Digital Signal Processing Reference
In-Depth Information
a non-parametric way, smoothed histograms in the YUV color space [
64
], has been
proposed. It learns the histograms from some labeled region and stored in 3D look-
up tables with smoothing. Then the value of
p
(
x
i
|
z
i
)
is searched from the histogram
tables.
In the MRF model,
p
is used to enforce the Markov properties of the labels.
In the Bayesian view, the prior
p
(
Z
)
(
Z
)
does not depend on the observed data
X
.Itis
assumed to be an Potts model, i.e.,
exp
s
i
∈
S
∑
p
(
Z
)=
(
=
)
,
s
j
∈N
i
λ
T
z
i
z
j
(3.16)
where
N
i
is the neighborhood system of
s
i
,
λ
is a negative constant, and
T
(
·
)=
1if
its argument is true and
T
0 if false. In video segmentation, the neighborhood
system includes two parts, the spatial and temporal neighborhoods. The prior in the
spatial neighborhood system incorporates the spatial smoothness constraint, which
can reduce the effect of noise. The prior in the temporal neighborhood system is
used to incorporate the inter-frame information. In the case of binary class problem
(e.g., in foreground/background segmentation,
z
i
∈{
(
·
)=
1
,−
1
}
), the prior
p
(
Z
)
can be
transformed as an isotropic Ising model, i.e.,
exp
s
i
∈
S
∑
p
(
Z
)=
s
j
∈N
i
λ
z
i
z
j
.
(3.17)
does not depend on the observed data. But in the
applications of video segmentation, observed data-dependent prior is necessary. In
the part of spatial neighborhood system, the contrast information is incorporated by
modulating the prior according to the intensity gradients. In the temporal part, the
intensity difference is used to control the probability of
s
i
and
s
j
having the same
label. Therefore, in video segmentation, the prior is expressed as
As noted above, the prior
p
(
X
)
exp
exp
−
Δ
/
σ
s
i
∈
S
∑
2
i
,
j
p
(
Z
)=
s
j
∈N
i
λ
T
(
z
i
=
z
j
)
·
,
(3.18)
where
is a positive constant.
From the equation we can see that if
s
i
and
s
j
have a larger intensity difference, then
they have a higher probability of being different labels.
Combining (
3.14
), (
3.15
), and (
3.18
), the posterior in MRF model is expressed as
Δ
i
,
j
is the intensity difference between
s
i
and
s
j
and
σ
C
exp
1
s
i
∈
S
log
(
p
(
x
i
|
z
i
))+
s
i
∈
S
∑
p
(
X
|
Z
)=
s
j
∈N
i
λ
m
T
(
z
i
=
z
j
)
,
(3.19)
i
where
C
is the partition function and
λ
m
=
λ
exp
(
−
Δ
/
σ
)
.
,
j
