PhD Thesis Semi-Supervised Ensemble Methods for Computer Vision

48 Chapter 4. SemiBoost and Visual Similarity Learning
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SemiBoost
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Figure 4.2: SemiBoost combined with a learned similarity measure from given labeled
samples.
symmetric with respect to its arguments. The learning problem can be defined as a learning
problem on the product space as F d : X × X → Y = [−1 1]. To train a maximum
margin classifier the training set
D d = {(x, x ′ , +1)|y = y ′ , x, x ′ ∈ D L } ∪
∪{(x, x ′ , −1)|y ≠ y ′ , x, x ′ ∈ D L } (4.15)
is built by taking pairs of images of “same” and “different” class. Using pairs allows
us to create a large number of training samples while having only a few labeled starting
samples. In particular, if we omit pairs with self-similarities such as (x, x), we can create
n·(n−1)
2
training pairs out of n positive samples. The symmetry of the distance is not
satisfied automatically, therefore it has to be enforced by introducing each pair twice,
i.e., both (x, x ′ ) and (x ′ , x). This also means that the number of training samples in fact
becomes n · (n − 1). Then, as in [Hertz et al., 2004] we use boosting to learn a classifier.
The trained and normalized classifier F d (x, x ′ ) ∈ [−1 1] is interpreted as a distance
d(x, x ′ ) = 1 2 (F d (x, x ′ ) + 1). (4.16)
The conversion into a similarity s(x, x ′ ) can be done as proposed in Equation (4.14),
i.e., using a radial basis function. This learned similarity can now be used as a prior for
SemiBoost as depicted in Figure 4.2.
4.2.3 Using Arbitrary Classifiers as Similarity-Priors
Above, we have discussed the issue of training a pair-wise classifier which can serve as a
distance measure for SemiBoost. However, sometimes it is the case that we have already