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Similarly, weights and bias in hidden layer neuron can be updated as follows
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6.2 Learning 3D Motion
This section presents different 3D motions of objects in the physical world. The
motion in a 3D space may consists of a 3D
scaling, rotation, and translation
and
composition of these three operations. These three basic class of transformations
convey dominant geometric characteristic of mapping. Such a mapping in space
preserves the angles between oriented curves, and the phase of each point on the curve
is also maintained during motion of points. As described in [
1
,
14
], the complex-
valued neural network enables to learn 2D motion of signals, hence generalizes
conformal mapping on plane. Similarly, a 3D vector-valued neural network enables
to learn 3D motion of signals, and will provide generalization of mappings in space.
In contrast, a neural network in a real domain administers 1-D motion of signals,
hence does not preserve the amplitude as well as phase in mapping. This is the main
reason as to why a high dimensional ANN can learn high dimensional mapping,
while equivalent real-valued neural network cannot [
11
].
In order to validate the proposedmotion interpretation system, various simulations
are carried out for learning and generalization of high dimensional mapping. It has
the capacity to learn 3D motion patterns using set of points lying on a line in the 3D
space and generalize them for motion of an unknown object in the space. We have
used a 2-6-2 structure of 3D vector-valued neural network in all experiments of this
section, which transform every input point (
x
x
,
y
,
z
)
in
the 3D space. First input of input layer takes a set of points lying on the surface of
a object and second input is the reference point of input object. Similarly, the first
neuron of output layer gives the surface of transformed object and second output is
its reference point. Empirically, it is observed that considering reference point yields
better testing results. The input-output values are with in the range
,
y
,
z
) into another point
(
1.
In all simulations, the training input-output patterns are the set of points lying on a
straight line with in the space of unit sphere (0
−
1
≤
x
,
y
,
z
≤
≤
≤
radius
1), centered at the origin
ˀ
and all the angles vary from 0 to 2
. Figure
6.1
a presents an example input-output
mapping of training patterns. The following few examples depict the generalization
ability of such trained network over standard geometric objects.
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