Image Processing Reference
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the union of all the other classes considered). On the contrary, the homogenous area
detector is not very precise because it uses filters with size 9
9 windows, which erases
all of the roads that cut through homogenous areas. It is then natural to choose as the
focal elements the set “roads
×
homogenous areas” and the set “non-homogenous”.
This is an example of a detector for which no focal element is a singleton. Urban area
detectors are modeled in a similar fashion. This example shows that knowledge about
the detectors and their behaviors is what allows us to conduct the modeling phase and
choose the focal elements.
∪
The mass functions are then learned from the response histograms of the various
detectors, by minimizing a distance between these histograms and trapezoidal para-
metric functions.
The fusion is then performed according to Dempster's non-normalized orthogo-
nal rule (conjunctive fusion) since all the imprecisions and ambiguities regarding the
detectors are explicitly taken into account in the modeling. This makes it possible to
reduce the focal elements to singletons or union of two classes only. Furthermore, this
is a typically open world application: it is not possible to predict all the classes that
may show up in the image and only those for which detectors have been designed can
be detected. The non-normalized combination allows us to represent in the mass of
the empty set anything that is not predicted.
Finally, the decision phase is conducted in a Markovian framework, ensuring the
addition of spatial consistency between the areas, hence an additional level of spatial
information. The pignistic probabilities (see Chapter 7) make it possible to go back to
probabilities for singletons, which are then combined to a spatial regularization term.
The result of an interpretation is shown in Figure 9.4.
9.4.3.
The modeling level: fuzzy fusion of spatial relations
In this last example, the spatial information we are considering is structural infor-
mation, involving no longer the local consistency of the classes or areas, but instead
the relations between the objects we are looking for. The application involves recog-
nizing internal structures of the brain in MRI images, using an anatomical atlas as our
guide [BLO 00c, GER 00].
A cross-section extracted from the 3-D volume of the atlas is shown in Figure 9.5;
the view of the corresponding cross-section in the 3-D MRI acquisition which needs
to be processed is represented in Figure 9.6.
The recognition is performed progressively, with one structure detected at each
step. Each step relies on the objects obtained in the previous steps and on various kinds
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