Digital Signal Processing Reference
In-Depth Information
In the video image sequence, according to the Gaussian function can modeling for
each pixel (x
0
, y
0
) from 1 to t. Set to the time t, the finite set of pixel is {x
1
,. . . , x
t
} =
{I (x
0
, y
0
, s) | 1≤s≤t}, where I is the video frame. If all the historical values of the
pixel are approximated Through the K Gaussian functions, Then at time t, the
probability of pixel value x
t
belongs to the background is :
K
=
(3)
P
(
x
)
=
ω
*
η
(
x
,
μ
,
)
t
i
,
t
t
i
,
t
i
,
t
i
1
Where x
t
is the pixel values of the time t, usually constituted by the three channels'
color values of red, green and blue. K is the number of mixture Gaussian model. The
value of K generally depends on the available memory size and the computing power
of system, under the normal circumstances, values are between 3 and 5. The greater
the value of K, the stronger the ability to handle fluctuations, but the more time. ω
i,t
is
weights of the model i in the mixture Gaussian model at the time t.η( ) is i-th
Gaussian distribution at the time t. Defined as follows:
1
1
−
1
T
−
(
x
−
μ
)
(
x
−
μ
)
t
i
,
t
t
i
,
t
(4)
η
(
x
,
μ
,
)
=
e
,
i
=
1
2
...,
K
2
i
,
t
t
i
,
t
n
1
i
,
t
(
2
π
)
|
|
2
2
i
,
t
Assuming the pixels of each color channel independently of each other and have same
|
x
−
μ
|
≤
λσ
σ
2
i,t
I. If
variance. So the covariance matrix is∑
i,t
=
i
,
t
−
1
i
,
t
−
1
, means the
x
t
match the Gaussian model, update the model parameters:
(
)
ω
=−
1
1
α ω
+
α
it
,
it
,
−
1
(5)
(
)
μ
=−
ρ
μ
+
ρ
X
t
t
−
1
t
T
(
)
(
) (
)
2
2
σ
=−
1
ρ σ
+
ρ
X
−
μ
X
−
μ
t
t
−
1
t
t
t
t
Where α is the weight update rate, ρas a parameter update rate, ρ = αη (x
t
, μ
t
, σ
t
). The
K Gaussian distributions arrangement in decreasing order of ω/σ, Meet the following
type of pre-B models as the background:
b
ω
i
(6)
B
=
arg min
i
=
1
>
T
b
K
ω
i
i
=
1
