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T
H T ( x )=
( h t ( x )) .
(1)
t =1
The weak hypotheses often internally operate with discrete values corresponding
to partitions of the object space χ . Such weak hypotheses are called by Schapire
and Singer [9] space partitioning weak hypotheses. Moreover, the weak hypothe-
ses usually make their decision based only on a single image feature which is
either discrete (e.g. LBP) or is quantized (e.g. Haar-like features and a thresh-
old function). In the further text, such functions f : χ
are reffered to in
general simply as features and the weak hypotheses are combinations of such
features and a look-up table functions l :
N
N R
h t ( x )= l t ( f t ( x )) .
(2)
In the further text, c ( j t specifies the real value assigned by l t to output j of f t .
The decision strategy S of a soft cascade is a sequence of decision functions
S = S 1 ,S 2 ,...,S T ,where S t :
1. The decision functions S t are evaluated
sequentially and the strategy is terminated with negative result when any of
the decision functions outputs
R
,
1. If none of the decision functions rejects the
classified sample, the result of the strategy is positive.
Each of the decision functions S t bases its decision on the tentative sum of
the weak hypotheses H t , t<T which is compared to a threshold θ t :
S t ( x )= ,
if H t ( x ) t
.
(3)
1 , if H t ( x )
θ t
In this context, the task of learning a suppression classifier can be formalized
as learning a new soft cascade with a decision strategy S and hypotheses h t =
l t ( f t ( x )), where the features f t of the original classifier are reused and only the
look-up table functions l t are learned.
2.1 Learning Suppression with WaldBoost
Soft cascades can be learned by several different algorithms [1,2]. We chose the
WaldBoost algorithm [11,13] by Sochman and Matas which is relatively simple
to implement, it guarantees that the created classifiers are optimal on the train-
ing data, and the produced classifiers are very fast in practice. The WaldBoost
algorithm described in the following text is a slightly simplified version of the
original algorithm. The presented version is specific for learning of soft cascades.
Given a weak learner algorithm, training data
{
( x 1 ,y 1 ) ..., ( x m ,y m )
}
,x
χ, y
and a target miss rate α , the WaldBoost algorithm solves a
problem of finding such decision strategy that its miss rate α S is lower than α
and the average evaluation time T S = E (arg min i ( S i
∈{−
1 , +1
}
= )) is minimal:
T S ,s.t. α S <α.
S =argmin
S
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