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and height of an object with four parameters, i.e., the top, bottom, left and right
coordinates
.
The sliding window approach is most widely used in object localization with
bounding boxes [9, 2]. To find the bounding box where the quality function
f
reaches its global maximum, we need evaluate the function on all possible rect-
angles in the image, whose number is on the order of
O
(
t
,
b
,
l
,
r
)
n
4
(
)
×
n
image.
To reduce the computational cost, usually only rectangles at a coarse location grid
and of a small number of possible widths and heights are considered. On the other
hand, different approaches can be adopted to use a local optimum to approximate
the global one, when the quality function
f
has certain properties, such as smooth-
ness. All these approaches make detection tractable at the risk of missing the global
optimum, and with demand for well informed heuristics about the possible location
and sizes of the object.
In recent years, the most popular technique in the sliding window approach is the
cascade [10]. The cascade technique decomposes a strong object/non-object classi-
fier into a series of simpler classifiers. These classifiers are arranged in a cascade, so
that the simpler and weaker classifiers will eliminate most of the candidate bound-
ing boxes, before the more powerful and complicated classifiers will make finer
selection. However, the cascade of classifiers is slow to train. Moreover, it unfor-
tunately involves many empirical decisions, e.g., choosing the false alarm rate and
missing rate at each stage of the cascade. The cascade technique always reduces the
performance compared with the original strong classifier.
The branch-and-bound search scheme was recently introduced [5] to find the
globally optimal bounding box without the heuristics and assumptions about the
property of the quality function. It hierarchically splits the parameter space of all
the rectangles in an image, and discards large parts if their upper bounds fall lower
than an examined rectangle.
For localization based on bounding boxes, a set of rectangles is encoded with
for an
n
[
T
,
B
,
L
,
R
]
, each indicating a continual interval for the corresponding parameter in
(
. The approach starts with a rectangle set containing all the rectangles in
the image, and terminates when one rectangle is found that has a quality function
no worse than the bounds
f
of any other rectangle set.
At every iteration, the parameter space
t
,
b
,
l
,
r
)
is split along the largest of the
four dimension, resulting in two rectangle sets both pushed into a queue together
with their upper bounds. The rectangle set with the highest upper bound is retrieved
from the queue for the next iteration.
The steps of the branch-and-bound search scheme can be summarized as follows:
1. Initialize an empty queue
Q
of rectangle sets. Initialize a rectangle set
R
to be
all the rectangles:
T
and
B
are both set to be the complete span from zero to the
height of the image.
L
and
R
are both set to be the complete span from zero to
the width of the image.
2. Obtain two rectangle sets by splitting the parameter space
[
T
,
B
,
L
,
R
]
[
T
,
B
,
L
,
R
]
along the
largest of the four dimension.
3. Push the two rectangle sets in Step 2 into queue
Q
with their respective quality
bound.
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