Page 131 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

120 STATE ESTIMATION

this problem is to embed the measurement z(i) in the state variable
x(i). This can be done by encoding the state variable as integers from 1
up to 16. If i is not a license plate pixel, we define the state as
x(i) ¼ z(i). If i is a license plate pixel, we define x(i) ¼ z(i) þ 8. With
that, K ¼ 16. Figure 4.16 shows these states for one video line.
The embedding of the measurements in the state variables is a form
of state augmentation. Originally, the number of states was 2, but after
this particular state augmentation, the number becomes 16. The advan-
tage of the augmentation is that the dependence, which does exist
between any pair z(i), z(j) of measurements, is now properly modelled
by means of the transition probability of the states. Yet, the model still
meets all the requirements of an HMM. However, due to our definition
of the state, the relation between state and measurement becomes
deterministic. The observation probability degenerates into:

1 if n ¼ k and k 8
(
P z ðnjkÞ¼ 1 if n ¼ k 8 and k > 8
0 elsewhere
In order to define the HMM, the probabilities P 0 (k) and P t (kj‘) must
be specified. We used a supervised learning procedure to estimate
P 0 (k) and P t (kj‘). For that purpose, 30 images of 30 different vehicles,
similar to the one in Figure 4.14, were used. For each image, the
license plate area was manually indexed. Histogramming was used to
estimate the probabilities.
Application of the online estimation to the video line shown in
Figures 4.15 and 4.16 yields results like those shown in Figure 4.17.
The figure shows the posterior probability for having a license plate.
According to our definition of the state, the posterior probability of
having a license plate pixel is P(x(i) > 8jZ(i)). Since by definition
online estimation is causal, the rise and decay of this probability
shows a delay. Consequently, the estimated position of the license
plate is biased towards the right. Figure 4.18 shows the detected
license plate pixels.

4.3.3 Offline state estimation

In non-real-time applications the sequence of measurements can be
buffered before the state estimation takes place. The advantage is that
not only ‘past and present’ measurements can be used, but also ‘future’