For pixel i we define the model:
2.2 Data
p(y,l,l',b',b)= n QAQAQAQ-, a)
k = 1
e,= P (Y k iL k )
(2)
Qt = P( L k 1 L 'k )
(3)
e 3 = P (L' k i L' t , )
(4)
e 4 =p(L' k iN-k(i))
(5)
Q 5 = p(L' k 1 B' k )
(6)
e 6 =p(B'kiB k )
(7)
Qi = p(B k iB;).p(B;)
(8)
where term (2) is modeled with multivariate guassian densities
and terms (3),(4),(6) and (7) represent contingency tables. Term
(8) is a degenerate probability term in the sense that it is
generated from a deterministic mechanism, to be described below.
For (5), p(L' k I N k (i)) oc e, where C is the number of
the 8 nearest neighbouring pixels having the same label as pixel
i from the previous iteration, and Ot and /3 are user specified
parameters, taken to be 0 and 1 respectively in the experiments in
sections 3 and 4.
For (8), p(B k ) is taken to be a uniform distribution.
The term p(B k IB t ) is calculated by making use of
P(B' k I Y, B*) (9)
obtained from (1), as follows:
1. According to (1), calculate (9).
2. From B’, calculate the distance transform (Borgfors,
1986) from the boundary pixels, denoted DT in the
following. As its name suggests, DT has increasing
value as distance from the boundary increases.
3. Update B by steepest descent (or otherwise) using DT.
4. Goto 1
We note that there are many alternatives to test at each of the
above 3 steps. Perhaps most important are:
1. The relative weighting of contribution from spectral,
neighbours and boundary
2. The choice of boundary updating algorithm and the
formation of p(B’IB).
We use orthorectified and calibrated Landsat TM satellite data
(Wu, 2001) to provide observations relating to land cover. The
data were prepared following the procedures described by Furby,
2002.
In section 3, we artificially created a boundary by image
interpretation, and specified a simple 2-class {forest, non-forest}
spectral classifier by specifying (for term (2) of equation (1)
below) a threshold on image band 5.
In section 4 we use forest inventory boundary information.
Typically these boundaries represent forests having various states
of growth and harvest. In the example, we present the results for a
relatively small area and the updating of the one existing
hardwood boundary for this area.
In section 4, the (multivariate gaussian) spectral specification of
the model was taken from the analysis described in Caccetta and
Chia, 2002, which provided the class mean and covariance
estimates used. Seven classes were defined: Softwood 1 (mature);
Softwood 2 (juvenile), Karri 1 (mature), Karri 2 (juvenile),
Hardwood, Jarrah, and Bare (cleared). The error rates derived
from validation data were used to specify term (3) in the model.
“Sensible” values for the remaining terms were specified, with
“sensible” being an area which requires further investigation.
In the example given in 3 and 4, n was specified to be 1.
3 Examination of model properties using a contrived example
Here we test the model on a simple case of 1 image, 1 boundary
and a 2-class problem. Two boundary starting positions were tried.
Results for boundary starting position 1 are depicted in figure la-f
below. Results for boundary starting position 2 are depicted in
figure 2a-i below.
Some observations derived from the experiment were:
1. False edges result in many false minima which the
boundary can get stuck on.
2. Ideally, the boundary prior should reduce the number of
initial false edges, say produced from systematic
mislabelling of pixels (and thus has potential to remove
large regions of misclassified pixels), though this would
suggest that the boundary would need to be close to the
correct position to start with.
3. After 2, one would hope that false edges produced from
individual (or small groups of) misclassified pixels
would be removed by successive application of the
neighbourhood prior.
4. One can observe the effect of the boundary prior in
figures lb and 2b. The relative weighting of this prior
needs to be considered carefully. In initial testing (not
illustrated here), the boundary did not update as the
prior was too strong and the boundary defined its own
position.
