Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

3.3 Classification
(a) linear separable
(b) not linear separable
Figure 3.2: The SVM classifier for two different 2-dimensional feature spaces.
The feature vectors lying on the boundary are called support vectors, hence the name
support vector machines. If all feature vectors were removed except the support vectors, the
resulting classifying hyperplane will remain the same. This is why those feature vectors are
called support vectors.
SVM training For a classification problem using two classes the training data consists of
feature vectors ⃗x i ∈ R n and the associated class labels y i ∈ {−1, 1} where n is the number
of elements in the feature vectors. All feature vectors which lie on the separating hyperplane
satisfy the equation
⃗x i ⃗w + b = 0 (3.7)
where ⃗w is the normal to the hyperplane and b/‖⃗w‖ is the perpendicular distance from the
hyperplane to the origin. The training data must satisfy the equations
and
⃗x i ⃗w + b ≥ 1 for y i = 1 (3.8)
⃗x i ⃗w + b ≤ 1 for y i = −1 (3.9)
which can be combined into
y i (⃗x i ⃗w + b) − 1 ≥ 0 ∀i (3.10)
We now consider the feature vectors which lie on the boundaries and satisfy the following
equations
⃗x i ⃗w + b = 1 for y i = 1 (3.11)
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