Digital Signal Processing Reference
In-Depth Information
of CS in order to reduce the quantity of data that must be processed. Specifically,
recent methods using CS to perform background subtraction, more general signal
tracking, multi-view visual tracking, and particle filtering will be discussed.
4.1.1
Compressive Sensing for Background Subtraction
One of the most intuitive applications of compressive sensing in visual tracking
is the modification of background subtraction such that it is able to operate on
compressive measurements. Background subtraction aims to segment the object-
containing foreground from the uninteresting background. This process not only
helps to localize objects, but also reduces the amount of data that must be processed
at later stages of tracking. However, traditional background subtraction techniques
require that the full image be available before the process can begin. Such a scenario
is reminiscent of the problem that CS aims to address. Noting that the foreground
signal (image) is sparse in the spatial domain, [33] have presented a technique via
which background subtraction can be performed on compressive measurements
of a scene, resulting in a reduced data rate while simultaneously retaining the
ability to reconstruct the foreground. More recently, a modification which adaptively
adjusts the number of collected compressive measurements based on the dynamic
foreground sparsity typical to surveillance data has been proposed in [152].
Denote the images comprising a video sequence as
N
is the
vectorized image captured at time
t
. Cevher
et al.
[33] model each image as the sum
of foreground and background components
f
t
and
b
t
, respectively. That is,
x
t
=
x
t
}
t
=
0
,where
x
t
∈
R
{
f
t
+
b
t
.
(4.1)
M
×
N
Assume
x
t
is sensed using
Φ
∈
C
to obtain compressive measurements
y
t
=
Φ
represents a CS decoding procedure, then the proposed method for
estimating
f
t
from
y
t
is
x
t
.If
Δ
(
Φ
,
y
)
f
t
=
Δ
(
Φ
,
y
t
)
,
y
−
(4.2)
where it is assumed that
y
t
=
Φ
b
t
is known via an estimation and update procedure.
begin,
y
0
To
is
initialized
using
a
sequence of
N
compressively sensed
y
j
}
j
background-only frames
1
that appear before the sequence of interest begins.
These measurements are assumed to be realizations of a multivariate Gaussian
random variable, and the maximum likelihood (ML) procedure is used to estimate
its mean as
y
0
=
{
=
1
N
j
1
y
j
. This estimate is used as the known background for
t
0
in (
4.2
). Since the background typically changes over time, a method is proposed
for updating the background estimate based on previous observations. Specifically,
the following is proposed:
=
∑
=
y
t
+
1
=
α
(
y
ma
t
y
t
y
t
−
ΦΔ
(
Φ
,
))+(
1
−
α
)
(4.3)
+
1
y
ma
t
y
ma
t
=
γ
y
t
+(
1
−
γ
)
,
(4.4)
+
1
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