5.2.2
The MAP configuration 
derived in the above is the optimal mapping
from the scene to the model object under consideration. In other words,
it is the optimal labeling of the scene in terms of the model object.
Suppose there are L potential model objects. Then L MAP solutions,
, can be obtained after matching the scene to
each of the models in turn (Fast indexing of model objects (Lamdan and Wolfson 1988) is a topic not studied in this work). However, any
feature in the scene can have only one match of model feature. To
resolve this, we use the following method of cost minimization.
Rewrite 
in (5.18) into the
following form
where
and
are local posterior energy of order one and two, respectively. 
is the cost
incurred by the local match 
given the rest of the matches. It
will be used as the basis for selecting the best matched objects for i
in matching to multiple model objects. The image feature i is
considered to come from object 
if the decision incurs the least
cost
The final label for i is feature number 
of object
. Note, however, the MAP principle is applied to matching to a
single model object not to multiple objects; the simple rule
(5.22) does not maximizes the posterior since at least
the partition functions are different for matching to different objects.
Figure 5.5: Mapping from the scene to multiple model objects. Different
textures represent different structures. Bold lines represent
sub-mappings. Note that the background and un-model structure are mapped
to the NULL
structure (the blank square).
From (Li 1992c) with permission; © 1992 Elsevier.
Applying (5.22) to every i yields an overall mapping from the scene to the models, composed of several sub-mappings, as illustrated in Fig. 5.5. On the right, the squares with different textures represent the candidate model structures to which the scene is to be matched. Among them, the blank square represents the NULL model. On the left is the scene structure. The regions, each corresponding to a subpart of a model structure, are overlapping (un-separated). The recognition of the overlapping scene is (1) to partition the scene structure into parts such that each part is due to a single object and (2) to find correspondences between features in each part and those in the corresponding model object. The parts of the scene corresponding to the background and un-modeled objects should be mapped to the NULL , or in other words assigned the NULL label.
