The proposed proce-dure is as follows:

� For each point i ¼ 1; . . . ;n a weighting function iscomputed:

Wi ¼Wh X�i � x0;Y�i � y0

� �¼ exp �ðX

�i � x0Þ2 þ ðY�i � y0Þ

2

h

( )ð3Þ

Note that, the weight Wi depends on the Euclidian dis-tance between ðx0; y0Þ and X�i ;Y

�i

� �and, in addition, con-

tains a smoothing parameter h. In order to simplify thenotation, it will be assumed that weights Wi have beenrecentered so

Pni¼1Wi ¼ 1.

� Compute the weighted sample covariance matrix

R ¼r2

X rXY rXZ

rXY r2Y rYZ

rXZ rYZ r2Z

0B@

1CA ð4Þ

with

r2X ¼

Xn

i¼1

WiðX�i Þ2 � �X�2; r2

Y

¼Xn

i¼1

WiðY�i Þ2 � �Y�2; r2

Z ¼Xn

i¼1

WiðZ�i Þ2 � �Z�2

F. de Asís López et al. / Measurement 51 (2014) 259–264 261

rXY ¼Xn

i¼1

WiX�i Y�i � �X� �Y�; rXZ

¼Xn

i¼1

WiX�i Z�i � �X� �Z�; rYZ ¼

Xn

i¼1

WiY�i Z�i � �Y� �Z�

being the weighted sample means

�X� ¼Xn

i¼1

WiX�i ;

�Y� ¼Xn

i¼1

WiY�i ; and �Z�

¼Xn

i¼1

WiZ�i

� Obtain the estimated vector ða; b; cÞ as the third linearprincipal component obtained from the covariancematrix R defined above. Then, a plane normal to thisvector and passing through ð �X�; �Y�; �Z�Þ is calculated:

aðx� �X�Þ þ bðy� �Y�Þ þ cðz� �Z�Þ ¼ 0

Finally, evaluating that plane in ðx0; y0Þ and solving in z,an estimation of mðx0; y0Þ is obtained:

mðx0; y0Þ ¼ �Z� � aðx0 � �X�Þ þ bðy0 � �Y�Þc

It is well known that the nonparametric estimatesmðx; yÞ heavily depend on the bandwidth h used in thekernel weights in (3). The bandwidth is a trade-off be-tween the bias and the variance of the resulting estimates.We used cross-validation for the automatic determinationof the bandwidth,that is, h is obtained by minimizing

CVðhÞ¼Xn

i¼1

minx;y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX�i �x0Þ2þðY�i �y0Þ

2þðZ�i � m�iðx;yÞÞ2q

ð5Þ

where m�iðx; yÞ indicates the fit of mðx; yÞ leaving out thei-sample data ðX�i ;Y

�i ; Z

�i Þ.

2.3. Uncertainty estimation

Uncertainty on a point cloud provides a measure of itsquality. We are interested in determining the spatial distri-bution of the uncertainty through the measured object, notjust in a single point, since it can be different depending onfactors such as the distance to the object, the angle of inci-dence or the density of points [17].

As we have pointed above, each laser point ðX�;Y�; Z�Þcan be understood as a measure of the real pointðX;Y ;mðX;YÞÞ with associated errors ðeX ; eY ; eZ¼mðX;YÞÞ.Therefore, the precision of the measurement in each pointðX;YÞ can be given by

DðX;YÞ ¼ EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieXð Þ2 þ ðeYÞ2 þ ðemðX;YÞÞ2

q� �ð6Þ

Note that DðX;YÞ represents the mean Euclidean distancebetween the observed and the true point (the standarddeviation of the measurements considering that the laserhas no systematic errors). We decided to use this approachsince it is not possible to know which point on the surfaceof the object corresponds to a point on the point cloud.

In practice, DðX;YÞ need to be estimated given that m isunknown. Given a point ðx0; y0Þ, an estimation of Dðx0; y0Þcan be obtained by means of the Nadaraya–Watson esti-mator [18,19]:

Dðx0; y0Þ ¼Xn

i¼1

Wh X�i � x0;Y�i � y0

� �di ð7Þ

where

di ¼minx;y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX�i � xÞ2 þ ðY�i � yÞ2 þ ðZ�i � mðx; yÞÞ2

q

represents the minimum distance between the ithobserved point and the estimated surface.

The estimation Dðx0; y0Þ in (7) is obtained as a weightedmean of the euclidean distances d1; . . . ; dn, where theweight Wh X�i � x0;Y

�i � y0

� �given by the kernel function

in (3) associated to each di depends on the smoothingbandwidth h.

According to the definition of Wh, the values di closed toðx0; y0Þ have more influence on the estimate Dðx0; y0Þ thanthose farther away. The amount of relative influence iscontrolled by the bandwidth h. On the one hand, if h issmall the resulting estimate Dðx0; y0Þ heavily depends onthose observations that are closest to ðx0; y0Þ and tends toyield a more wiggly estimate; and if h becomes closer tozero the estimate tends to adjust too much to the dataand, as a consequence, have a very high variance. On theother hand, if bandwidth is too large the estimated surfacewill be a plane and will not adjust. That shows the signifi-cance of arranging a tool for the automatic choice of themost appropriate smoothing bandwidth. As is was men-tioned in Section 2.2, in this paper we propose to usecross-validation for the automatic choice of the bandwidth.So, the bandwidth h will be selected by minimizing theexpression in (5).

Although the summation in (7) extends to the n pointson the point cloud, in practice it is possible to reduce thenumber of terms by limiting it to those points with a Wh

significantly different from zero. This is especially impor-tant, from a computational point of view, when massivepoint clouds are evaluated.

2.4. Point cloud comparison

The aim of this section is to compare the precision oftwo point clouds

X1�i ;Y

1�i ; Z

1�i

� �and X2�

j ;Y2�j ; Z

2�j

� �for i ¼ 1; . . . ;n and j ¼ 1; . . . ;m, coming from a uniqueobject represented by ðx; y;mðx; yÞÞ being mðx; yÞ an un-known function. Specifically, we are interested in testing,for a given ðx0; y0Þ, the null hypothesis

H0ðx0; y0Þ : D1ðx0; y0Þ ¼ D2ðx0; y0Þ

where D1 and D2 represents the mean Euclidean distanceto cloud points 1 and 2 respectively. Clearly, we will acceptequality of precision at the point ðx0; y0Þ if the difference

T ¼ D1ðx0; y0Þ � D2ðx0; y0Þ ð8Þ

Fig. 1. Point cloud of the studied beam.

Table 1Summary of the standard deviations of both point clouds.

262 F. de Asís López et al. / Measurement 51 (2014) 259–264

is null. In another case, if T > 0 precision of point cloud 2 isgreater than precision of point cloud 1 and vice versa.

In practice we do not know the value of T, so it must beestimated. The estimate T is obtained replacing in (8)D1ðx0; y0Þ by D1ðx0; y0Þ (obtained with the cloud 1) andD2ðx0; y0Þ by D2ðx0; y0Þ (obtained with the cloud 2). Ofcourse, since T is only an estimate of the true T, the sam-pling uncertainty of these estimates should be acknowl-edged. Therefore a ð1� aÞ (e.g. a ¼ 0:05) simulationinterval CI ¼ ða; bÞ around T is derived, and it is checkedwhether the 0 lies (equality of precision) or not (differentprecision) within this interval.

2.4.1. Bootstrap procedureIn the construction of the simulations intervals for T it is

necessary to know the percentiles distribution of T . Never-theless, it is well known that in the nonparametric regres-sion context the asymptotic theory is of little help todetermine these percentiles, meanwhile resampling meth-ods like bootstrap introduced by Efron [20] are moreappropriate. The steps for the construction of the simula-tion interval for the true T are as follows:

Step 1. For b ¼ 1 to B (e.g. B = 1000),

simulate samples X1�b1 ;Y1�b

1 ; Z1�b1

� �; . . . ; X1�b

n ; Y1�bn ; Z1�b

n

� �and X2�b

1 ;Y2�b1 ; Z2�b

1

� �; . . . ; X2�b

m ;Y2�bm ; Z2�b

m

� �by randomly

sampling with replacement the n and m data set corre-sponding to each of the point clouds, and obtain the

bootstrap estimates Tb according to (6)

Step 2.

Calculate the ð1� aÞ100% limits for the simulationinterval of T:

Min. 1st Qu. Median Mean 3rd Qu. Max.

ðTa=2; T1�a=2Þ ð9Þ Cloud 1 0.70 2.06 3.13 5.44 4.34 88.97Cloud 2 1.29 2.37 3.95 7.96 6.06 230.80

where Tp represents the percentile p of the boot-strapped estimates T1; . . . ; TB.

3. Case study

The methodology explained in the previous sectionswas applied to the comparison of two point clouds ob-tained by measuring a bridge beam with two kinds of laserscanner systems. One of the scanners was a Riegl LMS-Z390i, which is a time-of-flight scanner [21]. This equip-ment has nominal accuracy and precision of 6 mm and4 mm (one sigma at 50 m range), respectively. The otherscanner was a Faro Focus3D, a phase-shift based laserscanner with nominal accuracy and precision of 2 mmand 0.95 mm (one sigma at 25 m range), respectively.

Both point clouds were measured from two stationsonly a few meters apart, in order to ensure that distanceand incidence angles on the bridge were similar. Point den-sity was also the same for both point clouds. Fig. 1 showsthe point cloud measured using the time-of-flight scanner(no visual difference with respect to the phase-shift basedscanner point cloud is appreciable after registration).

Taking into account the accuracy and precision specifi-cations of both scanners, the Faro Focus3D should providea point cloud closer to the real beam than the LMS-Z390i.However, as stated above, we are not interested in thequality of each isolated point but in the degree of adjust-ment of a surface model to the whole beam.

4. Results and discussion

Following the previously exposed methodology, thestandard deviations of the two point clouds, D1 (corre-sponding to the Riegl LMS-Z390i) and D2 (correspondingto the Faro Focus3D) and the differences between themin the 1200 points of the grid (60 � 20) considered, were

0

5

10

15

20mm

−10 0 10 20

−4

−2

0

2

4

6

cloud 1

X (m)

Y (m

)

0

5

10

15

20mm

−10 0 10 20

−4

−2

0

2

4

6

cloud 2

X (m)

Y (m

)

Fig. 2. Bootstrap 0.975 percentiles of the distributions of the estimated D1 (cloud 1) and D2 (cloud 2), respectively.

F. de Asís López et al. / Measurement 51 (2014) 259–264 263

calculated. Table 1 shows minimum and maximum valuesof the standard deviation as well as the four quartiles.Notice that, contrary to expectations, given the technicalcharacteristics of both scanners, D2 is greater than the D1,that is, the point cloud measured with the phase-shiftbased scanner has more noise than the point cloudmeasured with the time-of-flight scanner. A similar unex-pected result, attributed to a bad calibration of the FaroFocus3D scanner, was obtained in [22] where a FaroFocus3D and a Leica HDS6100 terrestrial laser scannerwere compared.

Fig. 2 shows the standard deviations obtained in bothpoint clouds, and the difference between them. Greaterdeviations correspond to areas where the angle of

Fig. 3. 3D rendering of the standard deviations of point cloud measured with t

incidence of the scanning beam on the surface is high, alsoto areas near the corners and those of low density of pointsdue to the distance to the scanner.

Fig. 3 shows the standard deviation represented on a 3Dmodel of the beam bridge. Again, greater standard devia-tions can be appreciated in point cloud 2.

In Fig. 4, a set of points of the beam bridge showingsignificant differences between both point clouds arerepresented. In order to reduce computational costs andto simplify the figure, a grid of 20 � 20 points wasanalyzed. For a total of 400 points, there were significantdifferences in 146 of them. Therefore, the representationof the beam bridge obtained from each of the two pointclouds is different in some areas.

he time-of-flight scanner (left) and the phase-shift based scanner (right).

−10 0 10 20 30−4

−20

24

6X (m)

Y (m

)

Fig. 4. A grid of points where significant differences between both point clouds exits. In red when D2 > D1 while blue corresponds to points where D1 > D2.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

264 F. de Asís López et al. / Measurement 51 (2014) 259–264

5. Conclusions

In this paper the problem of comparing two pointclouds has been tackled from a new perspective. The meth-odology proposed for point cloud comparison takes intoaccount that point coordinates have errors. This methodol-ogy is based on computing distances between points oftwo point clouds in the normal direction of one of themand repeating the calculations by resampling the pointclouds. This allows determining simulation intervals forthe distances between both point clouds, instead of just asingle value of the distance. Consequently, it is possibleto establish, at a specific simulation level, if the differencesbetween both point clouds are significant.

The application of the proposed methodology to com-pare two point clouds of a beam bridge measured withtwo different scanner systems, showed significant differ-ences in parts of the beam. This is important in inspectionworks since different conclusions could be reacheddepending on the measuring instrument.

Acknowledgements

The authors gratefully acknowledge the financialsupport from Projects BIA2011-26915 and MTM2011-23204 (FEDER support included) of the Spanish Ministryof Science and Innovation, and project 10 PXIB 300 068PR of the Xunta de Galicia.

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