Page 51 - Dynamic Vision for Perception and Control of Motion

2.1 Three-dimensional (3-D) Space and Time 35

workload in computation, efficient evaluation of all these traverses is very impor-
tant. This is the subject of the next section.
Again, all of this is independent of the special application, once the scene tree
for the set of problems has been defined.

2.1.2 Jacobian Matrices for Concatenations of HTMs

To be flexible in coding and to reduce setup time for new task domains, the ele-
ments of Jacobian matrices may be determined by numerical differencing instead
of fully analytical derivations; this is rather computer-intensive. An analytical deri-
vation based on the factored matrices may be most computer-efficient. Figure 2.11
shows in the top row the total HTM for the homogeneous feature vector e of the
example Equation 2.10 from road vehicle guidance with scene perception through
an active camera. The two translations T have two and one unknown components
here (maximally six are possible in total); all three rotation-angles have to be de-
termined from vision as well. So there are six unknowns in the problem, for which
the entries in the Jacobian matrix have to be computed.
To obtain these elements, nominal value and the systematically perturbed values
due to changes in each unknown state variable or parameter dx i have to be com-
puted for each feature point, just one at a time to obtain partial derivatives. If there
are n state variables and q parameters to be iterated during recursive estimation, (n
+ q + 1) total transformations have to be computed for each feature point. To be ef-
ficient, the nominal values should be exploited as much as possible also for com-
puting the n + q perturbed values. This procedure for the unknown state compo-
nents is sketched in the sequel; adaptable parameters will be discussed in Section
2.2 and Chapter 6.

2.1.2.1 Partial Derivatives of Homogeneous Transformation Matrices

The overall transformation matrix for each feature point x Fk on the object (the rod
in our case) in 3-D space was given by Equation 2.10:
e = [ PR ˜ ˜ R ˜ T ˜ T ˜ R ] x = T ˜ x .
˜
p șc ȥc Rc Ro ȥo o tot o
(2.10)
unknowns: ș , ȥ , y gc (x co ,y oR ) ȥ !
c
o
c
To describe the state of the rod relative to the camera, first its state relative to
the road CS has to be specified by (\ o , x co , y oR). Then the state of the camera rela-
tive to the road is given by [y gc , (the known elevation H c), \ c , 4 c]. Under the as-
sumption of a planar straight road, the geometric arrangement of the objects “cam-
era” and “rod” is fully described by specifying these “state components”. The
unknown state vector x S of this problem with six components thus is
T
x = (ȥ , x , y , y , ȥ , ș ). (2.12)
S o co oR gc c c
These components have to be iterated during the vision process so that the un-
derlying models yield a good fit to the visual features observed. In Equation 2.10
each R represents a single rotational and each T up to three translational compo-