range determination with waveform recording laser systems using a wiener filter

emote Sensing 61 (2006) 95–107www.elsevier.com/locate/isprsjprs

ISPRS Journal of Photogrammetry & R

Range determination with waveform recordinglaser systems using a Wiener Filter

Boris Jutzi a,⁎, Uwe Stilla b

a FGAN-FOM Research Institute for Optronics and Pattern Recognition, 76275 Ettlingen, Germanyb Photogrammetry and Remote Sensing, Technische Universitaet Muenchen, 80290 Muenchen, Germany

Received 28 December 2005; received in revised form 31 August 2006; accepted 1 September 2006

Abstract

Current pulsed laser scanning systems determine the range to an object surface by a time-of-flight measurement. Criticalmeasurement situations occur in discriminating the ranges of surfaces close to their edges or of small objects within the beamfootprint which are closely located in range. Capturing the complete waveform of the laser pulse allows discriminating differencesin a range smaller than the length of the laser pulse. The capabilities of this technique can be predicted by modeling the emittedpulse, the surface, and the backscattered pulse. Due to the varying waveforms of the emitted pulses each individual emitted pulse isrecorded and considered for the determination of the surface features. A deconvolution is used to remove the characteristic of thetransmitted waveform from the received waveform to obtain a surface response. A Wiener Filter reduces the noise of thedetermined surface response. For extraction of temporal position, length, and amplitude the corresponding surface features areapproximated by Gaussians using the Levenberg–Marquardt Method. Experiments have shown that a stepped surface within thebeam with a step smaller than ten times of the pulse length can be distinguished.© 2006 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rightsreserved.

Keywords: Laser scanning; Waveform analysis; Signal processing; Feature extraction

1. Introduction

The automatic generation of 3-d models describingman-made objects like buildings is of great interest inphotogrammetric research (Stilla et al., 2005). Inphotogrammetry the distance to a surface is classicallyderived from a triangulation of corresponding imagepoints from two or more pictures of the surface. Thepoints are chosen manually or detected automatically byanalyzing image structures. Besides this indirect mea-

⁎ Corresponding author. Tel.: +49 7243 992 337; fax: +49 7243 992299.

E-mail address: [email protected] (B. Jutzi).

0924-2716/$ - see front matter © 2006 International Society for PhotogrammAll rights reserved.doi:10.1016/j.isprsjprs.2006.09.001

surement using object characteristics, which depends onnatural illumination, active laser scanning systems allow adirect and illumination-independent measurement of therange. Laser scanners capture the range of 3-d objects in afast, contactless, and accurate way. An overview ofairborne laser scanning systems is given in Huising andGomes Pereira (1998), Wehr and Lohr (1999), Baltsavias(1999).

Current pulsed laser scanning systems for topographicmapping are based on time-of-flight ranging techniques todetermine the distance to the illuminated object. The time-of-flight is derived by the elapsed time between the emittedand backscattered laser pulses. The signal analysis todetermine this time typically operateswith analog threshold
etry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V.
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96 B. Jutzi, U. Stilla / ISPRS Journal of Photogrammetry & Remote Sensing 61 (2006) 95–107

detection (e.g. peak detection, leading edge detection,constant fraction detection) (Jutzi and Stilla, 2003). A fewsystems capture multiple reflections caused by objects atdifferent ranges with surfaces smaller than the footprint ofthe beam. Most systems capture the first and the lastbackscattered laser pulse. First pulse as well as last pulseexploitation is used for different applications like urbanplanning or forestry surveying. While first pulse registra-tion is the optimal choice to measure the hull of partiallypenetrable objects (e.g. canopy of trees), last pulseregistration should be chosen to measure non-penetrablesurfaces (e.g. ground surface). Fig. 1a shows a section of animage taken in first pulse mode. The foliage of the trees isvisible. Fig. 1b was taken in last pulse mode. The branchesand foliage are not visible anymore.

Critical measurement situations can occur if a singlepulse is strongly deformed or more than one pulse is

Fig. 1. Section of an urban scene. a) Elevation images captured byfirst pulse mode, b) elevation images captured by last pulse mode,c) difference image of first and last pulse mode.

backscattered (Fig. 2). The following examples give aselection of these situations:

i. A sloped surface covers a range interval within thefootprint of the beam and leads to a deformation(widening) of the backscattered pulse dependingon the slope and the size of the footprint. Analogdetectors, which are for example measuring theleading edge of the pulse, only find a single rangevalue which is typically shifted from the meandistance of the surface towards the sensor.

ii. Two different elevated areas within the footprint ofthe beam, as occurring at building walls (groundand roof) lead to two pulses with a temporal offset.In this case, analog detectors can measure twodifferent range values, but the coverage of the areaswithin the footprint cannot be determined withoutany further information (Vosselman, 2002).Depending on the processing of first or last pulsedata, building areas dilate or erode. To visualize thevarious sizes of the building footprints in first andlast pulse image, a difference image was calculated(Fig. 1c). The ambiguous pixels of the building arevisible as a bright stripe along the buildingcontours. A section of this ambiguous area wasenlarged and is depicted in Fig. 1c (bottom left).

iii. Two areas at slightly different elevations within thefootprint of the beamwill lead to two superimposedbackscattered pulses (Wagner et al., 2006) if therange difference of the areas is smaller than thelength of the laser pulse (Katzenbeisser, 2003). Ingeneral an analog detector measures a single rangevalue.

iv. Small objects within the footprint distributedrandomly around a mean elevation value (e.g.average height of crop plants on a field) lead to awidening of the backscattered pulse depending onthis distribution. This results in a widened pulsewith a low intensity which is difficult to detectwith a fixed threshold of an analog system.

These examples suggest that the complete waveformin between the first pulse and last pulse might be ofinterest, because it includes the backscattering charac-teristic of the illuminated field.

Prior work on analyzing the full waveform wascarried out by NASA to examine vegetation with respectto bio mass, foliage, or density (e.g. trees, bushes, andground). The Laser Vegetation Imaging Sensor (LVIS)was used to record the waveform and to determine thevertical density profiles in forests (Blair et al., 1999). Thespaceborne Geoscience Laser Altimeter System (GLAS)

Fig. 2. Surface characteristic and pulse form. a) Plane surface, b) sloped surface, c) two significantly different elevated areas, d) two slightly differentelevated areas, e) randomly distributed small objects.

97B. Jutzi, U. Stilla / ISPRS Journal of Photogrammetry & Remote Sensing 61 (2006) 95–107

determines the distance to the Earth's surface, and aprofile of the vertical distribution of clouds and aerosols(Brenner et al., 2003). In the work of both systems thesurface characteristics were determined by comparing aparametric description of the transmitted and receivedwaveform. The shape of the surface response was notdetermined.

Apart from the range measurement of laser scanningsystems, some systems additionally deliver a singlereflectance value derived from the amplitude or thepower of the backscattered laser light. The amplitude isdefined as the signal maximum and the power by signalintegration of the measured laser light. These values giveradiometric information about the surveyed area. Thisamplitude value or power value can be used for separatingsegments of artificial objects from vegetation (Hug andWehr, 1997; Maas, 2001), classify individual trees orforest stands according to species (Moffiet et al., 2005) orto perfectly texture 3-d scene models (Sequeira et al.,1999). Vosselman (2002) suggested to use the intensity ofthe laser beam response in order to estimate and improvethe position of the edge in between areas with differentreflectance properties. Examining the power of thebackscattered pulse allows increasing the positionaccuracy of an object edge, too (Jutzi et al., 2005).

To enable an interpretation of the measured waveformof a backscattered pulse, understanding the physicalbackground of pulse propagation and surface interactionis necessary (Der et al., 1997; Jutzi et al., 2002; Steinvalland Carlsson, 2001; Wagner et al., 2004, 2006).Especially in the case of multiple backscatters, thereceived waveform may be complex. For analysis of thewaveform and to separate fine structures, models for pulsepropagation and surface interaction have to be introduced.

In this paper, we propose a method for a detailedanalysis of the full waveform of laser pulses. The

presented approach improves the range resolution andallows discriminating multiple surface responses. InSection 2, the waveform of the emitted and back-scattered pulse, the laser beam, and the illuminatedsurface are modeled for capturing the measurementsituation. The algorithms for gaining and discriminatingsurface responses are described in Section 3. In Section4 outdoor experiments with different surface configura-tions are presented. Finally in Section 5, the receiveddiscrimination results of the surface responses arediscussed concerning accuracy and reliability.

2. Modeling

In this section, a model for the waveform of thebackscattered laser pulse is derived. This waveformdepends on the transmitted waveform of the emittedlaser pulse, the spatial energy distribution of the beam,and the material and geometric reflectance properties ofthe surface.

2.1. Modeling the transmitted waveform of the emittedpulse

Depending on the laser system, the waveform of apulse may appear in different shapes. Different modelsof the waveform are known from the literature. Brenneret al. (2003) proposed a simple temporal symmetricGaussian distribution for modeling the waveform of thespaceborne Geoscience Laser Altimeter System(GLAS). A waveform with an exponential distributionis applied by Steinvall (2000), while Wagner et al.(2004) uses a rectangular distribution. The laser source'smodulation effect was not captured in the mentionedmodels. Some laser systems, like the used multi modeErbium fiber laser, typically show variations of the

Fig. 3. Examples of the transmitted waveform.


waveform, which cannot be neglected for a detailedanalysis. Fig. 3a–c gives examples of emitted pulses ofour laser system.

Generally, the basic waveform depends on the pulsegeneration process of the laser source. A strong randomintensity fluctuation is caused by the mode-beating ofthe multi mode laser light, which is generated byinterference of neighboring longitudinal modes withinthe laser resonator. Furthermore, external disturbances,fluctuations of relaxation oscillation, and inharmonicoscillation (spiking) randomly modulate the waveform.

The basic waveform s(t) of the used laser system canbe described by a time delayed Gaussian with theamplitude a and the pulse length w:

sðtÞ ¼ 2aw

d

ffiffiffiffiffiffiffiffiln 2p

rd expf−4d ln 2d ðt−sÞ

2

w2gd ð1Þ

The length of a pulse is defined by one-half of thepulse's maximum amplitude, known as Full-width-at-half-maximum (FWHM).

The random modulation of the basic waveform,caused by intensity fluctuations, is modeled by noisefollowing a Gaussian distribution m(t). This multipli-cative component is defined by the parameters μ forthe modulation offset and σ representing the modula-tion standard deviation. Finally, the modulated wave-form sm(t) is

smðtÞ ¼ sðtÞd mðtÞ with mðtÞfNðl; r2Þ: ð2ÞOther laser systems may generate waveforms which

appear with different shapes, e.g. a Q-switched laserwith an exponential shape. The exponential waveformcan be described by s(t)=t2 . exp{− t /w}. If the shape ofthe transmitted waveform differs from the presented themodel has to be adapted.

2.2. Modeling the spatial energy distribution of theemitted pulse

The spatial energy distribution of a laser pulse (beamprofile) depends on the used pump source, the opticalresonator, and the laser medium. Often, these profilesare modeled by a cylindrical distribution (top-hat form)or by a 2d-symmetric Gaussian distribution (Kamer-mann, 1993). Measurements of the beam profile in thenear field have shown that a cylindrical distribution fitsour data best (Fig. 4). For the treatment of monostaticsystems it is convenient to use a spherical coordinatesystem with origin in the detector and emitter lens. Wetake the optical axis as the polar axis of the coordinate

system and call the range r, the zenith angle ε, and theazimuth angle α. In this case the spatial irradiancedistribution K of the laser beam over the angle ε isdescribed by

KðeÞ ¼ Ece20k if 0V eV e0 ¼ beam radius;0 otherwise:

�ð3Þ

2.3. Modeling of the surface

In this section, we describe the influence of thesurface on the laser beam by geometrical shape andmaterial properties. Capturing the interaction of theemitted pulse with the illuminated surface is relevant toanalyze the received waveform. The effects on theobserved data of the surface structure are dependent ontheir scale. Structures with scales below the laserwavelength may result in interference effects (speckle).If the receiver aperture is large compared to thecorrelation size of the speckle pattern, the intensityfluctuations due to speckle will be smoothed by aperture

Fig. 4. Spatial energy distribution of the system: measured profile (left)and modeled profile (right).

Fig. 5. Schematic illustration of the surface geometry. a) Side view of asingle surface, b) oblique view of a single surface, c) oblique view oftwo differently elevated surfaces.


averaging. In this case only insignificant fluctuations ofthe waveform with slightly lower amplitude aremeasured. Because these conditions are often met,speckle effects will be neglected in this paper.

2.3.1. Geometrical reflectanceFor handling the imaging properties of extended

surfaces, we assume the following: (i) object and sensorare stationary, (ii) the illuminated part of the object islocally plane, (iii) the distance from the sensor is verylarge compared to the beam footprint. In this case weassume for the area of the footprint that the plane surfacediffers from the corresponding sphere only marginally.Let us denote the range of a unbounded surface A alongthe optical axis by r0A, the angle between the surfacenormal and the optical axis by φ, the slope direction byαA, and the beam radius by b. The observation geometryis depicted in Fig. 5a and b. The range rA(ε,α) of anilluminated surface point depends on zenith angle ε andazimuth angle α and is given by

rAðe; aÞ ¼ r0Að1þ e d tanðuÞ d cosða−aAÞÞ: ð4Þ

2.3.2. Material reflectanceUsing a monostatic system, the angles of the incident

and reflected light coincides, but the beam attenuation isstill related to the angle of incidence φm. A special caseis given for specular surfaces with material reflectanceρm=1:

qspecularðuÞ ¼ 1 if u ¼ 0;0 if u p 0:

�ð5Þ

For many surfaces the reflectance is assumed to beuniform and isotropic, also known as Lambertian

diffuser. In this case the observed reflectance dependson material reflectance ρm and the angle of incidence φ:

qdiffuseðuÞ ¼ qmd cos2ðuÞ: ð6Þ

A discussion of reflectance measurements for dif-ferent materials is shown in Jelalian (1992). There arestrong variations within the reflectance ρm for the samematerial, caused by the measurement situation. These


values should be treated with some caution if they are tobe used for material discrimination.

Combining the information presented in Sections2.3.1 and 2.3.2 we finally get the general expression forthe reflectance density function ρA of a diffuse orspecular surface with respect to the geometry andmaterial:

qAðr; e; aÞ ¼ qdiffuseðuÞd dðrAðe; aÞ; rÞ¼ qmd cos

2ðuÞd dðrAðe; aÞ; rÞ; orqAðr; e; aÞ ¼ qspecularðuÞd dðrAðe; aÞ; rÞ;

ð7Þ

where δ represents the delta function with dðrA;rÞ ¼1 if r ¼ rA;0 if r p rA:

�The delta function δ(rA,r) is used to

describe the surface in range rA.

2.3.3. Multiple surface reflectionsWe have assumed that just one plane object surface

occurs within the laser beam. If differently elevatedobject surfaces are partly illuminated by the laser beam,we will receive a mixture of reflections at differentranges. Multiple reflections of the surfaces A1, A2, …,An within the beam corridor are calculated by:

qmultipleðr; e; aÞ ¼Xni¼1

qAiðr; e; aÞ

�dðrAiðe; aÞ;minðfrAiðe; aÞji ¼ 1:::ngÞÞ:

ð8ÞThe delta function δ is used to describe each single

surface Ai in range rAi. Illuminated surface areas are

described by min({rAi(ε,α)|i=1… n}) which considers

the line of sight and disregards the occluded areas. Thecomplete reflection ρmultiple is given by the sum of allpartial surface reflections.

2.4. Atmospheric transmission

The one-way propagation path for the atmospherictransmission ηa with the atmospheric extinction coeffi-cient σa (Kamermann, 1993) is given by

gaðrÞ ¼ expf−rad rg: ð9ÞThis formula is seldom used, but might be of impor-

tance for measurements under bad weather conditions.

2.5. Receiver efficiency

The receiver attenuates the perceived radiant inten-sity which is described by S and depends on the aperture

diameter D and object range r. Assuming D≪ r, thereceiver characteristic is given by

SðD; rÞ ¼ kD2

4kr2dgr; ð10Þ

with ηr being the receiver efficiency.

2.6. Calculating the received waveform of the back-scattered pulse

The received power P(t) depends on the modulatedwaveform of the transmitted laser pulse (Eq. (2)), thespatial energy distribution of the laser beam (Eq. (3)),the atmospheric transmission on the way from the sensorto the surface (Eq. (9)), the reflectance density function(geometrical and material) (Eq. (7)), the atmospherictransmission on the way from the surface to the sensor(Eq. (9)), the receiver attenuation (Eq. (10)), and theilluminated surface area over the infinitesimal volumeelement r2sin(ε)dαdεdr:

PðtÞ ¼Z l

r¼0

Z k=2

e¼0

Z 2k

a¼0smðt− 2r

cÞ

�KðeÞdgaðrÞdqAðr; e; aÞd gaðrÞdS�ðD; rÞd r2sinðeÞdadedr; ð11Þ

where c is the speed of light. This can be written in theform

PðtÞ ¼Z l

r¼0smðt− 2r

cÞd HðrÞ dr; with

HðrÞ ¼Z p=2

e¼0

Za¼0

2p

KðeÞd gaðrÞdqAðr; e; aÞ

�gaðrÞd SðD; rÞd r2sinðeÞdade: ð12ÞUnder the assumption (iii) of Section 2.3.1, that the

general range r0A of the system origin to the surface ismuch greater than the geometrical reflectance propertiesinside the range interval [r0A−Δr, r0A+Δr] the formulaH(r) can be simplified

HðrÞcCr0A

Z p=2

e¼0

Z 2p

a¼0KðeÞdqAðr; e; aÞd r 2sinðeÞdade;

with Cr0A ¼ g2aðr0AÞdSðD; r0AÞ¼ expf−rad2r0Agd pD2

4pr20Agr:

ð13Þ

With this approximation, P(t) depends on the trans-mitted waveform sm(t), the spatial energy distribution of


the laser beam K(ε) and the reflectance density functionρA(r,ε,α) integrated over the illuminated surface area,only.

Assume the geometry according to Fig. 5c the twodifferently elevated specular plane plates A1 and A2 arepartly illuminated by the laser beam. Now let the surfaceA2 be bounded by a straight line and xd =[r0d, ε0d, α0d]be the point on the line with the lowest distance to theoptical axis we obtain

HðrÞ¼Cr0A

Z p=2

e¼0

Z 2p

a¼0KðeÞ

�½dðrA1ðe; aÞ; rÞddðrA1ðe; aÞ;minðfr0A2 ; rA1ðe; aÞgÞÞþdðr0A2 ; rÞddðr0A2 ;minðfr0A2 ; rA1ðe; aÞgÞÞ��r2sinðeÞdade with

rA1ðe; aÞ ¼r0A1 if e Nasin

sinðe0dÞcosða−a0dÞd cosðe0dÞ

� �

ce0d

cosða−a0dÞ ;

l otherwise:

8>>>>>>><>>>>>>>:

ð14Þ

3. Analyzing the waveform

Analyzing the waveform of the backscattered pulseis useful to obtain information about the illuminatedsurface. For this analysis, very short pulses comparedto the illuminated structures in range are advanta-geous, e.g. for resolving fine structures of branches thepulse length has to be shorter than some nanoseconds.In general, short pulses have a lower pulse power andare more difficult to detect. In the case of a badsignal to noise ratio, a matched filter approach can beused (Jutzi and Stilla, 2004). However, this results inonly a single estimate of the travel time of the pulsewithout further features. In this paper we focus onpulses wider than a nanosecond and a moderate pulsepower (10 kW).

For analyzing the surface response it has to beextracted from the measured signal. The following stepshave to be carried out: (i) detection of the backscatteredpulse, (ii) deconvolution of the transmitted with thereceived waveform, and (iii) Wiener filtering to estimatethe surface function.

3.1. Detection of the backscattered pulses

First, in order to analyze the waveform, the back-scattered pulse of interest has to be detected and extracted.

For pulse detection, a noise dependent threshold isestimated to discriminate a single pulse from thebackground noise. A signal interval without pulses isprocessed to estimate the noise and to characterize it byits mean and its standard deviation. In practice, this canalso be done from a signal including pulses, as thepulse duration is negligible compared to the signalduration. If the signal is higher than three times of thenoise's standard deviation for at least 5 ns, a pulse willbe assumed to have been found and a waveform in-terval including the pulse will be accepted for furtherprocessing.

3.2. Deconvolution

The received waveform p(t) corresponds to aconvolution of the transmitted waveform sm(t) and thesurface response h(t):

pðtÞ ¼ smðtÞ⁎hðtÞ: ð15Þ

By transforming p(t) into the Fourier domain andsolving the resulting equation for the spectral surfacefunction H

¯( f ) we obtain

H�ð f Þ ¼P� ð f ÞS�mð f Þ

: ð16Þ

For calculating H¯( f ), the functions P

¯( f ) and S

¯m( f )

have to be known, which means that p(t) and sm(t) haveto be measured. Measurements of the waveform arealways associated with a receiver noise term n(t), whichis added to the signal

sm;nðtÞ ¼ smðtÞ þ nðtÞ ¼ sðtÞd mðtÞ þ nðtÞ ð17Þ

and

pnðtÞ ¼ pðtÞ þ nðtÞ: ð18Þ

Depending on the receiver noise, large numericalerrors may appear for H

¯( f ). Therefore, we need a filter

that reduces the noise, without smearing the surfacefunction.

3.3. Wiener Filter

For the estimation of the surface function, we use theWiener Filter (Wiener, 1949). This Optimal Filter mini-mizes the mean squared error between the uncorrupted


surface function H¯( f ) and the estimated surface function

H¯ˆ ( f ):Z þl

−ljhðtÞ− h ðtÞj2dt ¼

Z þl

−ljH� ð f Þ−H

� ð f Þj2df

is minimized:

ð19Þ

A solution has to be found where the Wiener FilterW( f ) is a real function

W ð f Þ ¼jP�ð f Þj2

jP�ð f Þj2 þ jN�ð f Þj2:ð20Þ

N¯( f ) can be easily estimated from the background

noise. P¯( f ) depends on the received waveform and the

surface function. For designing a Wiener Filter, |P¯( f )|2

has to be estimated.Assuming that a plane surface is perpendicular to the

pulse propagation direction and the surface is illumi-nated by an infinitesimal footprint, then h(t)=δ(t) (Diracdelta function) and P

¯( f )=S

¯m( f ). Instead of sm(t)

(Eq. (2)), we measure sm(t)+n(t). For the estimation ofsm(t), we low-pass-filter the received signal in timedomain using a linear binominal filter. In this case, wereceive for the Wiener Filter

W ð f Þ ¼jS�mð f Þj2

jS�mð f Þj2 þ jN�ð f Þj2: ð21Þ

The estimation of the surface function is then given by

H� ð f Þ ¼P�nð f ÞS�mð f Þ dW� ð f Þ ð22Þ

and the estimated surface response ĥ(t) is obtained bytransforming H

¯ˆ ( f ) into time domain.

3.4. Approximation of the surface response

By determining the temporal position, length, andamplitude of the waveform, surface features like therange, elevation variations, and reflectivity of thesurface can be derived from the extracted surfaceresponse. These surface features are extracted by fittinga Gaussian h(t) to the estimated surface response

h ðtÞY hðtÞ ¼Xni¼1

hAiðtÞ

¼ aAi d expf−4dln2dðt−tAiÞ2w2Ai

g: ð23Þ

For the estimation of the three parameters (i) timevalue tAi

, (ii) temporal pulse length wAi, and (iii)

maximum amplitude aAiof the Gaussian, the iterative

Levenberg–Marquardt Method is used (Marquardt,1963). For n total number of responses, i presents thecurrent response to process. The generalized iterationrule to estimate the parameters qAi

=[tAi, wAi

, aAi] is

described by

fqAi;k−qAi;kþ1g¼ fJTR−1Jg−1JTR−1f hðtÞ−hAi;kðtÞg; ð24Þ

where qAi,k are the pulse parameters tAi,k, wAi,k, and aAi,k

of the current iteration step k, J is the Jacobian matrix ofthe estimated Gaussian h(t), and Σ is the covariancematrix of the estimated pulse parameters.

The iteration starts with tAi,1 as the temporal positionof the surface response maximum, wAi,1 as the length ofthe surface response, and aAi,1 as the surface responsemaximum. The iteration is repeated until the change inhAi,k(t) is below a specified tolerance. From the estimatedposition tAi

the range value rAito the object is determined

by evaluating rAi= tAi

c / 2, where c is the speed of light.From the estimated wAi

the varying elevations of theobject can be derived by dAi

=wAic / 2, where the

determined elevation variations due to vegetation andslopes depend on the lasers footprint size. Skewwaveforms of the surface response are not consideredin this approach.

A measure describing the quality of the estimatedparameters tAi

, wAi, and aAi

is given by their variancesr2tAi , r

2wAi

, and r2aAi . These variances can be determineddirectly from the main diagonal of the covariancematrix. If the variances are all below a given threshold,the surface response will be accepted. Then theestimated Gaussian h(t) will be subtracted from theestimated surface response h (t) and the remainingwaveform will be processed again in the same manner.This kind of processing is repeated until all responseswith high quality (low values for the variances r2tAi , r

2wAi

,and r2aAi ) are detected.

If the temporal distance ΔT in between the singleresponses h Ai

(t) and h Ai+1(t) is close to 0.85 wc, then the

estimated parameters for each single surface will have tobe revised, because of the overlapping single responses.wc is derived from the system and will be explained inthe next paragraph. Fig. 6 shows a simulation of twooverlapping single responses (dotted lines) for differentdistances together with the received surface response(solid line). In Fig. 6a the temporal distance ΔT inbetween the single responses is 2 wc (wc=7.5 cm), thenthe maxima position of the surface response and thesingle responses are very close to each other. In Fig. 6bthe temporal distance ΔT of the single responses is wc.

Fig. 6. Overlapping responses (dotted lines) and received surfaceresponse (solid line) with different temporal distances ΔT betweensingle responses. a) ΔT=2 wc, b) ΔT=wc, c) ΔT=0.85 wc.

Fig. 7. Impulse response. a) Averaged impulse response estimated by1000 samples, b) single impulse response with noise.


The surface response has two small peaks, whereas themaxima positions of the peaks are closer to each otherthan the maxima positions of the single responses. For atemporal distance 0.85 wc the surface response showsonly a single maximum (Fig. 6c). In the case of twoclose located responses, the estimation is performed bytwo Gaussians with 6 parameters (tA1

, wA1, aA1

, tA2, wA2

,aA2

); for three responses, 9 parameters (tA1, wA1

, aA1, tA2

,wA2

, aA2, tA3

, wA3, aA3

) have to be estimated; and so on.

3.5. Impulse response of the measurement unit

If data from a real system is used the impulseresponse of the measurement unit has to be taken intoaccount in the parameter estimation, as the properties ofthe measurement unit affect the detected signal. Thismeans that the measured waveform, which is originallydefined by a convolution of the transmitted waveformand the surface response is additionally convolved with

the impulse response. The length of the impulseresponse is influenced by the bandwidth of the usedreceiver unit. A given bandwidth for a Gaussianfrequency function with a cutoff frequency fc leads toan impulse response length wc:

wc ¼ 2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln

ffiffiffi2

pdln2d

1fc

s: ð25Þ

wc is the impulse response length of the system anddefines the length of the signal measured by a givensystem when the surface response is an ideal Dirac deltafunction δ(t).

4. Experiments

An experimental setup with a pulsed Erbium fiberlaser (wavelength: 1.55 μm, beam divergence: 1 mrad,pulse length: 5 ns at FWHM) was built up for exploringthe capabilities of waveform analysis for discriminatingobjects close in range. For capturing the transmitted andthe received waveforms, two InGaAs detectors and twoamplifiers with an overall bandwidth fc=1 GHz wereused. Both analog signals were sampled with a rate of 20GSample/s.

The used bandwidth results in an impulse responselength ofwc=0.3 ns (Eq. (25)). An example of an impulseresponse derived by averaging 1000 samples of a signal


scattered by a specular surface (plane metal plate) isshown in Fig. 7a.

For reducing the noise of a single impulse response(Fig. 7b), a low pass filter is used. As the low pass filterreduces the signal bandwidth, the length of the shownimpulse response increases fromwc=0.3 ns to w c=0.5 ns.The length w c of the low pass filtered impulse response is

Fig. 8. Example of the measurements and results for the response of asingle surface. a) Transmitted waveform of the emitted pulse, b) receivedwaveform of the backscattered pulse, c) estimated surface response,d) approximated surface response.

the lower bound of the length of the surface response toresolve a single surface. This length is relevant todiscriminate multiple surfaces which are located close toeach other.

The investigations explore the reliability of a rangemeasurement for a single surface (Section 4.1) and therange discrimination of two surfaces (Section 4.2). Forboth experiments, 500 samples of the transmitted andreceived waveform from a plane plate with specularscattering characteristic are captured. The range accu-racy is measured by the standard deviation σrA of therange estimation. The ability of discriminating twonearby surfaces is tested by using two plates arrangedbehind each other in propagation direction of the laserbeam (Eq. (14)) with a distance below the pulse length.Each surface was illuminated partly by half of the beamfootprint. The range of the plates to the laser was about100 m and the distance in between them 0.15 m, whichcorresponds to 2w c of the example in Fig. 7a.

4.1. Single surface

For estimating the range accuracy to a plane surfaceat the range of 100 m a single surface response isestimated. Fig. 8a and b show an example of a singlemeasurement illustrating the emitted and the back-scattered pulse. In this figure, the depicted area ofinterest has a range interval of 2 m. Fig. 8b shows theshape of the received waveform which is similar to thetransmitted waveform in Fig. 8a, but different inamplitude. The estimated surface response obtained bythe Wiener Filter is depicted in Fig. 8c. It shows a singlestrong peak with a maximum at the range of 100 msurrounded by small signal ripples. The result of Fig. 8cis further processed by the Levenberg–MarquardtMethod for gaining the parameters of the surfaceresponse. From the estimated parameters, the waveformof the surface response is calculated. An example of thiswaveform is shown in Fig. 8d.

The distribution of the range values is depicted inFig. 10a by a histogram. All of the 500 emitted pulseswhich were backscattered at the plane surface locatedapproximately at 100 m in range were detected and thesurface responses were calculated. The range distribu-tion shows a standard deviation of σrA=6.2 mm.

4.2. Two surfaces

Two plane plates are positioned at 100 m and100.15 m, and the transmitted and the received waveformare captured. In this case, the goal of processing thewaveform is to detect, discriminate and estimate the range


of the two surface responses (i=1,2) as accurate aspossible.

Fig. 9a and b shows examples of a single measure-ment with the emitted and the backscattered pulse. Thesmall range shift of the plates generates an overlap of thereturned waveforms. This mixed signal is slightly longerthan the transmitted one with a dissimilar shape. Theresult of the Wiener Filter is depicted in Fig. 9c. It shows

Fig. 10. Histograms of the estimated range values. a) Single surface, b) twosurfaces (distance 0.15 m).

Fig. 9. Example of the measurements and results for the response of twosurfaces (distance 0.15m). a) Transmittedwaveformof the emitted pulse,b) received waveform of the backscattered pulse, c) estimated surfaceresponse, d) approximated surface responses.

two strong peaks at the range of 100 m and 100.15 m.The two estimated surface responses are visualized inFig. 9d. The curves of the estimated surface responses getin touch at the bottom, and the single responses overlapslightly in between each other. The overlapping of thesingle surface responses is a problem for accurate rangeestimation. Because the overlap of the responsesgenerates an overlay of the surface responses, the originalmaxima position of the single responses are shifted apartfrom each other. Thus the maximum position which isusually used for range estimation gives inaccurate rangevalues. By using an improved estimation (Section 3.4) ofthe single surface responses, this inaccuracy can beeliminated.

The amount of 500 pulses was measured from twoplane surfaces with a range distance of 0.15 m located at100m in range. The histogram of the range values shows abimodal distributionwith two significant peaks (Fig. 10b).The left range distribution shows a standard deviation ofrrA1 ¼ 5:5 mm and the right range distribution shows astandard deviation of rrA2 ¼ 6:6 mm. The averagedistance for 500 measurements is Δr =0.149 m with thestandard deviation of σΔr=5.9 mm.

5. Discussion

In general, analog techniques for range determinationby measuring the time-of-flight of laser pulses have astrong limitation regarding their capability of discriminat-ing surfaces which are very close located in range.


Katzenbeisser (2003) has shown that the discrimination ofnearby pulses will only be possible if the distance of twobackscattering object surfaces is at least greater than half ofthe pulse length. By recording the waveform with a highbandwidth of the receiver unit, an adequate sampling rate ofthe analog-to-digital converter, and the proposed algo-rithms for discriminating multiple surface responses, therange resolution can be significantly improved. Therefore,the received as well as the transmitted waveform have to berecorded. We have shown that surfaces with a distancecorresponding to a tenth of the emitted pulse length can beresolved. The used laser system allows discriminating twosurface responses with a distance of 0.15 m. It has to bementioned that in the experiments specular sufaces wereused. Only a few natural surfaces are specular. But, forinvestigating the limitations of the laser system, e.g.discriminating stepped surfaces (objects with smalldistances) specular surfaces seems to be the best choicedue to the proper signal-to-noise-ratio.

Furthermore, recording of the transmitted waveformenables to isolate the surface response from the receivedwaveform. By applying the Levenberg–MarquardtMethod to the surface response, three properties of theresponse can be estimated, namely time value, temporalpulse length, and maximum amplitude. From theseproperties we derive the corresponding surface features,namely range, elevation variations, and reflectivity.

The spatial energy distribution was modeled andmeasured in the near field. For the far field differentprofiles for modeling are used (Section 2.2). Thetransition from near-field to far-field is given by theRayleigh range zRay

kx20

k , where the wavelength is λ andthe beam waist is ω0 (Siegman, 1986). Assuming awavelength of λ=1.55 μm and a beam-waist diameterof D=2ω0≈60 mm we receive a Rayleigh range ofapproximately 1800 m.

For investigating the limits in resolving smalldistances in range within the beam we have chosenplane metal plates with specular surfaces. These specificsurfaces with high reflectivity and minimum surfaceslope were used for the experiments to capture awaveform with high quality. Diffuse surfaces with highmaterial reflectance might deliver similar results butwere not investigated in this paper. For the rangemeasurements in 100 m, the results of the experimentsshow a high reliability (σrA≤6.6 mm). The plates whichwere shifted by 0.15 m were supposed to reflect half ofthe pulse energy, each. The configuration was notadjusted in an optimal way so that the maximaamplitudes of the surface responses differ (Fig. 9c).The measurement of the average distance Δr =0.149 mwas obtained with a high reliability (σΔr=5.9 mm).

The limits to resolve small distances in range generallydepend on the length of the impulse response wc and thecorresponding bandwidth fc of the used measurementunit. Discriminating two responses by peak detection willbe possible if the temporal distance in between singleresponses is greater than 0.85 wc. Otherwise, the shape ofthe waveform of the overlapping responses resembles asingle widened peak. If the temporal distance in betweensingle responses is close to 0.85 wc, range estimation bythe maxima positions of the surface response will beinaccurate, because of the overlapping single responses(Section 3.4). The range estimation can be revised withthe Levenberg–Marquardt Method considering multiplesurfaces to increase the accuracy.

The temporal pulse length and amplitude of thewaveform were not investigated further in this paper.The temporal pulse length seems to be of special interestfor distinguishing natural from man-made surfaces.

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