The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
to detect and parameterize the peaks. This approach uses the 
Expectation Maximation (EM) algorithm (Dempster et al., 
1977). A detection algorithm based on the 
expectation-maximization (EM) algorithm is used to estimate 
the number of echo pulses of the waveforms. 
2. GENERATION AND DECOMPOSITION OF 
WAVEFORM 
The power entering the receiver is 
4 <A 2 
J?(0*ri(0*r(0 
Fig.2 System waveform of RIEGL LM-5600 
N is distinct targets within the travel path of the laser pulse, P ^ 
is the transmitted power; />(*) is the received power; D is the 
aperture diameter of the receiver optics; R j is the range 
between scanner and object i; <j\{t) is the so-called backscatter 
cross-section; r(?) is the receiver impulse function;* is the 
convolution operator. In practice, P ^ and T(t) cannot be easily 
determined independently. Therefore it is advantageous to 
rewrite the convolution term by making use of the commutative 
property of the convolution operator: 
where we introduce the system waveform S(t) of the laser 
scanner, defined as the convolution of the transmitted pulse and 
the receiver response function. It can be measured 
experimentally and is shown in Figure. 2 for the Riegl 
LMS-Q560. It can be seen that it is well described by a 
Gaussian function: 
S(t)=Se 2si 
S is the amplitude, S ? is the standard deviation. In order to 
come to an analytical waveform solution, let us assume that the 
scattering properties of a cluster of scatterers can be described 
by a Gaussian function: 
<j j (t) = à i e 2s ‘ ,<J i is the amplitude and S i the standard 
deviation. 
The convolution of two Gaussian curves gives again a Gaussian 
function, so that we obtain: 
('-O 2 
s 2 ,+*f,P,= 
Di 
S(Tr 
p,l 
Thus it can be seen the return waveforms are made up by 
Gaussians has proven to be a fairly good approximation. 
3. ALGORITHM OF DECOMPOSING WANEFORM 
To consistently geolocate the desired reflecting surface, for 
example, the underlying ground surface in vegetated regions, 
we need to be able to precisely identify the corresponding 
reflection within the waveform. Existing waveform processing 
methods generally do not take into account surface type nor its 
effect on the shape of the return laser pulse, and thus do not 
provide a consistent ranging point to a reflecting surface during 
data processing. These methods include finding the location of 
the peak amplitude within the waveform or the location of the 
centroid of the return waveform. Experiments illustrate that 
there is no such thing as a single best detector, rather the 
relative performance of the detectors depends on factors such as 
the characteristics of the effective scattering cross section, 
object distance and noise level. 
Thus, we propose to decompose a return waveform into 
components, the sum of which can be used to approximate the 
waveform and the locations of which can be used to improve 
the geolocation accuracy of the laser altimeter. We will assume 
that each mode represents the reflected distribution of laser 
energy from a reflecting surface within the footprint, and that 
the location of each mode can be used to geolocate the 
reflecting surface of interest in the vertical direction. Gaussian 
decomposition presents us with one possible model of the 
reflections contained in a complex, multi-modal waveform. 
As a first approximation we will assume that the laser output 
pulse shape or impulse response (i.e., the shape of the outgoing 
laser pulse after passing through the full detector and digitizer 
chain) is Gaussian. We further assume that the returning laser 
pulse is composed of a series of potentially-overlapping 
reflections similar in shape to the impulse response (i.e., in this 
case by a series of Gaussian-shaped reflections). As was shown 
in the theory part of the paper, the implicit assumption of 
Gaussian decomposition is that the cross-section profile can be 
represented by a series of Gaussian functions. 
It is of interest to extract more than the first and the last echo 
from each waveform. Also, the width of the echoes is of interest. 
A detection algorithm based on the improved 
expectation-maximization (EM) algorithm is used to estimate 
the number of echo pulses of the waveforms. The algorithm also 
outputs the width of the echo pulses. Unsupervised learning is a 
method of machine learning where a model is fitted to 
observations. An important part of the unsupervised learning 
problem is determining the number of components or classes 
which best describe the data.