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Terrestrial laser scanner self-calibration: Correlation sources and their mitigation
ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 93–102
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Terrestrial laser scanner self-calibration: Correlation sources and their mitigation Derek D. Lichti ∗ Department of Geomatics Engineering, Centre for Bioengineering Research and Education, The University of Calgary, 2500 University Dr NW, Calgary AB T2N 1N4, Canada
article
info
Article history: Received 22 June 2009 Received in revised form 8 September 2009 Accepted 18 September 2009 Available online 6 October 2009 Keywords: Laser scanning Calibration Error Modelling Correlation
abstract Instrument calibration is recognised as an important process to assure the quality of data captured with a terrestrial laser scanner. While the self-calibration approach can provide optimal estimates of systematic error parameters without the need for specialised equipment or facilities, its success is somewhat hindered by high correlations between model variables. This paper presents the findings of a detailed study into the sources of correlation in terrestrial laser scanner self-calibration for a basic additional parameter set. Several pertinent outcomes, resulting from experiments conducted with simulated data, and 12 real calibration datasets captured with a Faro 880 terrestrial laser scanner, are presented. First, it is demonstrated that panoramic-type scanner self-calibration from only two instrument locations is possible so long as the scans have orthogonal orientation in the horizontal plane. Second, the importance of including scanner tilt angle observations in the adjustment for parameter de-correlation is demonstrated. Third, a new network measure featuring an asymmetric distribution of object points that does not rely upon a priori observation of the instrument position is proposed. It is shown to be an effective means to reduce the correlation between the rangefinder offset and the scanner position parameters. Fourth, the roles of several other influencing variables on parameter correlation are revealed. The paper concludes with a set of recommended design measures to reduce parameter correlation in terrestrial laser scanner self-calibration. © 2009 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.
1. Introduction An important aspect of the quality assurance (QA) of threedimensional point clouds captured with terrestrial laser scanning (TLS) instruments is geometric calibration. Systematic errors inherent to such instruments can, if not corrected, degrade the accuracy of point clouds captured with a scanner. Systematic error modelling and a corresponding calibration methodology for estimating the model coefficients are therefore necessary components of a QA procedure. Furthermore, the calibration procedure must be as tractable as possible so that the time and effort required to perform it are minimal. Self-calibration approaches have recently been investigated by a number of researchers and can be categorised according to the type of targeting used, namely signalised point targets and planar features. The common thread between both approaches is the collection of a highly redundant set of observations from multiple instrument locations in a strong geometric configuration. The model variables comprising the scanner position and angular orientation elements, the target parameters (point co-ordinates or
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plane parameters) and the systematic error coefficients, called the additional parameters (APs), are simultaneously estimated from these observations. Point-target approaches have been used for the self-calibration of various TLS systems by Lichti (2007), Reshetyuk (2006, 2009), Schneider and Schwalbe (2008) and Schneider (2009). TLS self-calibration using planar features has been reported by Gielsdorf et al. (2004), Bae and Lichti (2007), Dorninger et al. (2008) and Molnár et al. (2009). The advantages of the self-calibration approach include optimal estimation of all model variables, and no special equipment or facilities, such as an electronic distance meter baseline, are required except for a room comprising some form of targeting (signalised points or planar features). It can yield very precise APs that have been demonstrated, through independent assessment, to improve the accuracy of subsequently-acquired point cloud data (e.g. Lichti, 2007). While one goal of self-calibration network design is to reduce the functional dependence between model variables, the mitigation of some correlation mechanisms remains problematic and is the subject of ongoing investigation. This paper presents an in-depth analysis of the correlation between some of these variables with emphasis on the basic additional parameter set comprising the rangefinder offset, collimation axis error, trunnion axis error and vertical circle index terms. Specifically, the following correlation mechanisms have been investigated for point-target self-calibration:
0924-2716/$ – see front matter © 2009 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved. doi:10.1016/j.isprsjprs.2009.09.002
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D.D. Lichti / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 93–102
Vertical axis
κ
Horizontal (Trunnion) Axis
φ
ω
Collimation Axis Fig. 1. Terrestrial laser scanner axes and orientation angles.
• Between the collimation axis error and the tertiary orientation angle, κ ; • Between the vertical circle index error and the scanner tilt angles, ω and φ , and the scanner position; and • Between the rangefinder offset and the scanner position. (See Fig. 1 for illustrations of these axes and orientation angles.) Investigations into their causes and mitigation have been conducted with both simulated and real datasets. The analyses of the results focus on solution quality in terms of parameter correlation and additional parameter precision. In the course of reporting on these, other secondary correlation mechanisms are revealed and analysed. Furthermore, the roles of several other influencing factors are reported to give a clear understanding of the correlation mechanisms and to develop network design measures for correlation mitigation. The factors investigated were:
• • • • • • • • •
The number of locations; Scan orientation in the horizontal plane; The presence/absence of instrument tilt angle observations; Angular observation parameterisation; Target field symmetry; The calibration room size; Observation precision; Tilt angle precision; and The number of observations.
2. Terrestrial laser scanner modelling 2.1. Scanner architectures It is important to first review the different terrestrial laser scanner architectures since they dictate how the observables are parameterised and they influence the correlation mechanisms encountered in self-calibration. Following the terminology of Staiger (2003), a TLS instrument can be categorised as either a camera scanner, having a limited field of view (FOV) of, say, 40◦ × 40◦ , a panoramic scanner or a hybrid scanner. This work is concerned with the latter two, which are capable of scanning through a full horizontal FOV, but differ in terms of their scanning mechanisms and their vertical fields of view. For both types, scanning in the horizontal direction is performed by rotation of the instrument head about its vertical axis. A panoramic scanner rotates through a horizontal range of 180◦ and features a rotating, single-facet mirror on the end of a shaft that deflects the laser beam from a lower vertical limit, −α0 (a few tens of degrees above nadir), up through zenith and down again to the lower limit of 180◦ + α 0 . Data are thus acquired in front of and behind the instrument along each vertical scanning profile. The so-called first
and second layers correspond to data captured below and above the zenith, respectively (Abmayr et al., 2005). The scanned field of view is spherical except for a small cone beneath the instrument. Examples of panoramic scanners include the Faro LS 880 and the Leica HDS 6000, which have respective vertical fields of view of 320◦ and 310◦ (POB, 2009). A hybrid scanner rotates through a horizontal range of 360◦ and acquires data in a vertical FOV that ranges from a minimum value below the instrument’s horizontal plane, αmin , to some maximum value, αmax , which is less than or equal to the zenith direction. Hybrid scanners feature either a rotating polygonal mirror, e.g., the Riegl LMS-Z620, or an oscillating mirror, e.g., the Trimble GX 3D Scanner, which provide smaller vertical fields of view of 80◦ and 60◦ , respectively (POB, 2009). The Leica Scan Station 2 also uses an oscillating mirror. Its vertical FOV extends up to the zenith by scanning through two separate windows (one on the front, one on the top). Though its coverage is nearly spherical—like a panoramic scanner—its construction and its 360◦ horizontal angular range are such that it is classified as a hybrid scanner. The rotation angle ranges for both the horizontal (θ ) and vertical (α ) observables, for panoramic and hybrid scanners, are pictured in Fig. 2. 2.2. Point target observation models The observation of a point i from scanner location j can be expressed in terms of range, ρij , horizontal direction, θij , and elevation angle, αij ,
ρij + ερij =
q
x2ij + y2ij + zij2 + ∆ρ
θij + εθij = arctan
yij
xij
(1)
+ ∆θ
(2)
αij + εαij = arctan q
zij
x2ij + y2ij
+ ∆α
(3)
where (x, y, z )ij are the Cartesian co-ordinates of point i in scannerspace j, which is related to object space by the rigid body transformation xij yij zij
" #
" # Xsj Xi Yi − Ysj = R3 κj R2 φj R1 ωj Z Z
i
(4)
sj
(X , Y , Z )i and (Xs , Ys , Zs )j are the object-space co-ordinates of point i and scanner location j, respectively; (ω, φ, κ)j are the rotation angles from object space to scanner space j that, coupled with the scanner location co-ordinates, comprise the exterior orientation parameters (EOPs); R1 , R2 , R3 are the matrices for rotation about the X -, Y - and Z -axes, respectively; ∆ρ , ∆θ and ∆α represent the respective systematic error correction models for the observations; and the ε terms are the respective random errors. If the scanner can be levelled via two orthogonal inclinometers, then the following two observation equations can be written for each scan location
ω j + εω j = 0
(5)
φj + εφj = 0.
(6)
The principal benefit of these observations is de-correlation of the tilt angles and the vertical circle index error parameter (Lichti, 2007). An additional benefit will be demonstrated later in this paper. The majority of instruments (25 out of 29) from a recent TLS hardware survey feature some form of tilt compensation mechanism (POB, 2009).
D.D. Lichti / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 93–102
a
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b
Fig. 2. Angular observation ranges. (a) Hybrid scanner. (b) Panoramic scanner.
2.3. Panoramic-specific equations
2.5. Self-calibration solution
Naturally, the observation equations developed for TLS instruments should model the acquisition geometry as closely as possible, which includes the scanning FOV. For panoramic scanners, the angular observations computed using Eqs. (2) and (3) must be modified so that they to conform to the limits depicted in Fig. 2. If the calculated horizontal direction using Eq. (2) fulfils the inequality θ < 0◦ , then the following modified observation equations are used for a panoramic scanner
A self-calibration using point targets is performed by simultaneously estimating all model variables (EOPs, APs and object point co-ordinates) in a parametric-model, free-network least-squares adjustment with inner constraints imposed on the object points. A minimally-constrained datum is necessary to prevent potential biases in the object point co-ordinates from propagating into other model parameters, particularly the APs. The APs are generally assumed to be block or network invariant. Each group of observations (ρ , θ , α and ω and φ ) is assigned an a priori variance. Baarda’s data snooping is used to identify gross errors and iterative variance component estimation is used to optimise the contribution of each observation group. Additional details about the estimation procedure can be found in Lichti (2007).
θij + εθij = 180 + arctan ◦
yij
xij
+ ∆θ
(7)
αij + εαij = 180 − arctan q ◦
zij
x2ij + y2ij
+ ∆α.
(8)
3. The basic additional parameter set and its estimation
If θ ≥ 0◦ , then no change is required. Though this is essentially a software implementation issue, the details are provided here since the angle parameterisation, as governed by scanner architecture, is a significant factor governing self-calibration solution strength as initially demonstrated by Lichti (2009) and shown in more detail herein. 2.4. Additional parameter models Guided by similarities in construction to theodolites and total stations, TLS systematic error model selection is driven by the requirements for a particular instrument, i.e. model identification from highly-redundant datasets. The following models, comprising the rangefinder offset, a0 , the collimation axis error, b1 , the trunnion axis error, b2 , and the vertical circle index error, c0 , can quite reasonably be described as a basic set of APs for TLS instruments.
∆ρ = a0
(9)
∆θ = b1 sec αij + b2 tan αij
∆α = c0 .
(10) (11)
The model terms and the issues surrounding their estimation are described in detail in Section 3. Though more extensive sets of APs are reported in the literature, this constitutes a common set of parameters in the models of Gielsdorf et al. (2004), Lichti (2007), Reshetyuk (2006, 2009), Schneider and Schwalbe (2008) and Schneider (2009). Additionally, the model of Abmayr et al. (2005) includes the latter three parameters. These four terms can also be described as the basic parameters of total station instruments, which share many salient properties with TLS instruments in terms of their construction, and can be self-calibrated in a similar manner (Lichti and Lampard, 2008).
3.1. Rangefinder offset The rangefinder offset, a0 , models the offset between the range measurement origin and the scanner space origin. It can be precisely estimated by both the point- and plane-based approaches, though it is worth noting it cannot be determined by the very practical, on-site method for cyclic error estimation proposed by Dorninger et al. (2008). Its correlation with the scanner position can be quite high, with Lichti (2007) reporting up to 0.87. It will be shown that this correlation depends on a number of factors. Most TLS instruments available nowadays feature the ability to set up over a known point: 24 out of 29 from the aforementioned hardware survey (POB, 2009). Reshetyuk (2009) demonstrates with both simulated and real data that the imposition of weighted constraints on the scanner position is very effective at lowering the correlation to approximately 0.3. However, this measure requires extra effort to independently determine the co-ordinates of the scanner location relative to the target field and to optically-centre the instrument. A calibration room with larger dimensions may also offer a means of de-correlation. This possibility has been investigated for this paper. If neither independent position information nor a larger space for a calibration range are available, or the extra effort required for the former is not desired, then a new method proposed herein can be used to achieve the desired de-correlation. Its basis is the combination of two network design features. The first is an asymmetrical distribution of targets in the vertical dimension, e.g. points on the floor but not on the ceiling. It will be demonstrated that this measure on its own is not sufficient, so the second feature, the inclusion of inclinometer observations is required. Though a seemingly unusual combination of design measures, it will be shown to be effective for both simulated and real datasets.
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3.2. Collimation axis error The collimation axis error b1 is the non-orthogonality between the instrument’s collimation (line of sight) and horizontal (trunnion) axes. De-correlation of this parameter from the tertiary rotation angle, κ , has been shown to be a challenge for hybrid scanners by Reshetyuk (2006, 2009), Schneider and Schwalbe (2008) and Schneider (2009) since these instruments only collect one data layer. If the scanner is level, i.e. ω = φ = 0◦ , then Eq. (2) reduces to
θij + εθij
! − Xi − Xsj sin κj + Yi − Ysj cos κj + ∆θ = arctan Xi − Xsj cos κj + Yi − Ysj sin κj (12)
which can be shown to have the following equivalent form
θij + εθij = arctan
Yi − Ysj
!
Xi − Xsj
− κ j + ∆θ .
Fig. 3. Schematic layout of the simulated self-calibration room.
(13) 3.4. Vertical circle index error
Eq. (13) is the familiar form of the theodolite direction observation equation (e.g. Kuang, 1996), in which κ is the unknown orientation parameter. The cause of the correlation between κ and b1 can be determined from the analysis of the Maclaurin series of the collimation axis error model, i.e.
b1 sec αij = b1 1 +
1 2
αij2 +
5 24
αij4 + · · · .
(14)
If data are acquired in only layer then the functional dependence between the constant offset (i.e. zero-order) term of the series and κ is perfect. So, regardless of the observed elevation angle range, perfect correlation between b1 and κ exists for hybrid instruments. If data are acquired in two layers, as is the case for panoramic scanners, then the two are de-correlated due to the opposite sign of the secant function at α = 180◦ , even if the range of observed elevation angles is low (Lichti, 2009). This agrees well with basic surveying principles, from which it is known that the determination of the collimation axis error in a theodolite requires observation of a target on a horizontal line of sight on two faces (e.g. Davis et al., 1981). The latter point is the key since the sign of the error changes from one face to the next. Observing along a horizontal sight line simply eliminates the influence of the trunnion axis error. A large range of elevation angle observations is therefore not sufficient for b1 –κ de-correlation in hybrid scanner selfcalibration. It has therefore been recommended that b1 should be removed from the self-calibration AP model of hybrid scanners unless κ can somehow be independently observed and therefore constrained in the solution (Lichti, 2009). It has also been shown (Lichti, 2009) that the inclusion of orthogonal scans in the network effectively reduces this correlation for panoramic scanners. It will be shown herein that the correlation can even be reduced for a network comprising single scans at each nominal instrument location as long as there exists κ -angle diversity. 3.3. Trunnion axis error The trunnion axis error b2 is the non-orthogonality between the scanner’s trunnion and vertical axes. This term can be estimated for both panoramic and hybrid scanners. Its precise determination requires observation at high elevation angles since it is zerovalued at the horizon. Its precision is also improved by including orthogonal scans (Lichti, 2009). Its correlation with other model variables is not a problem for either scanner architecture provided there is vertical symmetry in the distribution of the observations. As will be shown, though, secondary correlations between b2 and other parameters can be introduced if an asymmetric target distribution exists.
The vertical circle error c0 models the constant offset between the scanner-space horizontal plane and the elevation-angle measurement origin and is highly or perfectly correlated with ω and φ . For an approximately-level scanner, re-arrangement of Eq. (3) and subsequent differentiation with respect to ωj , for example, leads to the following constant relationship, which illustrates the mathematical cause of the problem
∂ c0 = 1. ∂ωj
(15)
These parameters are de-correlated explicitly by observing ω and φ , even with low precision, and implicitly by the panoramic scanner angle parameterisation. The latter means makes sense since observations on two faces are needed to eliminate c0 in a theodolite (Davis et al., 1981). Precise estimation of c0 is independent of the elevation angle range for panoramic scanners but is mildly dependent for hybrid scanners (Lichti, 2009). 4. Experiment datasets 4.1. Simulated datasets Several TLS self-calibration networks were simulated with realistic parameters drawn from the author’s past experiences (e.g., Lichti, 2007). First, it was assumed that the calibration would be conducted indoors in a room with length, L, width, W and height, H. For most simulations these variables were set at 12 m, 9 m and 3 m, respectively, which correspond to values from previous experiments with real data. Two nominal scanner locations were simulated. Their separation was governed by the room dimensions and the minimum observable range, ρ0 , of 1.5 m, the suggested value of the Faro 880 scanner that is the subject of the real-data experiments, though it is recognised that newer scanners can operate at closer range (e.g., POB, 2009). It is demonstrated in Lichti (2007, 2009) that self-calibration of panoramic-type scanners from two locations is possible. It will be shown in this paper that two-location self-calibration for hybrid instruments without collimation axis error model is also possible. The simulated room layout is depicted in plan view in Fig. 3. The scanner was assumed to be nominally level at both locations and the number of orthogonal scans simulated from each was four. The number of object points was set to 180, the largest number that was available in previous experiments conducted by the author. Their locations, i.e. whether they were only on the walls or on the walls, the floor and the ceiling, was governed by the
D.D. Lichti / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 93–102
a
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b
Fig. 4. Vertical profiles showing the relationship between the elevation angle α and the elevation angle range ψ . (a) Hybrid scanner. (b) Panoramic scanner.
permitted range of observations above or below the horizon, denoted by ψ and called the elevation angle range hereafter, which was varied from ±10◦ to ±70◦ in increments of 10◦ . The former limit is a realistic minimum value and the latter is slightly higher than that average value of 66◦ computed from 12 real datasets. The relationship between the elevation angle α and the elevation angle range ψ is shown for both architectures in Fig. 4. Target fields, having an asymmetric vertical distribution, were also generated such that ψmin was −70◦ and ψmax was varied in 10◦ increments from 0◦ to 40◦ . If the elevation angle range ψ was such that all points were confined to the 4 walls, then the object points were distributed in proportion to the respective wall dimensions and uniformly within each wall. If the elevation angle range ψ was such that points also lay on the floor and ceiling, then one-half of the points were distributed on the 4 four walls as described above and the remaining half were equally divided among the floor and the ceiling. These point locations were randomly generated subject to the constraint that no point could lie within a certain distance of the instrument location as governed by the minimum observable range ρ0 and the elevation angle range ψ . This procedure resulted in a realistic distribution of observations (Lichti, 2009). Observations were weighted on a group-wise basis using realistic standard deviation estimates from previous experiments. For the range, ρ , the horizontal direction, θ , and elevation angle, α , these were ±2 mm, ±1800 and ±1800 , respectively. Both the ω and φ angle observations’ standard deviations were ±30 (±0.05◦ ). No parameter constraints were imposed on the scanner position. Self-calibration software composed by the author was used for all simulated and real data processing. 4.2. Real datasets Results from real self-calibration datasets are also presented. The instrument studied was a Faro 880 terrestrial laser scanner, a panoramic instrument that features a vertical FOV of 320◦ (−70◦ to 250◦ ) and a 180◦ horizontal FOV. Its rangefinder operates by the phase difference method in which the range is proportional to the phase difference between the received and emitted signals. It also features two orthogonal inclinometers that measure the instrument’s tilt. Their outputs are used by the accompanying software to correct the captured scan data, resulting in a ‘levelled’ dataset, which justifies the use of zero-valued ω and φ parameter observations. The system also corrects some systematic errors using manufacturer-determined coefficients. The effects of switching off these internal corrections have been reported by Lichti and Licht (2006). Twelve self-calibration datasets were captured over a 22-month period: eleven with the same Faro 880 scanner and the twelfth with a second instrument of the same variety. For each dataset a
point-target self-calibration was performed using proprietary Faro targets arranged on the walls, the floor and the ceiling, though the target field varied in terms of the number of targets and room size. Of the 12 datasets, the two for which the tilt angle observations were available, denoted simply as A and B, are of primary interest. For dataset A, a total of 8 scans were captured from two nominal locations on opposite sides of the room. For each scan the entire instrument was rotated about its vertical axis atop the tripod so as to introduce a 90◦ κ rotation angle relative to the previous scan. For dataset B, two orthogonal scans were captured from four nominal locations distributed throughout the room. The networks were set up in a 12 m × 9 m × 3 m room having 131 and 135 targets, respectively. The degrees-of-freedom from the basic AP set selfcalibration were 1768 and 1651 for datasets A and B, respectively. Though only the basic AP set was used, the observation group standard deviations were determined by variance component estimation in self-calibration adjustments with the full AP models in order to properly model the random error dispersion. See Lichti (2007) for a description of the full AP models. 5. Results and discussion 5.1. Collimation axis error (b1 ) and tertiary rotation angle (κ) correlation As shown in Lichti (2009), inclusion of scans captured at the same nominal location but with orthogonal κ rotation angles in a self-calibration network improves the solution in terms of the correlation between κ and the collimation axis error term b1 for panoramic-style scanners. That analysis is extended here to investigate the effect of κ angle diversity and the number of scan locations. Datasets A and B were used to compute several different adjustment solutions in which the angle parameterisation, number of scan locations and number of orthogonal scans were varied. The results are given in Table 1. First, the results from adjustments of the full datasets (i.e. all 8 scans each) using the hybrid and panoramic angle parameterisations are compared. Strictly speaking, use of the former model is incorrect for this type of scanner, but the results are presented to effectively demonstrate the problem with the hybrid model. Starting with rows 1 and 2 in Table 1, the poor precision of b1 and the near-perfect b1 –κ correlation (quantified by the maximum coefficient magnitude, |r |) confirms that reliable estimation of b1 is not possible when the hybrid angle parameterisation is used, as has been reported by Reshetyuk (2009) and Schneider and Schwalbe (2008). Analysis of the entries in rows 3 and 4, though, shows that reliable b1 estimation is possible for the panoramic case as the two variables are almost completely de-correlated and the precision of b1 is superior by at least an order of magnitude. In the remaining cases the influences of the number of scan locations, the number of orthogonal scans and the κ -angle diversity
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Table 1 Collimation axis error precision and its correlation with the tertiary rotation angle κ . Row
1 2 3 4 5 6 7 8 9
Angle parameterisation
Hybrid Hybrid Panoramic Panoramic Panoramic Panoramic Panoramic Panoramic Panoramic
# nominal locations
4 2 4 2 2 2 2 2 4
# orthogonal scans
2 4 2 4 3 2 1 1 1
κ angle diversity among scans (◦ )
0, 90 0, 90, 180, 270 0, 90 0, 90, 180, 270 0, 90, 270 0, 90 0 only 0, 90 0, 90, 270
a
Dataset A
Dataset B
σb1 (00 )
Maximum |rb1 −κ |
σb1 (00 ) Maximum |rb1 −κ |
–
– 1.00 – 0.10 0.17 0.28 0.87 0.64 –
±78.2 –
±1.2 ±1.5 ±2.3 ±12.7 ±6.9 –
±16.8 0.98 –
±1.6 – –
±2.6 ±7.9 ±6.6 ±3.0
– 0.22 – – 0.19 0.65 0.36 0.29
b
Fig. 5. Maximum correlation coefficients for simulated panoramic scanner self-calibration data. (a) With tilt angle observations. (b) Without tilt angle observations. (Note: ψmin = −ψmax .)
are considered for only the panoramic architecture. Decreasing the number of orthogonal scans per location from 4 to 3 (row 5) has a negligible effect on σb1 and the correlation coefficient. Further reduction to only 2 scans (row 6) has the effect of approximately doubling σb1 but the increase in the correlation is not of concern. Reduction to a single scan captured from 2 locations (row 7) increases σb1 by nearly an order of magnitude and increases the correlation up to 0.87. In this case, though, both κ angles were the same, 0◦ . The adjustment cases in row 8 also had single scans captured from two locations but, interestingly, due to their diversity of κ rotation angles, the solution is improved (over that of row 7) by up to 45% in terms of σb1 and correlation. So, panoramic scanner calibration from only two stations is possible as long as diversity exists among the tertiary rotation angles, but a four-scan configuration of two scans captured from two locations (row 6) or single scans taken from four locations (row 9) is preferable.
The corresponding results for the hybrid scanner are shown in Fig. 6. In this and all subsequent hybrid model simulations, b1 has been excluded from the AP model as recommended. No correlation of concern exists with c0 if the tilt angles are observed. If the tilt angles are not observed, though, then very high correlation exists with the tilt angles and with Zs , regardless of the elevation angle range ψ . Thus, observation of the scanner tilt angles is critical for de-correlation of the vertical circle index error and the exterior orientation for hybrid scanners and is also recommended for panoramic scanners. The angle parameterisation has a strong influence on the precision of both the vertical circle index error c0 and the trunnion axis error b2 . Fig. 7(a) shows that the panoramic parameterisation can achieve a much greater precision (σ ) for both APs in question and the precision of c0 is independent of the elevation angle range ψ . For the hybrid case (Fig. 7(b)), though, the precision of c0 does depend on ψ , though not as strongly as b2 does.
5.2. Vertical circle index error (c0 ), tilt angle (ω and φ ) and scanner height (Zs ) correlation
5.3. Rangefinder offset (a0 ) and scanner position correlation
Here the de-correlation of the tilt angles from the vertical circle index error c0 is examined in detail for both scanner architectures using simulated data. Fig. 5 shows the correlation coefficients from 2-station, 4-orthogonal-scan panoramic self-calibration datasets as functions of (symmetric) elevation angle range ψ for two mechanisms: between the tilt angles and c0 and between the Z component of the scanner position, Zs , and c0 . Results with and without the tilt observations are given in Fig. 5(a) and (b), respectively. As expected, the former results agree with those reported for real data (Lichti, 2007), i.e. no correlation whatsoever. The latter shows no significant correlation between c0 and the scanner position but correlation up to 0.55 between c0 and the tilt angles. Increasing the elevation angle range ψ has the effect of lowering this correlation.
5.3.1. Panoramic scanners—Simulated data The distribution of observations for all simulations to this point has been horizontally and vertically symmetric. Under such conditions the correlation between the rangefinder offset a0 and the scanner position co-ordinates is independent of both the angle parameterisation and the presence or absence of the tilt angle observations and is only slightly dependent on the elevation angle range ψ , as shown in Fig. 8. The precision of a0 is independent of all mentioned variables. The correlation exists with the planimetric position variables, Xs and Ys ; a0 is uncorrelated with Zs due to the vertically-symmetric point distribution. If a vertically-asymmetric target field is used, then the a0 position correlation is much lower, as shown in Fig. 9(a) where the largest magnitude coefficient is 0.34 instead of 0.77 (c.f. Fig. 8). The dependence increases as the upper limit of the elevation angle
D.D. Lichti / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 93–102
a
99
b
Fig. 6. Maximum correlation coefficients for simulated hybrid scanner self-calibration data. (a) With tilt angle observations. (b) Without tilt angle observations. (Note: ψmin = −ψmax .)
a
σ σ
b b2
σ σ
c0
b2 c0
Fig. 7. Precision of selected additional parameters for simulated scanner self-calibration data with tilt angle observations. (a) Panoramic. (b) Hybrid. (Note: ψmin = −ψmax .)
Fig. 8. Maximum correlation coefficients for simulated hybrid scanner selfcalibration data with tilt angle observations and a vertically-symmetric target distribution. ‘pos’ indicates scanner position co-ordinate parameters. (Note: ψmin = −ψmax .)
range ψ increases. This comes, however, at the cost of correlation of up to 0.45 between b1 and b2 —which are un-correlated in
a
the symmetric case (Fig. 8)—since only half of their respective functions (secant and tangent) is observed, albeit in two branches. As expected, though, this dependence decreases as the upper limit of ψ increases. No significant loss of AP precision results from the asymmetric design as σa0 , σb1 and σc0 are ±0.1 mm, ±0.800 and ±1.000 , respectively. Only σb2 was slightly dependent on elevation angle range ψ , rising from ±1.600 to ±2.000 . If the tilt angles are not observed then, as shown in Fig. 9(b), the a0 -position correlation remains high, despite the asymmetric target distribution, and the tilt angle-vertical circle index error correlation with similar magnitude exists. Thus for an asymmetric target field design, the ω and φ observations serve to de-correlate not only c0 and the tilt angles but also a0 and the scanner position. The values indicated on the x-axis of these figures are the upper limits of the elevation angle range, ψ . The maximum value of 40◦ corresponds to the practical limit of target placement on the walls for the realistic target field geometry described earlier. The lower limit for ψ used in all cases was −70◦ , which approximately corresponds to maximum data limits from previous tests with real
b
Fig. 9. Maximum correlation coefficients for simulated panoramic scanner self-calibration data with a vertically-asymmetric target distribution (ψmin = −70◦ ). (a) With tilt angle observations. (b) Without tilt angle observations. ‘pos’ indicates scanner position co-ordinate parameters.
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Fig. 10. Maximum correlation coefficients for simulated hybrid scanner selfcalibration data with tilt angle observations and a vertically asymmetric target distribution (b1 excluded from the AP model; ψmin = −70◦ ). ‘pos’ indicates scanner position co-ordinate parameters.
so as to make it asymmetric; ψmax was varied from 0◦ to 40◦ . The pertinent maximum correlation coefficients from the ensuing self-calibration adjustments are plotted in Fig. 11. The right-most samples of each plot represent the values from the full, symmetric dataset adjustments. The trends in these plots generally follow those of Fig. 9(a), but the a0 -position and the b1 –b2 curves intersect at lower values of ψ at approximately 10◦ and 0◦ instead of 30◦ . The two maximum a0 -position correlation coefficient curves agree quite closely, with respective mean and maximum differences of −0.03 and −0.10. The b1 –b2 coefficients differ more (mean 0.24 and maximum 0.28) but follow similar trends. In spite of these differences, there is a clearly an advantage to a vertically-asymmetric target field design, regardless of the value of ψmax . The causes of the differences between curves are, however, analysed in Section 5.4.4. 5.4. Other influencing variables
data. Testing showed that there were no differences in the results for target fields having the asymmetry above or below the horizon. There may, however, be practical considerations, i.e. it is easier to put targets on the floor than on the ceiling, but ceiling targets are more permanent. Furthermore, the elevation angle range above the scanner is greater since some scanners can capture data up to the zenith, whereas targets near the nadir can not be scanned due to the blind spot beneath the instrument. These aspects are also important for the estimation of other APs, such as the trunnion axis wobble parameters (e.g. Lichti, 2007). 5.3.2. Hybrid scanners—Simulated data As shown in Fig. 10, a vertically-asymmetric target field design also de-correlates the rangefinder offset a0 and scanner position co-ordinates for a hybrid scanner without b1 in the AP model and with tilt angle observations. As a result of the asymmetry, though, the trunnion axis error term b2 is correlated (by up to 0.45) with the tertiary angle κ since only half of one branch of the tangent function is observed. The rangefinder offset a0 and the vertical circle index error c0 are also correlated with each other by up to 0.55. The AP precision is slightly lower than that of the panoramic case with an asymmetric target field but is effectively independent of the degree of asymmetry. The respective maximum and minimum σa0 , σb2 and σc0 were ±0.1 mm and ±0.2 mm, ±7.500 and ±8.700 and ±14.100 and ±15.900 . Results without tilt angle observations are not shown. In this case the vertical circle index error c0 is nearly perfectly correlated (up to 0.98) to the tilt angles and the Zs scanner position co-ordinate, as are the rangefinder offset a0 and Zs (up to 0.81). 5.3.3. Panoramic scanners—Real data Datasets A and B were used to demonstrate this de-correlation method with real panoramic data. Each dataset was decimated
a
5.4.1. The effect of calibration room size—Simulated data Other parameters influencing the correlation between the rangefinder offset and the scanner position co-ordinates are reported in this section. In the first test, the dimensions of the simulated calibration room were varied so as to vary the maximum observable range. The maximum range for the 12 m × 9 m × 3 m room described earlier was 12.6 m. Both symmetric and asymmetric (with ψmax = 0◦ ) target field distributions were used. As shown in Fig. 12, increasing the room size has a minor influence in the symmetric case, reducing the a0 -position correlation from 0.9 to 0.7. For the asymmetric target field the effect is much greater with the correlation in question dropping below 0.15 at ρmax = 27.3 m. The accompanying b1 –b2 correlation only exceeds 0.5 (reaching 0.67) for the small room case (ρmax = 5.6 m). The precision of a0 was the same in all cases (±0.2 mm). Though this design measure does improve the a0 solution quality, it may not be a very practical one since a large room may be difficult to find. 5.4.2. The effect of basic observable precision—Real data Here, the role of the relative magnitudes of the precision of the three basic observables (ρ , θ and α ) is examined using all 12 real datasets. The maximum correlation coefficients were studied as a function the ratios of the a priori observation group standard deviations, i.e. σρ /σθ and σρ /σα . A more clear linear relationship can be seen in Fig. 13, which depicts the ratio of range observation variance and the product of the angular observation standard deviations, called here the precision ratio pr , pr = σρ2 / (σθ σα ) . (16) These results indicate that the relative precision of these three observation groups has a strong impact on the a0 -position correlation. Low precision ranges and/or high precision angular observations result in lower a0 -position correlation.
b
Fig. 11. Maximum correlation coefficients for real, panoramic scanner self-calibration data with a vertically asymmetric target distribution (ψmin = −70◦ ). ‘pos’ indicates scanner position co-ordinate parameters. (a) Dataset A. (b) Dataset B.
D.D. Lichti / ISPRS Journal of Photogrammetry and Remote Sensing 65 (2010) 93–102
a
101
b
Fig. 12. Maximum rangefinder offset-scanner position correlation coefficients for simulated, panoramic scanner self-calibration data as a function of maximum observed range. (a) Symmetric target distribution (ψmin = −70◦ , ψmax = 70◦ ). (b) Asymmetric target distribution (ψmin = −70◦ , ψmax = 0◦ ). ‘pos’ indicates scanner position co-ordinate parameters.
Fig. 13. Maximum rangefinder offset-scanner position correlation coefficients for real, panoramic scanner self-calibration data as a function of the precision ratio, pr , with the best-fit line shown.
5.4.3. The effect of tilt angle observable precision—Real data This investigation highlights another factor that influences the degree of rangefinder offset-scanner position correlation. Fig. 14 shows how the maximum correlation between a0 and the scanner position varies as a function of the tilt angle precision, σtilt , which was systematically varied from ±1.800 to ±5◦ . The curves of both real datasets follow similar trends in which the correlation decreases as the tilt angle observation precision does, with the dataset B curve decaying more slowly. Clearly, very precise tilt angle observations produce the same results as the self-calibration solution without their observation (horizontal lines), which seems to be a counter-intuitive outcome but is explained below. The observation of the tilt angles removes two datum defects, rotation about X and Y , so the number of required constraints is 4. If σtilt is large, then these datum elements are weakly defined. In this case, the precision of the estimated scanner EOPs, in particular
Zs , ω and φ , is very poor, but the APs are de-correlated from the EOPs, as is shown for a0 in Fig. 14. The AP precision is not affected (it is constant) but the EOP–EOP correlation coefficients are very high (i.e. ≥0.99). If, however, σtilt is low, then the two datum elements in question are tightly constrained and the a0 -position correlation increases and the EOP precision improves, though again the AP precision is unaffected. Intuitively, one might expect the effect of removing the tilt angle observations to be similar to that realised when a very large variance is used. This is not the case because in the absence of tilt angle observations, the datum elements for rotation about X and Y are defined by the approximate object point co-ordinates. These datum elements are well defined if the object points are well distributed in the horizontal plane, which should generally be true for panoramic or hybrid scanner self-calibration conducted in a room. This situation is, therefore, somewhat analogous to the case with high-precision tilt angle observations except without the benefit of the AP-EOP de-correlation. 5.4.4. The effect of the number of object points—Simulated data In this section the analyses focus on the differences in a0 position and b1 –b2 correlation between Fig. 9(a) and Fig. 11. Several possible causes were investigated, the first of which was basic observation precision. The 0.012 precision ratio (Eq. (16)) of the simulated data was much lower than that of the real datasets, which partly explains their differences. Observation precision does not however account for the differences between datasets A and B, whose respective precision ratios were 0.050 and 0.051. Though their respective tilt angle observation precisions differed at ±0.037◦ and ±0.06◦ , performing the adjustment of dataset B with the tilt precision of dataset A did not change the correlations significantly, so this factor was ruled out.
Fig. 14. Maximum rangefinder offset-scanner position correlation coefficients for real, panoramic scanner self-calibration data as a function of tilt angle observation precision.
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a
b
Fig. 15. Maximum correlation coefficients for simulated, panoramic scanner self-calibration data with a vertically asymmetric target distribution and variable number of object points. (a) Rangefinder offset-scanner position. (b) b1 –b2 .
Recalling that the simulations had a greater number of object points than the real datasets, an additional set of panoramic, asymmetric-distribution simulations were conducted in which the number of target points was varied. The results of these are shown in Fig. 15 for two upper elevation angle ranges only (ψmax = 0◦ and 40◦ ) since only these boundary values are necessary. It can be seen in Fig. 15(a) that the a0 -position correlation, contrary to what intuition would suggest, is in fact strongly influenced by the number of object points. The difference in correlation between the endpoints of each curve (i.e. at 30 points and 280 points) is 0.3. Such dependence does not exist in the vertically-symmetric case (figure not shown) where the correlation is constant. In contrast, the b1 –b2 correlation mechanism is independent of the number of object points, as shown in Fig. 15(b). The difference in correlation between each curve’s endpoints is only 0.08. This is also true for the symmetric case (not shown). It is quite likely that the unaccounted b1 –b2 correlation discrepancies are due to more subtle differences in the observation distributions of the two datasets, but this requires further investigation. 6. Conclusions This paper has examined the sources of correlation in terrestrial laser scanner self-calibration for a basic additional parameter set using both simulated and real datasets. It has been demonstrated that panoramic scanner self-calibration from only two instrument locations is possible so long as the scans have orthogonal orientation in the horizontal plane (i.e. κ angles). The importance of including tilt angle observations in the adjustment for parameter de-correlation has been shown for both symmetric and asymmetric target field design. The asymmetric design measure has been shown to be an effective means to reduce the correlation between the rangefinder offset and the scanner position parameters that does not rely upon a priori observation of the instrument position. The roles of other influencing variables on this particularly important correlation mechanism have also been studied. Some recommended TLS self-calibration network design measures therefore include an asymmetric target field design, the observation of the tilt angles, scans with orthogonal κ angles, use of as large a room as is possible and use of a large number of targets (i.e. ≥130). It should be stressed that this research has focused on networks in which the scanner is nominally vertical. While this is by far the most common operational configuration, some TLS instruments can operate at non-vertical orientations and, thus, the impact of this geometry on parameter correlation is an avenue of future research. Furthermore, the instrument orientation has been mod-
elled with a Cardan rotation angle sequence. Another focus of further investigation could be the effect of a different parameterisation such as the quaternion. Acknowledgement This research was supported by the Natural Science and Engineering Research Council of Canada (NSERC). References Abmayr, T., Dalton, G., Hätrl, F., Hines, D., Liu, R., Hirzinger, G., Frölich, C., 2005. Standardization and visualization of 2.5D scanning data and color information by inverse mapping. In: Optical 3-D Measurement Techniques VII, Vienna, Austria, 3–5 October, vol. I, pp. 164–173. Bae, K.-H., Lichti, D.D, 2007. On-site self-calibration using planar features for terrestrial laser scanners. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences 36 (Part 3/W52), 14–19. Davis, R.E, Foote, F.S, Anderson, J.M., Mikhail, E.M., 1981. Surveying: Theory and Practice. McGraw-Hill, New York, 992 pages. Dorninger, P., Nothegger, C., Pfeifer, N., Molnár, G., 2008. On-the-job detection and correction of systematic cyclic distance measurement errors of terrestrial laser scanners. Journal of Applied Geodesy 2 (4), 191–204. Gielsdorf, F., Rietdorf, A., Gruendig, L., 2004 A concept for the calibration of terrestrial laser scanners. In: Proceedings FIG Working Week, Athens, Greece, 22–27 May (on CD-ROM). Kuang, S., 1996. Geodetic Network Analysis and Optical Design: Concepts and Applications. Ann Arbor Press, Chelsea, Michigan. Lichti, D.D., 2007. Modelling, calibration and analysis of an AM-CW terrestrial laser scanner. ISPRS Journal of Photogrammetry and Remote Sensing 61 (5), 307–324. Lichti, D.D., 2009. The impact of angle parameterisation on terrestrial laser scanner self-calibration. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences 38 (Part 3/W8), 171–176. Lichti, D.D., Lampard, J., 2008. Reflectorless total station self-calibration. Survey Review 40 (309), 244–259. Lichti, D.D., Licht, M.G., 2006. Experiences with terrestrial laser scanner modelling and accuracy assessment. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences 36 (Part 5), 155–160. Molnár, G., Pfeifer, N., Ressl, C., Dorninger, P., Nothegger, C., 2009. Range calibration of terrestrial laser scanners with piecewise linear functions. Photogrammetrie, Fernerkundung, Geoinformation 1, 9–21. Point of Beginning (POB), 2009. 2009 laser scanner survey. http://laser. jadaproductions.net/ (accessed 04.03.09). Reshetyuk, Y., 2006. Calibration of terrestrial laser scanners callidus 1.1, Leica HDS 3000 and Leica HDS 2500. Survey Review 38 (302), 703–713. Reshetyuk, Y., 2009. Self-calibration and direct georeferencing in terrestrial laser scanning. Doctoral Thesis. Department of Transport and Economics, Division of Geodesy, Royal Institute of Technology (KTH), Stockholm, Sweden, January. Schneider, D., 2009. Calibration of a Riegl LMS-Z420i based on a multistation adjustment and a geometric model with additional parameters. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences 38 (Part 3/W8), 177–182. Schneider, D., Schwalbe, E., 2008. Integrated processing of terrestrial laser scanner data and fisheye-camera image data, The International Archives of the Photogrammetry. Remote Sensing and Spatial Information Sciences 37 (Part B5), 1037–1043. Staiger, R., 2003. Terrestrial laser scanning—Technology, systems and applications. In: Proceedings of 2nd FIG Regional Conference, Marrakech, Morocco, 2–5 December. http://www.fig.net/pub/morocco/index.htm (accessed 02.02.09).
Contents lists available at ScienceDirect
ISPRS Journal of Photogrammetry and Remote Sensing journal homepage: www.elsevier.com/locate/isprsjprs
Terrestrial laser scanner self-calibration: Correlation sources and their mitigation Derek D. Lichti ∗ Department of Geomatics Engineering, Centre for Bioengineering Research and Education, The University of Calgary, 2500 University Dr NW, Calgary AB T2N 1N4, Canada
article
info
Article history: Received 22 June 2009 Received in revised form 8 September 2009 Accepted 18 September 2009 Available online 6 October 2009 Keywords: Laser scanning Calibration Error Modelling Correlation
abstract Instrument calibration is recognised as an important process to assure the quality of data captured with a terrestrial laser scanner. While the self-calibration approach can provide optimal estimates of systematic error parameters without the need for specialised equipment or facilities, its success is somewhat hindered by high correlations between model variables. This paper presents the findings of a detailed study into the sources of correlation in terrestrial laser scanner self-calibration for a basic additional parameter set. Several pertinent outcomes, resulting from experiments conducted with simulated data, and 12 real calibration datasets captured with a Faro 880 terrestrial laser scanner, are presented. First, it is demonstrated that panoramic-type scanner self-calibration from only two instrument locations is possible so long as the scans have orthogonal orientation in the horizontal plane. Second, the importance of including scanner tilt angle observations in the adjustment for parameter de-correlation is demonstrated. Third, a new network measure featuring an asymmetric distribution of object points that does not rely upon a priori observation of the instrument position is proposed. It is shown to be an effective means to reduce the correlation between the rangefinder offset and the scanner position parameters. Fourth, the roles of several other influencing variables on parameter correlation are revealed. The paper concludes with a set of recommended design measures to reduce parameter correlation in terrestrial laser scanner self-calibration. © 2009 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.
1. Introduction An important aspect of the quality assurance (QA) of threedimensional point clouds captured with terrestrial laser scanning (TLS) instruments is geometric calibration. Systematic errors inherent to such instruments can, if not corrected, degrade the accuracy of point clouds captured with a scanner. Systematic error modelling and a corresponding calibration methodology for estimating the model coefficients are therefore necessary components of a QA procedure. Furthermore, the calibration procedure must be as tractable as possible so that the time and effort required to perform it are minimal. Self-calibration approaches have recently been investigated by a number of researchers and can be categorised according to the type of targeting used, namely signalised point targets and planar features. The common thread between both approaches is the collection of a highly redundant set of observations from multiple instrument locations in a strong geometric configuration. The model variables comprising the scanner position and angular orientation elements, the target parameters (point co-ordinates or
∗
Tel.: +1 403 210 9495; fax: +1 403 284 1980. E-mail address: [email protected].
plane parameters) and the systematic error coefficients, called the additional parameters (APs), are simultaneously estimated from these observations. Point-target approaches have been used for the self-calibration of various TLS systems by Lichti (2007), Reshetyuk (2006, 2009), Schneider and Schwalbe (2008) and Schneider (2009). TLS self-calibration using planar features has been reported by Gielsdorf et al. (2004), Bae and Lichti (2007), Dorninger et al. (2008) and Molnár et al. (2009). The advantages of the self-calibration approach include optimal estimation of all model variables, and no special equipment or facilities, such as an electronic distance meter baseline, are required except for a room comprising some form of targeting (signalised points or planar features). It can yield very precise APs that have been demonstrated, through independent assessment, to improve the accuracy of subsequently-acquired point cloud data (e.g. Lichti, 2007). While one goal of self-calibration network design is to reduce the functional dependence between model variables, the mitigation of some correlation mechanisms remains problematic and is the subject of ongoing investigation. This paper presents an in-depth analysis of the correlation between some of these variables with emphasis on the basic additional parameter set comprising the rangefinder offset, collimation axis error, trunnion axis error and vertical circle index terms. Specifically, the following correlation mechanisms have been investigated for point-target self-calibration:
0924-2716/$ – see front matter © 2009 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved. doi:10.1016/j.isprsjprs.2009.09.002
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Vertical axis
κ
Horizontal (Trunnion) Axis
φ
ω
Collimation Axis Fig. 1. Terrestrial laser scanner axes and orientation angles.
• Between the collimation axis error and the tertiary orientation angle, κ ; • Between the vertical circle index error and the scanner tilt angles, ω and φ , and the scanner position; and • Between the rangefinder offset and the scanner position. (See Fig. 1 for illustrations of these axes and orientation angles.) Investigations into their causes and mitigation have been conducted with both simulated and real datasets. The analyses of the results focus on solution quality in terms of parameter correlation and additional parameter precision. In the course of reporting on these, other secondary correlation mechanisms are revealed and analysed. Furthermore, the roles of several other influencing factors are reported to give a clear understanding of the correlation mechanisms and to develop network design measures for correlation mitigation. The factors investigated were:
• • • • • • • • •
The number of locations; Scan orientation in the horizontal plane; The presence/absence of instrument tilt angle observations; Angular observation parameterisation; Target field symmetry; The calibration room size; Observation precision; Tilt angle precision; and The number of observations.
2. Terrestrial laser scanner modelling 2.1. Scanner architectures It is important to first review the different terrestrial laser scanner architectures since they dictate how the observables are parameterised and they influence the correlation mechanisms encountered in self-calibration. Following the terminology of Staiger (2003), a TLS instrument can be categorised as either a camera scanner, having a limited field of view (FOV) of, say, 40◦ × 40◦ , a panoramic scanner or a hybrid scanner. This work is concerned with the latter two, which are capable of scanning through a full horizontal FOV, but differ in terms of their scanning mechanisms and their vertical fields of view. For both types, scanning in the horizontal direction is performed by rotation of the instrument head about its vertical axis. A panoramic scanner rotates through a horizontal range of 180◦ and features a rotating, single-facet mirror on the end of a shaft that deflects the laser beam from a lower vertical limit, −α0 (a few tens of degrees above nadir), up through zenith and down again to the lower limit of 180◦ + α 0 . Data are thus acquired in front of and behind the instrument along each vertical scanning profile. The so-called first
and second layers correspond to data captured below and above the zenith, respectively (Abmayr et al., 2005). The scanned field of view is spherical except for a small cone beneath the instrument. Examples of panoramic scanners include the Faro LS 880 and the Leica HDS 6000, which have respective vertical fields of view of 320◦ and 310◦ (POB, 2009). A hybrid scanner rotates through a horizontal range of 360◦ and acquires data in a vertical FOV that ranges from a minimum value below the instrument’s horizontal plane, αmin , to some maximum value, αmax , which is less than or equal to the zenith direction. Hybrid scanners feature either a rotating polygonal mirror, e.g., the Riegl LMS-Z620, or an oscillating mirror, e.g., the Trimble GX 3D Scanner, which provide smaller vertical fields of view of 80◦ and 60◦ , respectively (POB, 2009). The Leica Scan Station 2 also uses an oscillating mirror. Its vertical FOV extends up to the zenith by scanning through two separate windows (one on the front, one on the top). Though its coverage is nearly spherical—like a panoramic scanner—its construction and its 360◦ horizontal angular range are such that it is classified as a hybrid scanner. The rotation angle ranges for both the horizontal (θ ) and vertical (α ) observables, for panoramic and hybrid scanners, are pictured in Fig. 2. 2.2. Point target observation models The observation of a point i from scanner location j can be expressed in terms of range, ρij , horizontal direction, θij , and elevation angle, αij ,
ρij + ερij =
q
x2ij + y2ij + zij2 + ∆ρ
θij + εθij = arctan
yij
xij
(1)
+ ∆θ
(2)
αij + εαij = arctan q
zij
x2ij + y2ij
+ ∆α
(3)
where (x, y, z )ij are the Cartesian co-ordinates of point i in scannerspace j, which is related to object space by the rigid body transformation xij yij zij
" #
" # Xsj Xi Yi − Ysj = R3 κj R2 φj R1 ωj Z Z
i
(4)
sj
(X , Y , Z )i and (Xs , Ys , Zs )j are the object-space co-ordinates of point i and scanner location j, respectively; (ω, φ, κ)j are the rotation angles from object space to scanner space j that, coupled with the scanner location co-ordinates, comprise the exterior orientation parameters (EOPs); R1 , R2 , R3 are the matrices for rotation about the X -, Y - and Z -axes, respectively; ∆ρ , ∆θ and ∆α represent the respective systematic error correction models for the observations; and the ε terms are the respective random errors. If the scanner can be levelled via two orthogonal inclinometers, then the following two observation equations can be written for each scan location
ω j + εω j = 0
(5)
φj + εφj = 0.
(6)
The principal benefit of these observations is de-correlation of the tilt angles and the vertical circle index error parameter (Lichti, 2007). An additional benefit will be demonstrated later in this paper. The majority of instruments (25 out of 29) from a recent TLS hardware survey feature some form of tilt compensation mechanism (POB, 2009).
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a
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b
Fig. 2. Angular observation ranges. (a) Hybrid scanner. (b) Panoramic scanner.
2.3. Panoramic-specific equations
2.5. Self-calibration solution
Naturally, the observation equations developed for TLS instruments should model the acquisition geometry as closely as possible, which includes the scanning FOV. For panoramic scanners, the angular observations computed using Eqs. (2) and (3) must be modified so that they to conform to the limits depicted in Fig. 2. If the calculated horizontal direction using Eq. (2) fulfils the inequality θ < 0◦ , then the following modified observation equations are used for a panoramic scanner
A self-calibration using point targets is performed by simultaneously estimating all model variables (EOPs, APs and object point co-ordinates) in a parametric-model, free-network least-squares adjustment with inner constraints imposed on the object points. A minimally-constrained datum is necessary to prevent potential biases in the object point co-ordinates from propagating into other model parameters, particularly the APs. The APs are generally assumed to be block or network invariant. Each group of observations (ρ , θ , α and ω and φ ) is assigned an a priori variance. Baarda’s data snooping is used to identify gross errors and iterative variance component estimation is used to optimise the contribution of each observation group. Additional details about the estimation procedure can be found in Lichti (2007).
θij + εθij = 180 + arctan ◦
yij
xij
+ ∆θ
(7)
αij + εαij = 180 − arctan q ◦
zij
x2ij + y2ij
+ ∆α.
(8)
3. The basic additional parameter set and its estimation
If θ ≥ 0◦ , then no change is required. Though this is essentially a software implementation issue, the details are provided here since the angle parameterisation, as governed by scanner architecture, is a significant factor governing self-calibration solution strength as initially demonstrated by Lichti (2009) and shown in more detail herein. 2.4. Additional parameter models Guided by similarities in construction to theodolites and total stations, TLS systematic error model selection is driven by the requirements for a particular instrument, i.e. model identification from highly-redundant datasets. The following models, comprising the rangefinder offset, a0 , the collimation axis error, b1 , the trunnion axis error, b2 , and the vertical circle index error, c0 , can quite reasonably be described as a basic set of APs for TLS instruments.
∆ρ = a0
(9)
∆θ = b1 sec αij + b2 tan αij
∆α = c0 .
(10) (11)
The model terms and the issues surrounding their estimation are described in detail in Section 3. Though more extensive sets of APs are reported in the literature, this constitutes a common set of parameters in the models of Gielsdorf et al. (2004), Lichti (2007), Reshetyuk (2006, 2009), Schneider and Schwalbe (2008) and Schneider (2009). Additionally, the model of Abmayr et al. (2005) includes the latter three parameters. These four terms can also be described as the basic parameters of total station instruments, which share many salient properties with TLS instruments in terms of their construction, and can be self-calibrated in a similar manner (Lichti and Lampard, 2008).
3.1. Rangefinder offset The rangefinder offset, a0 , models the offset between the range measurement origin and the scanner space origin. It can be precisely estimated by both the point- and plane-based approaches, though it is worth noting it cannot be determined by the very practical, on-site method for cyclic error estimation proposed by Dorninger et al. (2008). Its correlation with the scanner position can be quite high, with Lichti (2007) reporting up to 0.87. It will be shown that this correlation depends on a number of factors. Most TLS instruments available nowadays feature the ability to set up over a known point: 24 out of 29 from the aforementioned hardware survey (POB, 2009). Reshetyuk (2009) demonstrates with both simulated and real data that the imposition of weighted constraints on the scanner position is very effective at lowering the correlation to approximately 0.3. However, this measure requires extra effort to independently determine the co-ordinates of the scanner location relative to the target field and to optically-centre the instrument. A calibration room with larger dimensions may also offer a means of de-correlation. This possibility has been investigated for this paper. If neither independent position information nor a larger space for a calibration range are available, or the extra effort required for the former is not desired, then a new method proposed herein can be used to achieve the desired de-correlation. Its basis is the combination of two network design features. The first is an asymmetrical distribution of targets in the vertical dimension, e.g. points on the floor but not on the ceiling. It will be demonstrated that this measure on its own is not sufficient, so the second feature, the inclusion of inclinometer observations is required. Though a seemingly unusual combination of design measures, it will be shown to be effective for both simulated and real datasets.
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3.2. Collimation axis error The collimation axis error b1 is the non-orthogonality between the instrument’s collimation (line of sight) and horizontal (trunnion) axes. De-correlation of this parameter from the tertiary rotation angle, κ , has been shown to be a challenge for hybrid scanners by Reshetyuk (2006, 2009), Schneider and Schwalbe (2008) and Schneider (2009) since these instruments only collect one data layer. If the scanner is level, i.e. ω = φ = 0◦ , then Eq. (2) reduces to
θij + εθij
! − Xi − Xsj sin κj + Yi − Ysj cos κj + ∆θ = arctan Xi − Xsj cos κj + Yi − Ysj sin κj (12)
which can be shown to have the following equivalent form
θij + εθij = arctan
Yi − Ysj
!
Xi − Xsj
− κ j + ∆θ .
Fig. 3. Schematic layout of the simulated self-calibration room.
(13) 3.4. Vertical circle index error
Eq. (13) is the familiar form of the theodolite direction observation equation (e.g. Kuang, 1996), in which κ is the unknown orientation parameter. The cause of the correlation between κ and b1 can be determined from the analysis of the Maclaurin series of the collimation axis error model, i.e.
b1 sec αij = b1 1 +
1 2
αij2 +
5 24
αij4 + · · · .
(14)
If data are acquired in only layer then the functional dependence between the constant offset (i.e. zero-order) term of the series and κ is perfect. So, regardless of the observed elevation angle range, perfect correlation between b1 and κ exists for hybrid instruments. If data are acquired in two layers, as is the case for panoramic scanners, then the two are de-correlated due to the opposite sign of the secant function at α = 180◦ , even if the range of observed elevation angles is low (Lichti, 2009). This agrees well with basic surveying principles, from which it is known that the determination of the collimation axis error in a theodolite requires observation of a target on a horizontal line of sight on two faces (e.g. Davis et al., 1981). The latter point is the key since the sign of the error changes from one face to the next. Observing along a horizontal sight line simply eliminates the influence of the trunnion axis error. A large range of elevation angle observations is therefore not sufficient for b1 –κ de-correlation in hybrid scanner selfcalibration. It has therefore been recommended that b1 should be removed from the self-calibration AP model of hybrid scanners unless κ can somehow be independently observed and therefore constrained in the solution (Lichti, 2009). It has also been shown (Lichti, 2009) that the inclusion of orthogonal scans in the network effectively reduces this correlation for panoramic scanners. It will be shown herein that the correlation can even be reduced for a network comprising single scans at each nominal instrument location as long as there exists κ -angle diversity. 3.3. Trunnion axis error The trunnion axis error b2 is the non-orthogonality between the scanner’s trunnion and vertical axes. This term can be estimated for both panoramic and hybrid scanners. Its precise determination requires observation at high elevation angles since it is zerovalued at the horizon. Its precision is also improved by including orthogonal scans (Lichti, 2009). Its correlation with other model variables is not a problem for either scanner architecture provided there is vertical symmetry in the distribution of the observations. As will be shown, though, secondary correlations between b2 and other parameters can be introduced if an asymmetric target distribution exists.
The vertical circle error c0 models the constant offset between the scanner-space horizontal plane and the elevation-angle measurement origin and is highly or perfectly correlated with ω and φ . For an approximately-level scanner, re-arrangement of Eq. (3) and subsequent differentiation with respect to ωj , for example, leads to the following constant relationship, which illustrates the mathematical cause of the problem
∂ c0 = 1. ∂ωj
(15)
These parameters are de-correlated explicitly by observing ω and φ , even with low precision, and implicitly by the panoramic scanner angle parameterisation. The latter means makes sense since observations on two faces are needed to eliminate c0 in a theodolite (Davis et al., 1981). Precise estimation of c0 is independent of the elevation angle range for panoramic scanners but is mildly dependent for hybrid scanners (Lichti, 2009). 4. Experiment datasets 4.1. Simulated datasets Several TLS self-calibration networks were simulated with realistic parameters drawn from the author’s past experiences (e.g., Lichti, 2007). First, it was assumed that the calibration would be conducted indoors in a room with length, L, width, W and height, H. For most simulations these variables were set at 12 m, 9 m and 3 m, respectively, which correspond to values from previous experiments with real data. Two nominal scanner locations were simulated. Their separation was governed by the room dimensions and the minimum observable range, ρ0 , of 1.5 m, the suggested value of the Faro 880 scanner that is the subject of the real-data experiments, though it is recognised that newer scanners can operate at closer range (e.g., POB, 2009). It is demonstrated in Lichti (2007, 2009) that self-calibration of panoramic-type scanners from two locations is possible. It will be shown in this paper that two-location self-calibration for hybrid instruments without collimation axis error model is also possible. The simulated room layout is depicted in plan view in Fig. 3. The scanner was assumed to be nominally level at both locations and the number of orthogonal scans simulated from each was four. The number of object points was set to 180, the largest number that was available in previous experiments conducted by the author. Their locations, i.e. whether they were only on the walls or on the walls, the floor and the ceiling, was governed by the
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Fig. 4. Vertical profiles showing the relationship between the elevation angle α and the elevation angle range ψ . (a) Hybrid scanner. (b) Panoramic scanner.
permitted range of observations above or below the horizon, denoted by ψ and called the elevation angle range hereafter, which was varied from ±10◦ to ±70◦ in increments of 10◦ . The former limit is a realistic minimum value and the latter is slightly higher than that average value of 66◦ computed from 12 real datasets. The relationship between the elevation angle α and the elevation angle range ψ is shown for both architectures in Fig. 4. Target fields, having an asymmetric vertical distribution, were also generated such that ψmin was −70◦ and ψmax was varied in 10◦ increments from 0◦ to 40◦ . If the elevation angle range ψ was such that all points were confined to the 4 walls, then the object points were distributed in proportion to the respective wall dimensions and uniformly within each wall. If the elevation angle range ψ was such that points also lay on the floor and ceiling, then one-half of the points were distributed on the 4 four walls as described above and the remaining half were equally divided among the floor and the ceiling. These point locations were randomly generated subject to the constraint that no point could lie within a certain distance of the instrument location as governed by the minimum observable range ρ0 and the elevation angle range ψ . This procedure resulted in a realistic distribution of observations (Lichti, 2009). Observations were weighted on a group-wise basis using realistic standard deviation estimates from previous experiments. For the range, ρ , the horizontal direction, θ , and elevation angle, α , these were ±2 mm, ±1800 and ±1800 , respectively. Both the ω and φ angle observations’ standard deviations were ±30 (±0.05◦ ). No parameter constraints were imposed on the scanner position. Self-calibration software composed by the author was used for all simulated and real data processing. 4.2. Real datasets Results from real self-calibration datasets are also presented. The instrument studied was a Faro 880 terrestrial laser scanner, a panoramic instrument that features a vertical FOV of 320◦ (−70◦ to 250◦ ) and a 180◦ horizontal FOV. Its rangefinder operates by the phase difference method in which the range is proportional to the phase difference between the received and emitted signals. It also features two orthogonal inclinometers that measure the instrument’s tilt. Their outputs are used by the accompanying software to correct the captured scan data, resulting in a ‘levelled’ dataset, which justifies the use of zero-valued ω and φ parameter observations. The system also corrects some systematic errors using manufacturer-determined coefficients. The effects of switching off these internal corrections have been reported by Lichti and Licht (2006). Twelve self-calibration datasets were captured over a 22-month period: eleven with the same Faro 880 scanner and the twelfth with a second instrument of the same variety. For each dataset a
point-target self-calibration was performed using proprietary Faro targets arranged on the walls, the floor and the ceiling, though the target field varied in terms of the number of targets and room size. Of the 12 datasets, the two for which the tilt angle observations were available, denoted simply as A and B, are of primary interest. For dataset A, a total of 8 scans were captured from two nominal locations on opposite sides of the room. For each scan the entire instrument was rotated about its vertical axis atop the tripod so as to introduce a 90◦ κ rotation angle relative to the previous scan. For dataset B, two orthogonal scans were captured from four nominal locations distributed throughout the room. The networks were set up in a 12 m × 9 m × 3 m room having 131 and 135 targets, respectively. The degrees-of-freedom from the basic AP set selfcalibration were 1768 and 1651 for datasets A and B, respectively. Though only the basic AP set was used, the observation group standard deviations were determined by variance component estimation in self-calibration adjustments with the full AP models in order to properly model the random error dispersion. See Lichti (2007) for a description of the full AP models. 5. Results and discussion 5.1. Collimation axis error (b1 ) and tertiary rotation angle (κ) correlation As shown in Lichti (2009), inclusion of scans captured at the same nominal location but with orthogonal κ rotation angles in a self-calibration network improves the solution in terms of the correlation between κ and the collimation axis error term b1 for panoramic-style scanners. That analysis is extended here to investigate the effect of κ angle diversity and the number of scan locations. Datasets A and B were used to compute several different adjustment solutions in which the angle parameterisation, number of scan locations and number of orthogonal scans were varied. The results are given in Table 1. First, the results from adjustments of the full datasets (i.e. all 8 scans each) using the hybrid and panoramic angle parameterisations are compared. Strictly speaking, use of the former model is incorrect for this type of scanner, but the results are presented to effectively demonstrate the problem with the hybrid model. Starting with rows 1 and 2 in Table 1, the poor precision of b1 and the near-perfect b1 –κ correlation (quantified by the maximum coefficient magnitude, |r |) confirms that reliable estimation of b1 is not possible when the hybrid angle parameterisation is used, as has been reported by Reshetyuk (2009) and Schneider and Schwalbe (2008). Analysis of the entries in rows 3 and 4, though, shows that reliable b1 estimation is possible for the panoramic case as the two variables are almost completely de-correlated and the precision of b1 is superior by at least an order of magnitude. In the remaining cases the influences of the number of scan locations, the number of orthogonal scans and the κ -angle diversity
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Table 1 Collimation axis error precision and its correlation with the tertiary rotation angle κ . Row
1 2 3 4 5 6 7 8 9
Angle parameterisation
Hybrid Hybrid Panoramic Panoramic Panoramic Panoramic Panoramic Panoramic Panoramic
# nominal locations
4 2 4 2 2 2 2 2 4
# orthogonal scans
2 4 2 4 3 2 1 1 1
κ angle diversity among scans (◦ )
0, 90 0, 90, 180, 270 0, 90 0, 90, 180, 270 0, 90, 270 0, 90 0 only 0, 90 0, 90, 270
a
Dataset A
Dataset B
σb1 (00 )
Maximum |rb1 −κ |
σb1 (00 ) Maximum |rb1 −κ |
–
– 1.00 – 0.10 0.17 0.28 0.87 0.64 –
±78.2 –
±1.2 ±1.5 ±2.3 ±12.7 ±6.9 –
±16.8 0.98 –
±1.6 – –
±2.6 ±7.9 ±6.6 ±3.0
– 0.22 – – 0.19 0.65 0.36 0.29
b
Fig. 5. Maximum correlation coefficients for simulated panoramic scanner self-calibration data. (a) With tilt angle observations. (b) Without tilt angle observations. (Note: ψmin = −ψmax .)
are considered for only the panoramic architecture. Decreasing the number of orthogonal scans per location from 4 to 3 (row 5) has a negligible effect on σb1 and the correlation coefficient. Further reduction to only 2 scans (row 6) has the effect of approximately doubling σb1 but the increase in the correlation is not of concern. Reduction to a single scan captured from 2 locations (row 7) increases σb1 by nearly an order of magnitude and increases the correlation up to 0.87. In this case, though, both κ angles were the same, 0◦ . The adjustment cases in row 8 also had single scans captured from two locations but, interestingly, due to their diversity of κ rotation angles, the solution is improved (over that of row 7) by up to 45% in terms of σb1 and correlation. So, panoramic scanner calibration from only two stations is possible as long as diversity exists among the tertiary rotation angles, but a four-scan configuration of two scans captured from two locations (row 6) or single scans taken from four locations (row 9) is preferable.
The corresponding results for the hybrid scanner are shown in Fig. 6. In this and all subsequent hybrid model simulations, b1 has been excluded from the AP model as recommended. No correlation of concern exists with c0 if the tilt angles are observed. If the tilt angles are not observed, though, then very high correlation exists with the tilt angles and with Zs , regardless of the elevation angle range ψ . Thus, observation of the scanner tilt angles is critical for de-correlation of the vertical circle index error and the exterior orientation for hybrid scanners and is also recommended for panoramic scanners. The angle parameterisation has a strong influence on the precision of both the vertical circle index error c0 and the trunnion axis error b2 . Fig. 7(a) shows that the panoramic parameterisation can achieve a much greater precision (σ ) for both APs in question and the precision of c0 is independent of the elevation angle range ψ . For the hybrid case (Fig. 7(b)), though, the precision of c0 does depend on ψ , though not as strongly as b2 does.
5.2. Vertical circle index error (c0 ), tilt angle (ω and φ ) and scanner height (Zs ) correlation
5.3. Rangefinder offset (a0 ) and scanner position correlation
Here the de-correlation of the tilt angles from the vertical circle index error c0 is examined in detail for both scanner architectures using simulated data. Fig. 5 shows the correlation coefficients from 2-station, 4-orthogonal-scan panoramic self-calibration datasets as functions of (symmetric) elevation angle range ψ for two mechanisms: between the tilt angles and c0 and between the Z component of the scanner position, Zs , and c0 . Results with and without the tilt observations are given in Fig. 5(a) and (b), respectively. As expected, the former results agree with those reported for real data (Lichti, 2007), i.e. no correlation whatsoever. The latter shows no significant correlation between c0 and the scanner position but correlation up to 0.55 between c0 and the tilt angles. Increasing the elevation angle range ψ has the effect of lowering this correlation.
5.3.1. Panoramic scanners—Simulated data The distribution of observations for all simulations to this point has been horizontally and vertically symmetric. Under such conditions the correlation between the rangefinder offset a0 and the scanner position co-ordinates is independent of both the angle parameterisation and the presence or absence of the tilt angle observations and is only slightly dependent on the elevation angle range ψ , as shown in Fig. 8. The precision of a0 is independent of all mentioned variables. The correlation exists with the planimetric position variables, Xs and Ys ; a0 is uncorrelated with Zs due to the vertically-symmetric point distribution. If a vertically-asymmetric target field is used, then the a0 position correlation is much lower, as shown in Fig. 9(a) where the largest magnitude coefficient is 0.34 instead of 0.77 (c.f. Fig. 8). The dependence increases as the upper limit of the elevation angle
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b
Fig. 6. Maximum correlation coefficients for simulated hybrid scanner self-calibration data. (a) With tilt angle observations. (b) Without tilt angle observations. (Note: ψmin = −ψmax .)
a
σ σ
b b2
σ σ
c0
b2 c0
Fig. 7. Precision of selected additional parameters for simulated scanner self-calibration data with tilt angle observations. (a) Panoramic. (b) Hybrid. (Note: ψmin = −ψmax .)
Fig. 8. Maximum correlation coefficients for simulated hybrid scanner selfcalibration data with tilt angle observations and a vertically-symmetric target distribution. ‘pos’ indicates scanner position co-ordinate parameters. (Note: ψmin = −ψmax .)
range ψ increases. This comes, however, at the cost of correlation of up to 0.45 between b1 and b2 —which are un-correlated in
a
the symmetric case (Fig. 8)—since only half of their respective functions (secant and tangent) is observed, albeit in two branches. As expected, though, this dependence decreases as the upper limit of ψ increases. No significant loss of AP precision results from the asymmetric design as σa0 , σb1 and σc0 are ±0.1 mm, ±0.800 and ±1.000 , respectively. Only σb2 was slightly dependent on elevation angle range ψ , rising from ±1.600 to ±2.000 . If the tilt angles are not observed then, as shown in Fig. 9(b), the a0 -position correlation remains high, despite the asymmetric target distribution, and the tilt angle-vertical circle index error correlation with similar magnitude exists. Thus for an asymmetric target field design, the ω and φ observations serve to de-correlate not only c0 and the tilt angles but also a0 and the scanner position. The values indicated on the x-axis of these figures are the upper limits of the elevation angle range, ψ . The maximum value of 40◦ corresponds to the practical limit of target placement on the walls for the realistic target field geometry described earlier. The lower limit for ψ used in all cases was −70◦ , which approximately corresponds to maximum data limits from previous tests with real
b
Fig. 9. Maximum correlation coefficients for simulated panoramic scanner self-calibration data with a vertically-asymmetric target distribution (ψmin = −70◦ ). (a) With tilt angle observations. (b) Without tilt angle observations. ‘pos’ indicates scanner position co-ordinate parameters.
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Fig. 10. Maximum correlation coefficients for simulated hybrid scanner selfcalibration data with tilt angle observations and a vertically asymmetric target distribution (b1 excluded from the AP model; ψmin = −70◦ ). ‘pos’ indicates scanner position co-ordinate parameters.
so as to make it asymmetric; ψmax was varied from 0◦ to 40◦ . The pertinent maximum correlation coefficients from the ensuing self-calibration adjustments are plotted in Fig. 11. The right-most samples of each plot represent the values from the full, symmetric dataset adjustments. The trends in these plots generally follow those of Fig. 9(a), but the a0 -position and the b1 –b2 curves intersect at lower values of ψ at approximately 10◦ and 0◦ instead of 30◦ . The two maximum a0 -position correlation coefficient curves agree quite closely, with respective mean and maximum differences of −0.03 and −0.10. The b1 –b2 coefficients differ more (mean 0.24 and maximum 0.28) but follow similar trends. In spite of these differences, there is a clearly an advantage to a vertically-asymmetric target field design, regardless of the value of ψmax . The causes of the differences between curves are, however, analysed in Section 5.4.4. 5.4. Other influencing variables
data. Testing showed that there were no differences in the results for target fields having the asymmetry above or below the horizon. There may, however, be practical considerations, i.e. it is easier to put targets on the floor than on the ceiling, but ceiling targets are more permanent. Furthermore, the elevation angle range above the scanner is greater since some scanners can capture data up to the zenith, whereas targets near the nadir can not be scanned due to the blind spot beneath the instrument. These aspects are also important for the estimation of other APs, such as the trunnion axis wobble parameters (e.g. Lichti, 2007). 5.3.2. Hybrid scanners—Simulated data As shown in Fig. 10, a vertically-asymmetric target field design also de-correlates the rangefinder offset a0 and scanner position co-ordinates for a hybrid scanner without b1 in the AP model and with tilt angle observations. As a result of the asymmetry, though, the trunnion axis error term b2 is correlated (by up to 0.45) with the tertiary angle κ since only half of one branch of the tangent function is observed. The rangefinder offset a0 and the vertical circle index error c0 are also correlated with each other by up to 0.55. The AP precision is slightly lower than that of the panoramic case with an asymmetric target field but is effectively independent of the degree of asymmetry. The respective maximum and minimum σa0 , σb2 and σc0 were ±0.1 mm and ±0.2 mm, ±7.500 and ±8.700 and ±14.100 and ±15.900 . Results without tilt angle observations are not shown. In this case the vertical circle index error c0 is nearly perfectly correlated (up to 0.98) to the tilt angles and the Zs scanner position co-ordinate, as are the rangefinder offset a0 and Zs (up to 0.81). 5.3.3. Panoramic scanners—Real data Datasets A and B were used to demonstrate this de-correlation method with real panoramic data. Each dataset was decimated
a
5.4.1. The effect of calibration room size—Simulated data Other parameters influencing the correlation between the rangefinder offset and the scanner position co-ordinates are reported in this section. In the first test, the dimensions of the simulated calibration room were varied so as to vary the maximum observable range. The maximum range for the 12 m × 9 m × 3 m room described earlier was 12.6 m. Both symmetric and asymmetric (with ψmax = 0◦ ) target field distributions were used. As shown in Fig. 12, increasing the room size has a minor influence in the symmetric case, reducing the a0 -position correlation from 0.9 to 0.7. For the asymmetric target field the effect is much greater with the correlation in question dropping below 0.15 at ρmax = 27.3 m. The accompanying b1 –b2 correlation only exceeds 0.5 (reaching 0.67) for the small room case (ρmax = 5.6 m). The precision of a0 was the same in all cases (±0.2 mm). Though this design measure does improve the a0 solution quality, it may not be a very practical one since a large room may be difficult to find. 5.4.2. The effect of basic observable precision—Real data Here, the role of the relative magnitudes of the precision of the three basic observables (ρ , θ and α ) is examined using all 12 real datasets. The maximum correlation coefficients were studied as a function the ratios of the a priori observation group standard deviations, i.e. σρ /σθ and σρ /σα . A more clear linear relationship can be seen in Fig. 13, which depicts the ratio of range observation variance and the product of the angular observation standard deviations, called here the precision ratio pr , pr = σρ2 / (σθ σα ) . (16) These results indicate that the relative precision of these three observation groups has a strong impact on the a0 -position correlation. Low precision ranges and/or high precision angular observations result in lower a0 -position correlation.
b
Fig. 11. Maximum correlation coefficients for real, panoramic scanner self-calibration data with a vertically asymmetric target distribution (ψmin = −70◦ ). ‘pos’ indicates scanner position co-ordinate parameters. (a) Dataset A. (b) Dataset B.
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b
Fig. 12. Maximum rangefinder offset-scanner position correlation coefficients for simulated, panoramic scanner self-calibration data as a function of maximum observed range. (a) Symmetric target distribution (ψmin = −70◦ , ψmax = 70◦ ). (b) Asymmetric target distribution (ψmin = −70◦ , ψmax = 0◦ ). ‘pos’ indicates scanner position co-ordinate parameters.
Fig. 13. Maximum rangefinder offset-scanner position correlation coefficients for real, panoramic scanner self-calibration data as a function of the precision ratio, pr , with the best-fit line shown.
5.4.3. The effect of tilt angle observable precision—Real data This investigation highlights another factor that influences the degree of rangefinder offset-scanner position correlation. Fig. 14 shows how the maximum correlation between a0 and the scanner position varies as a function of the tilt angle precision, σtilt , which was systematically varied from ±1.800 to ±5◦ . The curves of both real datasets follow similar trends in which the correlation decreases as the tilt angle observation precision does, with the dataset B curve decaying more slowly. Clearly, very precise tilt angle observations produce the same results as the self-calibration solution without their observation (horizontal lines), which seems to be a counter-intuitive outcome but is explained below. The observation of the tilt angles removes two datum defects, rotation about X and Y , so the number of required constraints is 4. If σtilt is large, then these datum elements are weakly defined. In this case, the precision of the estimated scanner EOPs, in particular
Zs , ω and φ , is very poor, but the APs are de-correlated from the EOPs, as is shown for a0 in Fig. 14. The AP precision is not affected (it is constant) but the EOP–EOP correlation coefficients are very high (i.e. ≥0.99). If, however, σtilt is low, then the two datum elements in question are tightly constrained and the a0 -position correlation increases and the EOP precision improves, though again the AP precision is unaffected. Intuitively, one might expect the effect of removing the tilt angle observations to be similar to that realised when a very large variance is used. This is not the case because in the absence of tilt angle observations, the datum elements for rotation about X and Y are defined by the approximate object point co-ordinates. These datum elements are well defined if the object points are well distributed in the horizontal plane, which should generally be true for panoramic or hybrid scanner self-calibration conducted in a room. This situation is, therefore, somewhat analogous to the case with high-precision tilt angle observations except without the benefit of the AP-EOP de-correlation. 5.4.4. The effect of the number of object points—Simulated data In this section the analyses focus on the differences in a0 position and b1 –b2 correlation between Fig. 9(a) and Fig. 11. Several possible causes were investigated, the first of which was basic observation precision. The 0.012 precision ratio (Eq. (16)) of the simulated data was much lower than that of the real datasets, which partly explains their differences. Observation precision does not however account for the differences between datasets A and B, whose respective precision ratios were 0.050 and 0.051. Though their respective tilt angle observation precisions differed at ±0.037◦ and ±0.06◦ , performing the adjustment of dataset B with the tilt precision of dataset A did not change the correlations significantly, so this factor was ruled out.
Fig. 14. Maximum rangefinder offset-scanner position correlation coefficients for real, panoramic scanner self-calibration data as a function of tilt angle observation precision.
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a
b
Fig. 15. Maximum correlation coefficients for simulated, panoramic scanner self-calibration data with a vertically asymmetric target distribution and variable number of object points. (a) Rangefinder offset-scanner position. (b) b1 –b2 .
Recalling that the simulations had a greater number of object points than the real datasets, an additional set of panoramic, asymmetric-distribution simulations were conducted in which the number of target points was varied. The results of these are shown in Fig. 15 for two upper elevation angle ranges only (ψmax = 0◦ and 40◦ ) since only these boundary values are necessary. It can be seen in Fig. 15(a) that the a0 -position correlation, contrary to what intuition would suggest, is in fact strongly influenced by the number of object points. The difference in correlation between the endpoints of each curve (i.e. at 30 points and 280 points) is 0.3. Such dependence does not exist in the vertically-symmetric case (figure not shown) where the correlation is constant. In contrast, the b1 –b2 correlation mechanism is independent of the number of object points, as shown in Fig. 15(b). The difference in correlation between each curve’s endpoints is only 0.08. This is also true for the symmetric case (not shown). It is quite likely that the unaccounted b1 –b2 correlation discrepancies are due to more subtle differences in the observation distributions of the two datasets, but this requires further investigation. 6. Conclusions This paper has examined the sources of correlation in terrestrial laser scanner self-calibration for a basic additional parameter set using both simulated and real datasets. It has been demonstrated that panoramic scanner self-calibration from only two instrument locations is possible so long as the scans have orthogonal orientation in the horizontal plane (i.e. κ angles). The importance of including tilt angle observations in the adjustment for parameter de-correlation has been shown for both symmetric and asymmetric target field design. The asymmetric design measure has been shown to be an effective means to reduce the correlation between the rangefinder offset and the scanner position parameters that does not rely upon a priori observation of the instrument position. The roles of other influencing variables on this particularly important correlation mechanism have also been studied. Some recommended TLS self-calibration network design measures therefore include an asymmetric target field design, the observation of the tilt angles, scans with orthogonal κ angles, use of as large a room as is possible and use of a large number of targets (i.e. ≥130). It should be stressed that this research has focused on networks in which the scanner is nominally vertical. While this is by far the most common operational configuration, some TLS instruments can operate at non-vertical orientations and, thus, the impact of this geometry on parameter correlation is an avenue of future research. Furthermore, the instrument orientation has been mod-
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