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4.3
Quality Bound
We define the “per feature vector contribution” as the contribution of each feature
vector in a subarea to the confidence that this subarea is the concerned object. In
particular, the “per feature vector contribution” is defined as in Equation 16.
K
k = 1
1
n k Pr
w k
2 Σ 1
t α t μ
t
k
W j =
(
k
|
z j )
z j
.
(16)
k
Therefore, Equation 15 can be rewritten as Equation 17, showing that the quality
function can be viewed as the sum of contribution from all involved feature vectors.
)= j
f
(
Z
W j
b
.
(17)
Given a test image, if we approximate the term n k with their values calculated on the
whole image, the per feature vector contributions W j
,
, ...,
H are independent
from the bounding box within the test image. This means that we can precompute
W j and evaluate the quality function on different rectangles by summing up those
W j that fall into the concerned rectangle.
We design a quality bound for the Gaussianized vector representation in a way
similar to the quality bound for histogram of keywords proposed in [5]. For a set
of rectangles, the quality bound is the sum of all positive contributions from the
feature vectors in the largest rectangle and all negative contributions from the feature
vectors in the smallest rectangle. This can be formulated as
j
1
f
W j 1 R max
(
R
)=
W j 1 × (
W j 1 >
0
)
W j 2 R min
+
W j 2 × (
W j 2 <
0
) .
(18)
where
[
T
,
B
,
L
,
R
]
are the intervals of t
,
b
,
l
,
r and R max and R min are the largest and
the smallest rectangles.
We demonstrate that Equation 18 satisfies the conditions of a qualify bound for
the branch-and-bound search scheme defined in Section 4.1.
First, the proposed f
is an upper bound for all rectangles in the set R .In
particular, the qualify function evaluated on any rectangle R can be written as the
sum of postive contributions and negative contributions from feature vectors in this
rectangle,
(
R
)
)= W j 1 R W j 1 × ( W j 1 > 0 )
+ W j 2 R W j 2 × ( W j 2 < 0 ) .
f
(
R
(19)
 
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