Graphics Reference
In-Depth Information
The mean
µ
B
and covariance matrix
B
can can computed from the collection of
N
B
background sample locations
{
B
i
}
in
B
using:
N
B
1
N
B
µ
=
I
(
B
i
)
B
i
=
1
(2.14)
N
B
1
N
B
B
i
)
−
µ
B
)
B
=
1
(
I
(
B
i
)
−
µ
B
)(
I
(
=
i
We can do the same thing for the foreground pixels in the trimap. Therefore, we
can obtain estimates for the prior distributions in Equation (
2.10
) as:
−
µ
B
)
−
1
B
log
P
(
B
)
≈−
(
B
(
B
−
µ
B
)
(2.15)
−
µ
F
)
−
1
F
log
P
(
F
)
≈−
(
F
(
F
−
µ
F
)
where we've omitted constants that don't affect the optimization. For the moment,
let's also assume
P
(α)
is constant (we'll relax this assumption shortly). Then sub-
stituting Equation (
2.12
) and Equation (
2.15
) into Equation (
2.10
) and setting the
derivatives with respect to
F
,
B
, and
α
equal to zero, we obtain the following
simultaneous equations:
F
B
−
1
F
2
2
d
I
3
×
3
2
d
I
3
×
3
+
α
/σ
α(
1
−
α)/σ
2
d
I
3
×
3
−
1
B
2
2
d
I
3
×
3
α(
1
−
α)/σ
+
(
1
−
α)
/σ
−
1
F
2
µ
+
α/σ
d
I
F
=
(2.16)
−
1
B
2
µ
+
(
1
−
α)/σ
d
I
B
(
I
−
B
)
·
(
F
−
B
)
α
=
(2.17)
(
F
−
B
)
·
(
F
−
B
)
Equation (
2.16
)isa6
×
6 linear system for determining the optimal
F
and
B
for
a given
α
;
I
3
×
3
denotes the 3
×
3 identity matrix. Equation (
2.17
) is a direct solu-
tion for the optimal
given
F
and
B
. This suggests a simple strategy for solving the
Bayesian matting problem. First, we make a guess for
α
at each pixel (for example,
using the input trimap). Then, we alternate between solving Equation (
2.16
) and
Equation (
2.17
) until the estimates for
F
,
B
, and
α
α
converge.
2.3.2
Refinements and Extensions
In typical natural image matting problems, it's difficult to accurately model the
foreground and background distributions with a simple pdf. Furthermore, these dis-
tributions may have significant local variation in different regions of the image. For
example, Figure
2.9
a illustrates the sample foreground and background distribu-
tions for a natural image. We can see that the color distributions are complex, so
using a simple function (such as a single Gaussian distribution) to create pdfs for the
foreground and background is a poor model. Instead, we can fit multiple Gaussians
to each sample distribution to get a better representation. These Gaussian Mixture
Models (GMMs) can be learned using the Expectation-Maximization (EM) algorithm
[
45
] or using vector quantization [
356
]. Figure
2.9
b shows an example of multiple
Gaussians fit to the same sample distributions as in Figure
2.9
a. The overlap between
