International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
how far two overlapped echoes separated will influence the 
detection results. 
2. WAVELET-BASED ECHO DETECTOR 
The wavelet transform is a tool to decompose a signal in terms 
of elementary contributions over dilated and translated wavelets. 
One of the continues wavelet transform(CWT) applications is 
resolving overlapped peaks in a signal(Jiao et al., 2008). The 
CWT at time uw and scale s can be represented as 
wf ws) =(fw..)=| fOw, Od (D 
where /{t) 1s the input signal, * denotes the complex conjugate, 
y, .(r) 1s the wavelet function controlled by a scale factor s and 
a translation factor uw. Wf{u, s) is the so called wavelet 
coefficients. Applying CWT to the waveforms can be 
considered as measuring the similarity between the waveform 
and the wavelets. If the chosen mother wavelet and the 
responded echo are similar in shape, then the locations where 
WC peaks occur imply the positions of the response echo in the 
waveform. Figure 2 shows an example of detecting echoes 
using the wavelet-based detector. A signal(waveform) is applied 
wavelet transform at two scales. Consequently we can obtain 
the WC (Wf (u, s4), Wf (u, s2)) at each scale. Taking the result 
at the smaller scale s; firstly, one can see that the locations 
where the WC peaks (Wf(u4,s41), Wf (u2, 43), Wf (us, $4) ) 
occur corresponding to the positions of echoes in the waveform. 
However for the case of larger scale s,, only two echoes are 
detected. This can be explained that the expanded wavelet 
cannot "see" the echoes whose size 1s smaller than the wavelet 
itself. 
          
signal 
wavelet 
(scale 1) 
OON REOR M OW NOM 
dbi m eomm mos ub oo 
xot XE ue $e oim m de de 
wow oul mom os xp mom 
ww wd an wees 
Hmmm 
SiO 
LB 
21 
HF s ; Vries, S MG s 
Winn) NE rd 
f IAN AAAS SAA 
     
® E 
E 9 
= = 
æ æ 
æ = 
signal 2 % 
wavelet 2 NUN 
(scale 2) 2 \ x 
» m 
i n 
FETT Wits) 
Hb, Sytem i "ers A Tr 
Tus, Lr v 
Figure 2. the interpretation of detecting echoes by CWT. 
Since the wavelet can be scaled by a scale factor, the wavelet- 
based detector is able to deal with different system with variant 
echo width, for example, the echo width is 5 ns for Leica 
ALS60 system and 8 ns for Optech ALTM 3100. Therefore to 
530 
optimize the detection results, an appropriate mother wavelet 
and a scale factor need to be prior determined. Many researches 
have considered the responded echo as a Gaussian function. For 
this reason the Gaussian wavelet is chosen as the mother 
wavelet in our study. Additionally by exploring some waveform 
samples, the scale parameter can be determined according to 
detection results. 
3. EXPERIMENT AND RESULTS 
3.1 Waveform simulation 
A received waveform is a power function of time and can 
be expressed as follows (Wagner et al., 2006; Mallet et al., 
2010): 
P(0- Y 5*0 @) 
where S(t) is the system waveform of the laser scanner, o;(t) is 
the apparent cross-section, N is the number of echoes and k; is a 
value varied by range between sensor and target. Eq. (2) 
indicates that a return echo is the convolution of system 
waveform and the apparent cross-section of a scatter. Wagner et 
al.(2006) have reported that the system waveform of Riegl 
LMS-Q560 can be well described by a Gaussian model. If the 
apparent cross-section of a scatter is assumed to be of Gaussian 
function, then the convolution of two Gaussian curves gives 
again a Gaussian distribution. The received waveform P.(t) 
can be rewritten as: 
ny 
N = 
P (t) = > Pe 255i (3) 
i=l 
where t; is the round-trip time, s, ; the standard deviation of the 
echo pulse, and P; the amplitude of cluster i. For this reason, the 
Gaussian function is chosen to simulate the return echo. The 
simulated waveform can be represented as follows: 
m 
w() - 3 g,() n 4) 
2 
G1) 
25° 
(5) 
  
| 
where w 1s the simulated waveform, m the number of return 
echoes, n the noises which have a normal distribution(Unoise» 
Onoise), 4 the location of time domain, and s the echo width 
which can be represented by full width at half maximum 
(FWHM =2+/2In2s). Figure 3 shows an example of a simulated 
waveform. 
g(t) 2 A-exp -