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In the following, we assume that the label of each local feature p x , belonging to
images in the training set Tr , is the label assigned to the image it belongs to (i.e., d x ).
Following the notation used in Section 4,
Tr,Φ ( p x )= Φ ( d x ) .
In other words, we assume that the local features generated over interest points of the
images in the training set can be labeled as the image they belong to. Note that the noise
introduced by this label propagation from the whole image to the local features can be
managed by the local features classifier. In fact, we will see that when very similar
training local features are assigned to different classes, a local feature close to them is
classified with a low confidence. The experimental results reported in Section 7 confirm
the validity of this assumption.
As we said before, given p x
p x
d x ,
d x
d x , a classifier Φ returns both a class Φ ( p x )= c i
C
to which it believes p x to belong and a numerical value ν ( Φ,p x ) that represents the
confidence that Φ has in its decision. High values of ν correspond to high confidence.
5.1 Local Feature Classifier
Among all the possible approach for assigning a label to a interest point, the simplest is
to consider the label of its closest neighbor in Tr . The confidence value can be evaluated
using the idea of the distance ratio discussed in Section 4.2.
We thus define a local feature based classifier Φ m ( p x ) that assign a candidate label
Φ m ( p x ) as the one of the nearest neighbor in Tr closest to p x (i.e., NN 1 ( p x ,Tr ) ):
Φ m ( p x )= Φ ( NN 1 ( p x ,Tr ))
The confidence here plays the role of a matching function, where the idea of the distance
ratio is used to decide if the candidate label is a good match:
ν ( Φ m ,p x )= 1 if σ ( p x ,t r ) <c
0 otherwise
The distance ratio σ here is computed considering the nearest local feature to p x and
the closest local feature that has a label different than the nearest local feature. This idea
follows the suggestion given by Lowe in [14], that whenever there are multiple training
images of the same object, then the second-closest neighbor to consider for the distance
ratio evaluation should be the closest neighbor that is known to come from a different
object than the first. Following this intuition, we define the distance ratio σ as:
σ ( p x ,T r )= d ( p x ,NN 1 ( p x ,Tr ))
d ( p x ,NN 2 ( p x ,Tr ))
where NN 2 ( p x ,Tr ) is the closest neighbor that is known to be labeled differently than
the first as suggested in [14].
The parameter c used in the definition of the confidence is the equivalent of the one
used in [14] and [5]. We will see in Section 7 that c =0 . 8 proposed in [14] by Lowe is
able to guarantee good effectiveness. It is worth to note that c is the only parameter to
be set for this classifier considering that the similarity search performed over the local
features in Tr does not require a parameter k to be set.
 
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