Graphics Reference
In-Depth Information
pixels
in the local neighborhood of an unknown pixel were used to build Gaussian mixture
models for the foreground and background. Instead, Wang and Cohen proposed
a non-parametric approach similar to a kernel density estimate. The basic idea is
to determine a set of candidate foreground
Recall that in Bayesian matting (Section
2.3
), samples of the known
F
and
B
,
B
C
}
samples for each pixel, which could be either original scribbles or estimates filled in
on a previous iteration. Then we compute a likelihood that pixel
i
has
{
F
1
,
...
,
F
C
}
and background
{
B
1
,
...
α
α
k
as
value
C
C
1
σ
2
2
1
C
2
w
m
w
n
e
−
d
I
i
−
(α
k
F
m
+
(
1
−
α
k
)
B
n
)
L
k
(
i
)
=
(2.57)
m
=
1
n
=
1
Here, the weights
w
of each foreground and background sample are related to the
spatial distance from the sample to the pixel under consideration and the uncertainty
of the sample, and the covariance
2
d
is related to the variances of the foreground and
background samples. Then the final formula for the
E
data
term is
σ
)
k
=
1
L
k
(
L
k
(
i
k
i
E
data
(α
)
=
1
−
(2.58)
i
)
values but not estimates of
F
and
B
. We discussed one method for getting such estimates in Section
2.4.4
. Wang and
Cohen proposed a slightly similar approach based on the foreground samples
After minimizing Equation (
2.54
), we obtain
α
{
F
m
}
and background samples
generated at pixel
i
during the creation of the data
energy. The idea is simply to estimate the foreground and background values as:
{
B
n
}
F
i
,
B
i
}=
2
{
arg min
{
B
n
}
I
i
−
(α
i
F
m
+
(
1
−
α
)
B
n
)
(2.59)
i
F
m
}
,
{
That is, we select the pair of foreground and background samples that gives the best
fit to the matting equation for the given
α
i
and
I
i
. The uncertainty of the pixel is
updated based on the weights
w
of the selected pair
F
i
,
B
i
}
.
Guan et al. [
182
] proposed an algorithm called
easy matting
that uses the same
MRF model with a few differences. They create
E
data
and
E
smoothness
using log likeli-
hoods instead of the exponential forms in Equations (
2.55
)-(
2.56
). The smoothness
term
E
smoothness
is also modulated by the image gradient; that is,
{
2
α
j
)
=
(α
−
α
)
i
j
E
smoothness
(α
i
,
(2.60)
I
i
−
I
j
The balance between
E
data
and
E
smoothness
is updated dynamically, so that the
smoothness term is weighted less as the iterations proceed. Finally, instead of using
belief propagation to solve for the matte, the minimization of Equation (
2.54
) with
respect to the user scribble constraints is posed as a variational problem that can be
solved directly as a linear system.
2.6
RANDOM-WALK METHODS
A family of successful matting algorithms is based on the concept of
random walks
for image segmentation as proposed by Grady [
176
]. As in a Markov Random Field,
we form a graph in which the set of vertices
V
contains all the image pixel locations
