Angular Resolution of Terrestrial Laser Scanners

The Photogrammetric Record 21114 141160 June 2006 ANGULAR RESOLUTION OF TERRESTRIAL LASER SCANNERS DereEK D Licut dlichticurtineduau Curtin University of Technology Perth Western Australia SONAM JAMTSHO sonamjamtshohotmailcom Ministry of Agriculture Thimphu Bhutan Abstract Knowledge of a laser scanner spatial resolution is necessary in order to prevent aliasing and estimate the level of detail that can be resolved from a scanned point cloud Spatial resolution can be decoupled into range and angular components The latter is the focus of this paper and is governed primarily by sampling interval and laser beamwidth However emphasis is often placed on one of thesetypically sampling intervalas an indicator of resolution Since both affect the resolution of a scanned point cloud consideration of only one factor independent of the other can lead to a misunderstanding of a systems capabilities This will be demonstrated to be inappropriate except under very specific conditions A new more appropriate resolution measure for laser scanners the effective instantaneous field of view EIFOV is proposed It is derived by modelling the shift variance of the equal angular increment sampling process laser beamwidthinduced positional uncertainty and observed angle quantisation with ensemble average modulation transfer functions AMTFs Several commercially available terrestrial laser scanner systems are modelled and analysed in terms of their angular resolution capabilities using the EIFOV KeEyworbs effective instantaneous field of view laser beamwidth laser scanning modulation transfer function resolution sampling INTRODUCTION LASER SCANNING INSTRUMENTS are increasingly being used for tasks traditionally performed using photogrammetric and surveying methods They provide users with a threedimensional 3D sampled representationa point cloudof an object or surface and are used in a diverse range of applications including metrology asbuilt surveys reverse engineering airborne topographic surveying cultural heritage recording and volume estimation Though the accuracy requirements for these applications may differ considerably spatial resolution is an important aspect of any laser scanner survey Spatial resolution governs the level of identifiable detail within a scanned point cloud and is particularly important for say recording of cultural heritage features with fine details For laser scanners it can be decoupled into range and angular components Range resolution is the ability of a rangefinder to resolve two objects on the same line of sight and is governed by pulse length for a pulsed system Kamerman 1993 Range measurement resolution is the ability to detect range 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd Blackwell Publishing Ltd 9600 Garsington Road Oxford OX4 2DQ UK and 350 Main Street Malden MA 02148 USA Licuti and JAMTsHo Angular resolution of terrestrial laser scanners differences between two angularly resolved objects and is directly proportional to timing resolution Wehr and Lohr 1999 Angular resolution the ability to resolve two objects on adjacent sight lines is a function of spatial sampling interval and the laser beamwidth For airborne laser scanner ALS systems the sampling interval is partially dependent upon aircraft motion whereas scanning mechanisms control it for terrestrial laser scanners TLSs Though resolution has many possible definitions it is commonly accepted that it is limited by both random and systematic errors den Dekker and van den Bos 1997 However resolution is a term that is often abused and misunderstood In terms of laser scanners emphasis is often placed on the finest possible sampling interval which is often much smaller than the laser beamwidth Since both factors influence the resolution of a scanned point cloud consideration of only one can lead to a misunderstanding of a systems capabilities To illustrate consider the article by Iavarone 2002 in which the author states that high scan resolution can be achieved by correlated sampling that is overlapping laser spots and therefore laser beam spot size is not a limiting factor While this is partially true in the sense that a fine sampling increment yields a high Nyquist frequency the benefit of correlated sampling is not fully realised because sampling is not the only factor that influences resolution A scanned point cloud may appear to have very high spatial resolution by virtue of a fine sampling interval and corresponding high point density The actual spatial resolution may be much lower if the beamwidth is large relative to the sampling interval because the fine details are effectively blurred It will be demonstrated in this paper that beamwidth can be a significant factor in reducing the spatial resolution of a scan cloud even in the presence of correlated sampling Though perhaps not an issue for smooth featureless surfaces it certainly could be for intricate surfaces with rapidly varying details that might be encountered in cultural heritage recording or asbuilt surveys of industrial plant The influence of finite laser beamwidth is discussed by several researchers Hebert and Krotkov 1992 Huising and Gomes Pereira 1998 Hodgson and Bresnahan 2004 but is not analytically quantified A new angular resolution measure for laser scanners the effective instantaneous field of view EIFOV is proposed It is derived from an ensemble average modulation transfer function AMTF that models the contributions of sampling beamwidth and angle quantisation The receiving system field of view is assumed equal to the beamwidth so it has no additional influence on resolution Its need is highlighted with real dataset examples that illustrate angular positional uncertainty due to beamwidth Following derivations of the AMTF and EIFOV and further experimental verification the angular resolution of several commercially available TLS systems is analysed Though the analyses focus on terrestrial systems the AMTF and EIFOV modelling approach can also be applied to ALS LASER SCANNER RESOLUTION Sampling Interval and Beamwidth Reporting Sampling and beamwidth reporting in sales literature tends to vary substantially from one vendor to the next which can cause further confusion about a systems capabilities Five examples are listed in Table I Four of the five manufacturers use angular measures for reporting angular sampling interval while one Leica uses a distance measure for the maximum sample density The Callidus and Optech specifications for beamwidth provide the most descriptive information in the form of initial diameter plus linear divergence as a function of range The Leica beam diameter specification is given for the range 0 to 50m while Trimble gives the diameter only at 50 m Riegl provides only diametric beam divergence Divergence may also 142 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record TABLE I Angular sampling interval and beamwidth reporting for some commercial TLS systems Leica 2004 2005 Lemmens 2004 Make Model Angular sampling interval Beamwidth Callidus CP 3200 00625 horizontal 12 mm 44 mrad 025 vertical Leica HDS 3000 Maximum sample density 12 mm 6 mm from 0 to 50m Optech ILRIS3D 20 pirad 12 mm exit diameter 170 rad divergence Riegl LMSZ420i 0004 horizontal 025 mrad divergence Trimble GS200 00018 Can be focused minimum is 3 mm at 50m be defined as the linear increase in radius see for example Weichel 1990 It is also worth noting that some instruments such as the Trimble GS200 offer a focusing capability that allows laser beam diameter optimisation for a given range Angular Resolution and Beamwidth Angular resolution is the ability to resolve two equally intense point sources on adjacent lines of sight den Dekker and van den Bos 1997 At the diffraction limit the classic Rayleigh criterion is often used Jelalian 1992 2442 1 7 D where 6 is the angular beamwidth D the illuminating aperture diameter and the wavelength The Rayleigh criterion gives the angle subtended by the first zeros of the Airy disk and encircles 84 of the illuminating beam power Kamerman 1993 However it is more relevant to uniformly illuminated beams than for laser beams with Gaussian profiles For a beam with a Gaussian wavefront the most common beam diameter definition is e which encircles 86 of the total beam power Marshall 1985 The diffractionlimited angular beamwidth for Gaussian beams is given by Jelalian 1992 42 A 6127 2 aD D 2 This measure is 52 of the Rayleigh criterion but in reality the beamwidth is larger than that indicated by the diffractionlimited criterion Kamerman 1993 In terms of divergence the beam radius w expands nonlinearly from the minimum radius the beam waist wo according to Weichel 1990 Ap wlan woylt 2s 3 TW where is range relative to the beam waist location For large ranges though a linear approximation of initial diameter or radius plus divergence is sufficient Positional Uncertainty Due to Beamwidth The TLS point clouds in Fig highlight the inherent angular positional uncertainty due to beamwidth In each case the coordinate system is externally defined the object space and the 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 143 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners 02 02 00 i OOF s 3 ar 02 Y 02 s E04 04 oS 06 06 é 5 08 a 08 tlle 2 10 10 0680 0675 0670 0665 0660 084 085 086 087 088 089 X m Xm a b Fic 1 a Leica HDS 2500 plumb line point cloud and best fit line b Rieg LMSZ210 plumb line point cloud and best fit line X axis scale has been greatly exaggerated Fig 1a is a point cloud of a plumb line 01 mm diameter scanned with a Leica HDS 2500 from a range of 55 m Also shown is the estimated plumb line determined by least squares 3D line fitting Twelve sampling profile lines intersect the plumb line as indicated by the 2 to 4mm long linear bands of points Angular measurement noise is apparent in the scatter of points about the centreline of each band The acute angle approximately 04note again the scale difference between axes between the plumb line and sampling profiles is due to the scanner not being levelled which is not possible with this instrument Fig 1b illustrates the point cloud and bestfit 3D line of a 1 mm diameter plumb line scanned with a Riegl LMSZ210 at 15m The horizontal axis scale differs from that of Fig 1b but the vertical scale is the same This scanner was purposely not levelled for data acquisition so several sampling profiles intersect the plumb line The bands of points are much longer due to the larger beamwidth of the Riegl instrument and quantisation noise is evident The correlated sampling of this scan is manifest as measurements recorded along three profile lines for any constant Z The band of points along each profile line is due to beamwidthinduced uncertainty in angular position Its cause is depicted schematically in Fig 2 For each point in this cloud the range measurement to the backscattering surface the plumb line is made to a point somewhere within the projected laser beam footprint Notwithstanding quantisation and other noise sources the apparent angular position of the range measurement is taken by convention to be the centre of the emitted beam Though a fine feature such as a plumb line can be resolved the actual angular position of the measured point may be biased by up to onehalf of the beam diameter The bias cannot be predicted because the scene phase that is the plumb line position is unknown The plumb line position can only be estimated with analytical techniques such as redundant geometric form fitting While this may appear to represent an extreme case it highlights very well the 144 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record Plumb line Sampling a profile line Position of range cee Circular measurement is gOS ircuvar taser somewhere in f beam footprint overlap region SS Apparent position of range measurement Sample locations Fic 2 Positional uncertainty of scanned plumb line samples due to laser beamwidth inherent positional ambiguity due to beamwidth that can exist in all point clouds but may be less obvious upon visual inspection Beamwidth uncertainty can also manifest itself at edges and tangent to curved objects such as cylindrical pipes A model that quantifies the ensemble average of beamwidthinduced phase shifts is proposed herein Equal Angular Increment Sampling Terrestrial laser scanner operation is depicted in Fig 3 in which the scanner is situated at the origin O of its internally defined Cartesian coordinate system A 3D scan of a scene can be compiled by mechanically deflecting the rangefinder laser beam in equal increments of arc in horizontal and vertical planes 4g and A respectively rather than an array of detectors Besl 1988 A scanned scene can thus be parameterised in terms of range p as a uniformly sampled function of horizontal direction 0 and elevation angle CO CO p500 S S pmdgnA50 mAg nA 4 MmO NOo where p is the sampled representation of the continuous scene p and 6 in this context represents the Dirac delta function The corresponding spatial sampling intervals are linearly dependent on range that is pA and pAy THE AMTF MOobDELLING APPROACH Ensemble Average Functions The sampled representation of a scene given by equation 4 is dependent upon the scene phase and thus is not shift invariant To cope with this for electrooptical imaging systems Park et al 1984 define the concepts of average system point spread function PSF and its 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 145 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners Z Circular beam envelope diameter 6 Za an ok Ge 4 f So X Ag Fic 3 Terrestrial laser scanner measurement Fourier transform the average system optical transfer function The average system PSF is an ensemble average function of randomly located point sources under the assumption that the independent variables 0 and in the present context are uniformly distributed on the sampling interval Park et al 1984 This permits application of modulation transfer function MTF analysisrestricted by definition to linear shiftinvariant systemsto sampled imaging systems Boreman 2001 Here the average MTF concept is applied to model the sampling process the laser beamwidth uncertainty and angular observation quantisation in order to derive a measure that accurately quantifies laser scanner angular resolution The physical bases for the sampling and beamwidth AMTFs are analogous to the sampling and detector footprint effects respectively in electrooptical imaging systems for example Boreman 2001 Scanner Sampling AMTF In the context of laser scanning the average PSF concept is applied to model the ensemble of random phase shifts of a scanned scene Taking the average over one element that is Ag x A of the sampling lattice in which the probability distribution is assumed to be uniform the resulting sampling average PSF APSF is a rectangular 2D step function 1 A A 0 Jal APSF 00 AgdAy 2 2 5 0 otherwise The normalising factor Ag A ensures that the transfer function magnitude is unity at the frequency origin The sampling AMTF is given by the modulus of the average point spread functions 2D Fourier transform sinzAg sinzAv AMTF uv sinn doy sinnAyv 6 TAG TAyV 146 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record where 4 and v are the horizontal and vertical spatial frequency domain variables respectively Functions for other sampling geometries such as hexagonal can also be derived Equation 6 represents the sampling AMTF corresponding to the directions of the spatial frequency domain coordinate system axes ff and v in this case Hadar et al 1997 demonstrate the angular dependence of the sampling AMTF for both rectangular and hexagonal sampling arrays Their work shows that resolution is considerably lower in directions other than parallel to the and v axes Though the present analysis is restricted to the directions of the coordinate axes 6 and yw and v attention is drawn to the fact that the resolution measures derived herein are not applicable in other directions Scanner Beamwidth AMTF An average PSF is used to model the ensemble of all possible phase shifts due to the finite beamwidth The probability governing the angular position of a range measurement is assumed to be uniform over the projected laser footprint Note that this does not refer to the irradiance distribution within the cross section which is typically Gaussian Assuming the beam cross section to be elliptical with axes dg and 6 the beamwidth average PSF is given by 4 PF ow 1 JDo5 typ APSF 0 a 169d dn 0 4 7 0 otherwise The corresponding AMTF is given by wi m 5pp2 a AMTF x v 8 1542 dev where J is the firstorder Bessel function of the first kind Equations 7 and 8 are easily modified for a circular beam and other crosssectional shapes can be modelled Quantisation AMTF Quantisation of the angular observations can also be modelled with the ensemble average approach Assuming that angular observations are rounded off as opposed to truncated and the quantisation steps are given by Qy and Q the uniform rectangular APSF is 1 x a APSF 00 4 QeQy 2 2 9 0 otherwise and the AMTF is AMTF 11 v Peet Smee 10 TOU TOxV Combined AMTF The product of equations 6 8 and 10 gives the combined AMTF for a rectangular sampling lattice and elliptical beam cross section and unequal angular quantisation increments 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 147 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners sinzAg sinnAv 2 ite d2v sin7Qgp sin7Qv AMTE gbq 14 11 TAG TA yV 7ydpu2 6 TOOL TOwV Equation 11 expresses the modulation loss as a function of spatial frequency Laser Scanner EIFOV Numerous measures exist for quantifying the resolution of imaging systems Holst 1998 gives a comprehensive treatment on the subject Since spatial domain metrics are often seen as easier to interpret than those in the frequency domain sampling interval and beamwidth might appear to be appropriate measures for laser scanners These will be demonstrated to be appropriate only under very specific conditions Use of the EIFOV is favoured for the analysis of electrooptical system resolution because it accounts for all factors such as the optics or electronics that may degrade image quality Slater 1975 It is a spatial domain measure of the average point spread function width Park et al 1984 and therefore a measure of system resolution Castleman 1996 The appropriateness of the EIFOV extends to laser scanners because it quantifies the combined effects of sampling beamwidth and quantisation The EIFOV is computed via the cutoff frequency EIFOV 12 675 2U The cutoff frequency j1 is the frequency at which the AMTF equals a threshold A that is AMTF s5q 4 13 The vertical angle dimension measures EIFOV and v are derived in the same manner Slater 1975 and Park et al 1984 use A 05 which was chosen as a compromise between several proposed thresholds Here 2 A x 06366 14 T is proposed to enforce the condition that the EIFOV A for A 6 ignoring quantisation and assuming equal sampling increments and a circular beam This simply reflects the fact that when the sampling interval is very coarse the beamwidth AMTF equation 8 main lobe is very broad and effectively has unit amplitude at the cutoff frequency The cutoff frequency is therefore governed solely by the sampling AMTF equation 6 and equals the Nyquist frequency In this situation a comparatively small beamwidth has no influence on angular resolution EXPERIMENTAL VALIDATION Experiment Description Experiments using plumb lines were conducted to validate the proposed modelling approach A 185 m long white plumb bob string 05 mm diameter was suspended from the end of a 2 m long steel bar centred atop a surveying tripod in a 100 m long corridor The string was tensioned with a plumb bob slightly offvertical to ensure that several vertical scan profile 148 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record lines intersected the line It was scanned at several ranges with a Leica HDS 2500 and a Riegl LMSZ210 The sampling interval for the Riegl was set to the finest possible 0072 For the Leica the sampling interval varied but was always less than 1 mm However in neither case was the observed point sampling interval uniform due to quantisation random angular measurement errors as well as systematic sinusoidal patterns in the Leica profile lines The returns from the plumb line were extracted from each point cloud for 3D bestfit line estimation Data near the floor had to be eliminated due to the integration of backscattered energy from the floor and from the plumb line Some data at close ranges had to be removed due to range biases caused by the high reflectivity of the plumb line Despite these problems data was available for at least 05 m of the plumb line length The number of points available for the line fitting ranged from 323 to 5681 for the Riegl and from 4439 to 27982 for the higher resolution Leica instrument The 3D line model is parameterised in scanner space coordinates as rrijs 15 where r is the 3D position vector for observed point i r is the position vector of the line s is the unit length direction vector of the line orthogonal to r and is the scale parameter for point observation 7 Following least squares estimation of the parameters on the right hand side of equation 15 3D residual vectors were computed Each points residual component in the direction mutually orthogonal to r and s that is in the 0 direction was calculated for the following analyses This procedure is analogous to the slit method for estimating the transfer function of optical imaging systems The slit method is a convenient technique that covers a broad range of spatial frequencies Williams 1999 One of its disadvantages stems from low amounts of optical energy passed through the slit However the plumb line technique proposed herein does not suffer from this drawback so long as there are many point observations of the plumb line as is the case here Recognising that both sampling interval and laser spot size influence resolution Bohler et al 2003 propose two means for TLS resolution assessment a test object comprising small elements such as a plumb line and a slotted target Focusing on the latter they constructed a star target of 300mm x 300mm with separation between the slotted front panel and the unslotted back panel of 55 mm They used it to test numerous scanners and the qualitative results they present give a good indication of each instruments capabilities A potential drawback to this design lies in the panel separation If it is less than half the pulse length for pulsed TLS systems then returns at the slot edgeswhich are important for resolution assessmentmay be biased due to the socalled mixedpixels effect Hebert and Krotkov 1992 Analysis Histograms of the orthogonal residual components are shown with their corresponding normalised magnitude spectra in Figs 4 and 5 for the Leica and Riegl data respectively The bin widths are 03 mm for the Leica and 10 mm for the Riegl Leica point cloud resolution is highest where the histogram width is minimal at 91 m The laser beamwidth of this instrument is 6mm for ranges up to 50 m but its exact behaviour is not known and would appear to be minimum in the vicinity of 91m The beam divergence beyond 50m is realised as the broadening histograms and corresponding narrowing of the spectra The Riegl beamwidth is known to be 40 mm at exit aperture Lemmens 2004 which closely corresponds to the histogram widths at close range The divergence is evident in the 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 149 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners o 1 3 200 3 05 0 0 15 10 5 0 5 10 15 0 01 02 03 500 g ro s 2 8 p91m 05 0 0 15 10 5 0 5 10 15 0 01 02 03 o 1 s 500 8 S 3 p248m 5 05 0 0 15 10 5 0 5 10 15 0 01 02 03 500 1 5 2 8 p508m 5 05 0 0 15 10 5 0 5 10 15 0 01 02 03 o 1 To 3 500 p608m 505 oO o 15 10 5 0 5 10 15 0 01 02 03 Orthogonal residual component mm Spatial frequency cycles mm Fic 4 Leica HDS 2500 plumb line scan orthogonal residual component histograms and spectra at different ranges histograms but fewer samples are available at longer ranges due to the coarser sampling increment Similar findings resulted from experiments conducted with other plumb lines black and green plumb lines for the Riegl and steel wire for the Leica At first glance it would seem that the histograms represent the ensemble average line spread functions LSFs of the respective instruments The relationship between the LSF and PSF is Williams 1999 Co LSF PSFO da 16 oo Ignoring quantisation and the finite thickness of the plumb lines assuming a circular beam and recognising that the sampling interval given by the histogram bin width is much smaller than the beamwidth in each case the circular beam LSF is given by 8 e LSF0 5 17 no VY 4 Clearly this semicircularshaped function does not match the histograms of Figs 4 and 5 150 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record o 1 3 c 8 0 0 50 0 50 0 001 002 003 004 200 g 1 e S 8 100 S 05 0 0 50 0 50 0 001 002 003 004 o 1 e s c 50 8 505 oN 0 0 50 0 50 0 001 002 003 004 o 1 e s Cc 10 0 0 50 0 50 0 001 002 003 004 o 1 To 10 2 505 6 5 S 0 0 50 0 50 0 001 002 003 004 Orthogonal residual component mm Spatial frequency cycles mm Fic 5 Riegl LMSZ210i plumbline scan orthogonal residual component histograms and spectra at different ranges The histograms are not the LSFs since the TLS response is effectively binary in so far as a range measurement is made if the backscattering surface falls within the beam footprint and enough energy is received at the detector If this condition is not fulfilled no observation is recorded For the plumb line scans the binary response process occurs in the direction orthogonal to the line This is inherently a nonlinear process and therefore equation 17 is not applicable A more representative analytical model for the histograms is that of ensemble average 1D cross sections of the 2D APSF in the direction orthogonal to the plumb line which under the conditions outlined earlier is given by the step function 4 6 Q APSF 00 762 2 18 0 otherwise The 1D Fourier transform of equation 18 is a sinc function Clearly the histograms and corresponding spectra in Figs 4 and 5 both conform to this model Distortions from the ideal forms exist due to random measurement noise finite plumb line width and histogram bin interval quantisation and slight curvature in some of the plumb lines The latter error was visible upon inspection of the point clouds but the maximum observed deflection was 1 mm over 175 m 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 151 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners As mentioned the rangefinding process described is nonlinear and as such linear system theory is not applicable Linear system theory has been used as a tractable and effective means to combine positional uncertainties due to sampling and quantisationboth of which are also nonlinear processesand beamwidth into one metric the EIFOV The potential exists to further develop the plumb line scanning technique into an effective method for beamwidth measurement A need for such a method clearly exists since some manufacturers do not provide complete laser beamwidth details AMTF ANALysIs OF TLS SysTEMS Resolution The angular resolution of 11 commercially available TLS systems is analysed using the EIFOV measure The selected systems are listed along with their pertinent resolution parameters in Table II and provide a good cross section of currently available instruments in terms of sampling interval and beamwidth To facilitate the comparison each vendors reported finest angular sampling interval and beamwidth and calculated EIFOV have been reduced to linear spatial units at a range of 50m This is made necessary by the variety of methods vendors use to report resolution information as discussed earlier but it is recognised that the relative resolutions of various instruments may be quite different at different ranges For the Callidus CP 3200 the finest horizontal sampling increment was used rather than the coarser vertical increment In all cases a circular laser beam cross section has been assumed Quantisation is neglected since this information was not readily available for all scanners though it will be demonstrated to be of only minor significance for two instruments The 11 systems can be classified into three groups according to EIFOV fine resolution scanners Leica HDS 2500 and HDS 3000 Trimble GS200 medium resolution instruments Faro LS 880 Optech ILRIS3D Riegl LMSZ420i and Zoller and Fréhlich Imager 5003 and coarse resolution instruments Callidus CP 3200 ISiTE 4400 Riegl LMSZ210 and its successor LMSZ210i All cases except for the Zoller and Frohlich instrument are examples of correlated sampling that is 4 6 in which the projected laser beam footprints of adjacent samples overlap While oversampling reduces aliasing by increasing the Nyquist frequency the benefit of doing so is diminished by resolution in terms of EIFOV reduction due to beamwidth In each case the more realistic EIFOV resolution measure is greater than the sampling interval by up to 21 times in the case of the Leica HDS 2500 At its finest sampling TABLE II Angular spatial resolution measures at 50 m for 11 commercial TLS systems Leica 2004 2005 Lemmens 2004 Make Model A mm 0 mm EIFOV mm EIFOVA Callidus CP 3200 545 2320 2056 38 Faro LS 880 07 155 133 19 ISiTE 4400 942 100 1246 13 Leica HDS 2500 025 60 52 21 Leica HDS 3000 12 60 53 44 Optech ILRIS3D 10 205 176 18 Riegl LMSZ210 628 1500 1415 23 Riegl LMSZ210i 87 1500 1291 15 Riegl LMSZ420i 35 125 112 32 Trimble GS200 16 30 30 19 Zoller and Frohlich Imager 5003 157 140 194 12 152 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record interval the point cloud resolution of the Zoller and Frohlich Imager 5003 is least influenced by beamwidth EIFOV 124 because it has the smallest beamwidthtosamplinginterval ratio 09 Similarly the resolution of the Callidus CP 3200 ISiTE 4400 Leica HDS 3000 Riegl LMSZ210 Riegl LMSZ420i and Trimble GS200 is not greatly affected by beamwidth as each have small EIFOVtosamplinginterval ratios Conversely the resolution of the Faro LS 880 Leica HDS 2500 Optech ILRIS3D and Rieg LMSZ210i scanners is greatly affected by beamwidth as indicated by their respective high ratios The highest resolution instrument is the Trimble GS200 not the Faro LS 880 Leica HDS 2500 Leica HDS 3000 or the Optech ILRIS3D that offer finer sampling increments due to their comparatively broader beamwidths The Callidus CP 3200 offers the lowest resolution due to its coarse sampling increment and broad beamwidth The finest available angular sampling interval of the Riegl LMSZ210i has been improved by 86 over the older LMS Z210 model However its resolution is not as significantly improved since the EIFOV is reduced by only 9 due to the large beamwidth common to both instruments The newer Leica model HDS 3000 offers a coarser sampling interval than the HDS 2500 but not at the expense of resolution as indicated by the nearly identical EIFOVs AMTF The AMTFs of the three fine resolution instruments are plotted along one frequency domain axis for positive spatial frequencies in Fig 6 Common to all functions is that they equal unity at the origin and are nonnegative for all frequencies due to the absolute value operation They decay rapidly to the first zero beyond which the secondary and higher side lobes have much lower amplitude than the main lobe Both sampling and beamwidth govern main lobe width which is of primary interest and the locations of the zeros whose spacing may not be uniform since equation 8 is aperiodic in m and v 1 vy vey Leica HDS 3000 09 vs Trimble 68200 we Cutoff frequency 7 08 07 woe ee eee ee Sutff frequency threshold 2 n 06 iN Ie Eos ry 1 04 on 03 7 02 01 Po 0 JO Np Se 0 01 02 03 04 05 06 Spatial frequency cycles mm Fic 6 Fine resolution instrument AMTFs at 50 m range 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 153 Licuti and JAMTsHo Angular resolution of terrestrial laser scanners The resolution hierarchy of the three TLS systems is clearly evident in the main lobe widths and cutoff frequencies For example the AMTF of the highest resolution instrument the Trimble GS200 has the broadest main lobe and highest cutoff frequency The narrowest main lobe and lowest cutoff frequency belong to the lowest resolution instrument of this group the Leica HDS 3000 despite it offering a slightly finer 12 mm versus 16 mm sampling interval than the Trimble GS200 The AMTFs of the medium and coarse resolution instruments are plotted in Figs 7 and 8 respectively Note the different frequency axis scales The highest and lowest resolution instruments of the medium group are the Riegl LMSZ420i and the Zoller and Frohlich Imager 5003 respectively For the coarse resolution group the ISiTE 4400 is the highest resolution scanner and the Callidus CP 3200 the lowest EIFOV versus Sampling Interval Fig 9 shows the EIFOV as a function of sampling interval for the fine and medium resolution instruments again in terms of linear units at 50 m to facilitate direct comparison Fig 10 with differs in scale by a factor of 10 presents the data for the coarse resolution instruments The line EIFOV A is plotted for reference in both figures The leftmost endpoint of each curve corresponds to the values given in Table II that is finest sampling interval and EIFOV Note that the Leica HDS 2500 and HDS 3000 curves are coincident except at small sampling intervals as are the Riegl LMSZ210 and LMSZ210i curves The curve hierarchy on the EIFOV axis corresponds to the beamwidth that is smaller beamwidth instruments have smaller EIFOVs and thus higher angular resolution Many of the curves have nearly constant trend at small sampling intervals where beamwidth most influences the EIFOV The trend gradient then increases and eventually converges to the line EIFOV 4 where the influence of beamwidth on resolution is negligible The convergence rate appears to be 1 eg te Say Optech ILRIS3D 09 So wens Riegl LMS2420i Faro LS 880 Not 7F Imager 5300 08 Cutoff frequency 11 we 07 Note oo NY LN fee Cutoff frequency threshold 27 06 WOON fe 2 iy PAL ON WL DA Yt E 05 it oN Ss rt boy xt tt 1 04 pS 4 e tt 1 03 4 AN 1 4 1 02 it tot mt os it 14 Decne 04 Tp eetet eee 0 7 K eat ae cee 0 005 01 015 Spatial frequency cycles mm Fic 7 Medium resolution instrument AMTFs at 50 m range 154 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record 1 Callidus CP 3200 Calliaus 09 Rieg LMS2210 sees Riegl LMSZ210i SITE 4400 08 Cutoff frequency 1 07 Ss eee eee NAL Cutoff frequeney threshold 27 06 NON 2 NS uw i E 05 i Voi ies Hl ol i Yt 04 PON ar A 1 03 i1 iM 02 i PL TA a 04 en UNS pee i yo SS mae 0 poo UN OS 8 ie 0 0005 001 0015 Spatial frequency cycles mm Fic 8 Coarse resolution instrument AMTFs at 50 m range inversely proportional to beamwidth For example the curves of the Trimble GS200 and Leica HDS 3000instruments with fine beamwidthsconverge very rapidly whereas those of the broad beamwidth instruments converge slowly Note also that 4 EIFOV that is the EIFOV is never less than the sampling interval under the constraint given by equation 14 Table II and Figs 9 and 10 demonstrate that neither sampling interval nor beamwidth adequately quantify angular resolution From Figs 9 and 10 it is clear that sampling interval is appropriate only when it equals the EIFOV that is when 4 6 Beamwidth is equivalent to EIFOV and thus an appropriate resolution measure for one multiple of 4 4 05456 This coefficient can be estimated by setting EIFOV 6 and solving equation 13 for the ratio Aé As an example consider the Trimble GS200 case in Table II where 6 EIFOV 30mm A 16 mm 05456 The Effect of Quantisation The influence of quantisation is analysed for the Riegl LMSZ420i in the vertical dimension Given the quantisation increment of Q 0002 Riegl 2005 or 17 mm at 50 m range and the information in Table II the three constituent and overall AMTFs are plotted in Fig 11 The quantisation AMTF has a very broad main lobe due to the small increment so it causes only minor modulation reduction The EIFOV is 113 mm which represents only a 09 reduction in resolution due to quantisation Quantisation has also been analysed for the older Riegl LMSZ210 scanner Using the data from Table II and the quantisation increments Qg 0018 and Q 0036 Riegl 2001 the respective horizontal and vertical EIFOVs at 50m range are 1423 mm and 1445 mm These represent 06 and 21 reductions in resolution respectively Clearly the effect of quantisation is only of minor significance and the dominant factors influencing the resolution of these two instruments are sampling and beamwidth 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 155 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners 50 45 ey 40 4 a i p oe WZ 35 aff 30 fy ig SF Pa ig a CG S eG Ww ae ZS i a 27 20 OLS Leica HDS 3000 pancccree 7 1 s Leica HDS 2500 7 o 15 C7 fo Trimble GS200 OO Optech ILRIS3D 10 fae Riegl LMSZ420i Ag Faro LS 880 w 5 Ta ZF Imager 5300 oe 77 7 EIFOVA 6 5 10 15 20 2 30 35 40 45 50 A mm Fic 9 EIFOV versus sampling interval at 50m range for fine and medium resolution instruments The line EIFOV A is plotted for reference Optimal Sampling Interval A question that naturally arises is what is the optimal sampling interval given a fixed beamwidth ignoring possible beam focusing capabilities If the sampling increment is set to be larger than the beamwidth then resolution will be poor If it is set to be finer than beamwidth then as demonstrated the full benefit of correlated sampling is not realised Setting the sampling interval equal to the beamwidth might appear optimal but this yields a mismatch in passbands because the main lobe of equation 8 is slightly broader than that of equation 6 The curves shown in Figs 9 and 10 indicate that a minimum EIFOV as a function of sampling interval does not exist even if subjected to a constraint such as minimum scan time which can be considered inversely proportional to the square of the sampling interval Therefore it is proposed that the optimal sampling interval be defined such that its AMTF matches that of the beamwidth at the cutoff frequency threshold Without loss of generality analysis is performed in one dimension and quantisation is ignored To match passbands and the cutoff frequency threshold equation 14 the cutoff frequency of the sampling AMTF equation 6 must be set equal to that of the beamwidth AMTF equation 8 Solving for 4 gives the optimal sampling interval of Aopt 085906 19 Thus the optimal sampling interval in the matched passband sense is 86 of the beamwidth 156 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record 500 7 Sa 7 fe 7 7 450 ALE S en 74 ie 400 MME 7 an 7 7 7 O7 350 oe Zz ta 7 if LY Za 0G 300 Me E a My 4 Or 6 250 oo Vv Tr 20 0 a en e 200 Ce 150 a 100 Callidus CP 3200 Vv ISiTE 4400 a Riegl LMSZ210 50 a tresses Riegl LMSZ210i EIFOVA ok 0 50 100 150 200 250 300 350 400 450 500 A mm Fic 10 EIFOV versus sampling interval at 50 m for coarse resolution instruments The line EIFOV A is plotted for reference SUMMARY AND CONCLUSIONS Sampling interval laser beamwidth and angular observation quantisation affect the spatial resolution of laser scanners though the former two factors are most significant for at least two of the instruments examined The effective instantaneous field of view has been proposed as a more appropriate measure of resolution since neither sampling interval nor beamwidth are adequate except under certain conditions To derive the EIFOV the shiftvariant sampling process beamwidth effect and quantisation have been modelled with ensemble average modulation transfer functions and combined into one AMTF The EIFOV is derived from this overall function Eleven commercially available TLS systems have been analysed in terms of their angular resolution capabilities Perhaps the most important result of this process is that a fine angular sampling interval does not necessarily produce a highresolution point cloud if the beamwidth is significant Even though a small in relation to beamwidth feature can be sensed its angular position may be significantly biased as indicated by the plumb line examples A fine angular resolution quoted for an instrument should be scrutinised critically unless it is much greater than the angular beamwidth since the actual resolution indicated by the EIFOV will be much larger The ratio of EIFOV to finest sampling interval reached up to 21 for one of the systems analysed The benefit of fine sampling intervalhigher Nyquist frequencytherefore may not be realised due to a comparatively large beamwidth This was confirmed by the TLS system analysis in which it was found that the instruments offering 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 157 Licuti and JAMTsHo Angular resolution of terrestrial laser scanners 1 SS 09 tt ae 08 te St 07 te 06 ea 05 se mh ft O4 mm 03 02 Sampling eeamwvicth Quantizai 01 Overall oo 0 Joe 0 002 004 006 008 01 O12 014 016 018 02 Spatial frequency cycles mm Fic 11 Riegl LMSZ420i constituent and overall AMTFs at 50 m with quantisation model the finest sampling interval did not possess the highest resolution in terms of EIFOV due to its comparatively broad beamwidth Some rules of thumb can be derived from the numerical results presented herein When the sampling interval is much larger than the beamwidth it is equal to the EIFOV This is the only condition under which sampling interval accurately represents resolution When the sampling interval is approximately 55 of the beamwidth the latter equals the EIFOV the only condition under which beamwidth accurately describes resolution The optimal sampling interval defined in terms of matched AMTF passbands at the cutoff frequency threshold was found to be 86 of the beamwidth These observations should not be interpreted as criticism of the scanner systems or their inventors The engineering skill required to develop a working scanning system is indeed impressive Rather the message to be gained by readers is that resolution is primarily a function of both sampling interval and beamwidth and as a result the attainable resolution of any system will invariably be coarser than is indicated by either of these measures Thus the more appropriate EIFOV should be used to measure resolution It is also important to bear in mind that resolution has been assessed only at one range 50 m which was the most practical choice to demonstrate the proposed analysis method given the available data However the relative resolution of the instruments analysed may be quite different at different ranges Thus to gain complete insight into a laser scanners capabilities the EIFOV must be computed as a function of range throughout its full operating range REFERENCES BESL P J 1988 Surfaces in Range Image Understanding SpringerVerlag New York 339 pages BOHLER W BoRDAS VICENT M and Marss A 2003 Investigating laser scanner accuracy International Archives of Photogrammetry Remote Sensing and the Spatial Information Sciences 345C15 696701 158 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd The Photogrammetric Record BoreMaNn G D 2001 Modulation Transfer Function in Optical and ElectroOptical Systems SPIE Optical Engineering Press Bellingham Washington 120 pages CASTLEMAN K R 1996 Digital Image Processing Prentice Hall Englewood Cliffs New Jersey 667 pages DEN DEKKER A J and VAN DEN Bos A 1997 Resolution a survey Journal of the Optical Society of America A 143 547557 Hapar O DoGariu A and BoREMAN G D 1997 Angular dependence of sampling modulation transfer function Applied Optics 3628 72107216 HEBERT M and Krorkov E 1992 3D measurements from imaging laser radars how good are they Image and Vision Computing 103 170178 Hopason M E and BRESNAHAN P 2004 Accuracy of airborne lidarderived elevation empirical assessment and error budget Photogrammetric Engineering Remote Sensing 703 331339 Ho st G C 1998 Sampling Aliasing and Data Fidelity for Electronic Imaging Systems Communications and Data Acquisition SPIE Optical Engineering Press Bellingham Washington 342 pages HulIsina E J and Gomes PEREIRA L M 1998 Errors and accuracy estimates of laser data acquired by various laser scanning systems for topographic applications SPRS Journal of Photogrammetry and Remote Sensing 535 245261 TAVARONE A 2002 Laser scanner fundamentals Professional Surveyor 229 httpwwwprofsurvcom psarchivhtm Accessed 4th March 2004 JELALIAN A V 1992 Laser Radar Systems Artech House Boston 292 pages KAMERMAN G W 1993 Laser radar Chapter 1 in Active ElectroOptical Systems Vol 6 Ed C S Fox In The Infrared and ElectroOptical Systems Handbook Infrared Information Analysis Center Ann Arbor Michigan 312 pages 176 LeIcA 2004 httpwwwcyracomproductshds2500specshtml Accessed 4th March 2004 Leca 2005 httphdsleicageosystemscomproductsHDS3000specshtml Accessed 6th May 2005 LeEMMENS M 2004 3D lasermapping GJM International 1812 4447 MARSHALL G F 1985 Gaussian laser beam diameters Chapter 6 in Laser Beam Scanning OptoMechanical Devices Systems and Data Storage Optics Marcel Dekker New York 448 pages 289301 Park S K SCHOWENGERDT R A and KAczyNskI MA 1984 Modulationtransferfunction analysis for sampled image systems Applied Optics 2315 25722582 RIEGL 2001 LMSZ210 Laser Mirror Scanner Technical Documentation and Users Instructions Rieg Laser Measurement Systems GmbH Austria 82 pages RIEGL 2005 httpwwwrieglcoatterrestrialscannersImsz420i420iallhtm Accessed 6th May 2005 SLATER P N 1975 Use of MTF in the specification and firstorder design of electrooptical and photographic imaging and radiometric systems Optica Acta 224 277290 Wenr A and Lour U 1999 Airborne laser scanningan introduction and overview JSPRS Journal of Photogrammetry and Remote Sensing 5423 6882 WEICHEL H 1990 Laser Beam Propagation in the Atmosphere SPIE Optical Engineering Press Bellingham Washington 108 pages WILLIAMS T L 1999 The Optical Transfer Function of Imaging Systems Institute of Physics Publishing Bristol 504 pages Résumé Il est nécessaire de connaitre la résolution spatiale des scanneurs a laser pour éviter des dégradations de type aliasing et estimer le niveau de détail que lon peut obtenir a partir dun nuage de points scannés On peut identifier dans la résolution spatiale une composante en distance et une angulaire Cette derniére qui fait lobjet de cet article est essentiellement tributaire de lintervalle déchantillonnage et de la largeur du faisceau laser Mais on considére généralement que cest seulement lintervalle déchantillonnage qui constitue lindicateur unique de résolution Or les deux facteurs affectent la résolution de tout nuage de points scannés de sorte que si lon en examine un indépendamment de lautre on peut aboutir a une mauvaise estimation des possibilités du systéme On a méme pu démontrer que cétait tout a fait inadéquat sauf dans certains cas particuliers Cest pourquoi on propose un nouveau moyen mieux adapté pour déterminer la résolution des scanneurs a laser en se basant sur le cone danalyse instantané efficace EIFOV On dérive ce dernier par 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd 159 LicuTi and JAmTsHo Angular resolution of terrestrial laser scanners une modélisation faisant intervenir la variance de glissement du processus incremental déchantillonnage équiangulaire lincertitude du positionnement provoqué par la largeur du faisceau laser et la quantification angulaire observée et en mélant les fonctions de transfert de modulation moyennes de lensemble AMTFs On a ainsi pu modeéliser et analyser plusieurs systemes de scanneurs terrestres a laser du commerce en ce qui concerne leurs possibilités de résolution angulaire grace a cette notion dEIFOV Zusammenfassung Mit der Kenntnis der rdumlichen Aufldsung eines Laserscanners lassen sich Aliasing Effekte vermeiden und der Detaillierungsgrad abschdtzen der aus einer gescannten Punktwolke abgeleitet werden kann Die rdumliche Aufldsung Idsst sich in eine Entfernungs und eine Winkelkomponente aufteilen Letztere steht hier im Vordergrund und wird vor allem durch das Abtastintervall und die Laserstrahlbreite bestimmt Oftmals wird jedoch als Indikator fiir die Aufldsung nur eines dieser Kriterienmeist das Abtastintervallherangezogen Da jedoch beide Kriterien die Auflosung der gescannten Punktwolke beeinflussen kann die von dem anderen Kriterium unabhdngige Betrachtung zu Fehlinterpretationen der Systemeigenschaf ten fiihren Es wird gezeigt dass dies bis auf wenige Spezialfalle zutrifft Ein besseres Maf fiir die Auflésung eines Laserscanners ist das effektive aktuelle Gesichtsfeld EIFOV Es wird durch eine Modellierung der Verschiebungsvarianz des Abtastprozesses auf der Basis gleicher Winkelelemente der durch die Laserstrahl breite induzierten Lageunsicherheit und der beobachteten Winkelquantisierung mit Hilfe von gemittelten Modulationstransferfunktionen AMTFs abgeleitet Mehrere kommerziell erhdltliche terrestrische Laserscanning Systeme werden hinsichtlich ihrer Winkelauflosungseigenschaften mit EIFOV modelliert und analysiert Resumen Para evitar el pixelado de bordes y estimar el nivel de detalle que se puede obtener de una nube de puntos escaneada es necesario conocer la resolucion espacial del escaner laser La resolucion espacial se puede descomponer en una componente de rango y una angular El objeto de este articulo es esta ultima componente que viene determinada principalmente por el intervalo de muestreo y la anchura del haz del laser Sin embargo con frecuencia se hace énfasis sdlo en uno de esos dos factores habitualmente el intervalo de muestreo como indicador de la resolucion Como ambos afectan a la resolucion de una nube de puntos escaneada considerar solo un factor independiente del otro puede conducir a una mala interpretacion de las capacidades de un sistema Se demostrard que este enfoque no es el mas adecuado excepto en circunstancias muy especificas Se propone el campo efectivo instantaneo EIFOV como una nueva medida de la resolucion mas adecuada para los instrumentos de barrido laser Esta medida se obtiene mediante un modelo de la varianza del desplazamiento de los incrementos angulares constantes del proceso de muestreo de la incertidumbre en la posicion inducida por la anchura del rayo laser y por la cuantizacion del angulo observado sobre la media poblacional de la funcion de transferencia de modulacién AMTF Se modelan varios sistemas comerciales de escaner laser terrestre y se analizan en funcion de su capacidad de resolucion angular utilizando el EIFOV 160 2006 The Authors Journal Compilation 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd