Object Detection Using Nested Cascades of Boosted Classifiers

5.2. Multiclass Adaboost and Multiclass Weak Classifiers
with (⃗a i ) m representing the value of the component m of the vector⃗a i .
It is important to note that the evaluation of {W j,m
l
} m, j,l takes order O(N), with N the number
of training examples, and does not depend on J (number of partitions) nor on M (number of
classes/components). This linear dependency, on N only, is very important to keep a low computational
complexity of the training algorithm.
In contrast to the one-class case, the output of the domain partitioning weak classifier also
depends on the component of the multiclass classifier, therefore the output of the classifier, for a
feature f , is: h( f(x),m) = c j,m ∈R such that f(x) ∈ F j .
For each weak classifier, the value associated to each partition block (c j,m ), i.e. its output, is
selected to minimize Z t :
min Z t = min ∑ W j,m
c j,m ∈R c j,m ∈R
+1 e−c j,m
+W j,m
−1 ec j,m
(5.6)
j,m
and the value of c j,m depends on the kind of weak classifier being used.
In the case of independent components, this minimization problem has an analytic solution
(using standard calculus):
(
c j,m = 1 j,m W
2 ln +1 + ε )
m
W j,m
−1 + ε (5.7)
m
with ε m a regularization parameter. Note that the output components are independent, i.e., the
values of c j,m1 and c j,m2 , for m 1 m 2 , are calculated independently and do not depend on each
other.
It is important to note how this regularization value, ε m , is selected. In [Schapire and Singer,
1999], in a two class classification problem, is shown that it is appropriate to choose ε ≪
2J 1 , with J
the number of partitions. The authors recommended to use ε on the order of 1/n, with n the number
of training examples. If the same analysis is done in the multiclass setting used here, the value of
ε m should be of the order of 1/n m , with n m the number of training examples for class m (this
analysis is straight forward from Equation (11) on [Schapire and Singer, 1999]). This corresponds
to smoothing the weighted histograms taking into account the number of training samples used to
evaluate it.
In the case of weak classifiers with joint components, the optimization problem also has an
analytical solution, but it requires to solve a combinatorial optimization problem. More formally,
in this case c j,m = β m c j , with β m ∈ {0,1}. The optimization problem to be solved is:
min Z t = min min ∑ W j,m
c j ∈R,β m ∈{0,1} β m ∈{0,1} c j ∈R
+1 e−c jβ m
+W j,m
−1 ec jβ m
(5.8)
j,m
In order to obtain the optimal value of this problem, 2 M problems of the form
min
ĉ j ∈R ∑ j
Ŵ j +1 e−ĉ j
+Ŵ j −1 eĉ j
(5.9)
must be solved, with each of these problems having a solution of the form
(Ŵ
ĉ j = 1 j
2 ln +1 + ε
)
Ŵ j −1 + ε , with Ŵ j
l
= ∑ W j,m
l
,l = {−1,+1} (5.10)
m:β m =1
70