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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
space. Based on the input vectors space, an initialized SOM will
be imposed for training process (c.f. step 3).
Step 2: Define the size, dimensionality, and shape of a SOM to
be used.
The size is actually the number of neurons for a SOM. It can be
determined arbitrarily, but one principle is that the size should
be easy enough to detect the pattern or structure of SOM
(Wilppu 1997). The number of neurons can be arranged in a 1-
or 2-dimensional space (dimensionality). Three kinds of shape
are allowed, i.e. sheet, cylinder or toroid, but usually sheet as
default shape.
Step 3: Initialize output vectors m randomly or linearly.
At the initialisation step, each neuron is assigned randomly or
linearly by some values for the d variables. Thus an initial SOM
is imposed in the input vectors space for the following training
process.
Step 4: Define the parameters that control the training process
involving map lattice, neighbourhood, and training rate
functions.
The number of neurons defined can be arranged in two different
map lattices, namely hexagonal and rectangular lattices.
However, hexagonal lattice is usually preferred because of
better visual effect according to Kohonen (2001).
Neighbourhood function has different formats such as 'bubbs',
‘gaussian’, ‘cutgauss’ and ‘ep’ (see Vesanto et al. 2000, pp. 10),
but gaussian function is usually adopted and it is defined by:
h(t) = oe [3]
where 0, is the neighbourhood radius at time t, a; is the
distance between neurons c and i on the SOM grid. It should be
noted that the size of the neighbourhood N,(/) reduces slowly as
a function of time, i.e. it starts with fairly large neighbourhoods
and ends with small ones (see figure 2).
The training rate function can be linear, exponential or
inversely proportional to time t (see Vesanto et al. 2000, pp.
10). For instance, et) a. /(1+100¢/T)is the option we
adopted in the following case study, where T is the training
length and Œ, is the initial learning rate. Usually the training
3. SOM-BASED CLUSTERING ANALYSIS FOR LASER
SCANNED POINT CLOUDS
3.1 Principle and overall procedure
Laser scanned point clouds are usually defined in a four
dimensional space with xyz coordinates and the return intensity.
For a given point cloud, all points constitute input vectors
which can be used for clustering analysis in order to distinguish
different points belonging to different objects. Depending on
the size of input cloud, an appropriate SOM will be decided
together with other parameter settings. Once all these are
determined, a SOM will be derived to represent the pattern or
structure of a point cloud. The SOM is organised in a grid in
which nearby neurons are more similar to those which are
length is divided into two periods: t, for the initial coarse
structuring period and t, for the fine structuring period.
Step 5: Select one input vector x, and determine its Best-
Matching Unit (BMU) or winning neuron using equation [1].
Although Euclidian distance is usually used in equation [1], it
could be various other measures concerning ‘nearness’ and
‘similarity’. Depending on the form of data measurement, other
measures are allowed as long as they represent the distance
between input and output vectors.
Step 6: Update the attributes of the winning neuron and all
those neurons within the neighbourhood of the winning neuron,
otherwise leave alone (c.f. equation [2]).
Step 7: Repeat steps 5 to 6 for a very large number of times
(training length) till a convergence is reached.
The convergence is set like this, 7 (r+1) = m (1), for t > 0 - M
i i >.
practice, the training length in epochs is determined by the size
of SOM (k) and the size of training data (n), for instance for
coarse period 4,,.
1 = —
I
H
After the above steps, all output vectors are projected on to a 1-
or 2-dimensional space, where each neuron corresponds to an
output vector that is the representative of some input vectors. A
2-dimensional hexagonal map lattice grid is shown in Figure 2
where each hexagonal cell has a uniform neighbourhood.
«- a ten-by-ten (one hundred neurons) lattice space -»
»
neighbouring ; 4
neurons at t1
neighbouring |
neurons at t2
ser Eum En mm: a = in
neighbouring
neurons att3 |
{winner neuron}
lorBMU |
A!
Figure 2: The characteristics of a 10x10 SOM (t1<t2<t3 with
h(n) in equation 3)
widely separated. With the SOM, various clusters can be
identified and they in essence represent different sets of points
with a certain similarity in terms of their coordinates and the
intensity. The process of the clustering analysis can be
described as follows. For a given point cloud, all the points
constitute input vectors in a four dimensional space. This space
is defined by three coordinates and the return intensity. These
vectors then are used to train a SOM as described in the above
section. With the SOM, points with similar attributes will
correspond to neurons that are grouped together. Various
clusters can be identified from the SOM, and finally the
derivation of various spatial object models is based on these
clusters.
From a more practical perspective, the points and their
corresponding attributes are used for creating input vectors in
Matlab. Then training process is performed on SOM Toolbox
