Graphics Reference
In-Depth Information
matrix on both sides, and denote the rows of this inverse by
r
's:
=[
α
i
]
−
1
α
β
F
2
−
B
2
F
1
−
F
2
B
1
−
B
2
(
I
i
−
B
2
)
i
i
(
1
−
α
)γ
i
i
(2.23)
r
1
r
2
r
3
(
=
I
i
−
B
2
)
Taking just the first element of the vector on both sides, we see that
r
1
I
i
r
1
B
2
α
=
−
(2.24)
i
r
1
B
2
.
which corresponds to Equation (
2.20
) with
a
=
r
1
and
b
=−
2.4.2
The Matting Laplacian
The assumption that the
values and colors inside a window are related by
Equation (
2.20
) leads to a natural cost function for the matting problem:
α
N
a
j
I
i
+
2
J
(
{
α
i
,
a
i
,
b
i
,
i
=
1,
...
,
N
}
)
=
w
j
(α
i
−
(
b
j
))
(2.25)
j
=
1
i
∈
This cost function expresses the total error of the linearity assumption in
Equation (
2.20
) over each window. We want to minimize
J
to find
i
at each pixel
as well as the coefficients
a
i
and
b
i
for every window
w
i
around pixel
i
. For brevity,
we'll write the left-hand side as
J
α
(
α
,
a
,
b
)
, where
α
is an
N
×
1 vector that collects all the
α
values in the image, and
a
and
b
represent the collections of affine coefficients for
each window. Since the windows between adjacent pixels overlap, the
estimates at
each pixel are not independent. We also add a
regularization term
to Equation (
2.25
):
α
i
2
N
a
j
I
i
+
+
ε
2
2
J
(
α
,
a
,
b
)
=
w
j
(α
i
−
(
b
j
))
a
j
(2.26)
j
=
1
∈
This term acts to bias the mattes toward being constant, since if
a
=
0,
α
i
=
b
i
is chosen to be on the order of 10
−
7
if each
ε
within the whole window
w
i
. Usually
[
]
color channel is in the range
.
Onfirst glance, this formulationdoesn't seemtohelpus solve themattingproblem,
since we still have many more equations than unknowns (i.e., the five values of
0, 1
,
a
,
and
b
at each pixel). However, by a clever manipulation, we can reduce the number
of unknowns to exactly the number of pixels. First, we rearrange Equation (
2.26
)asa
matrix equation:
α
2
I
j
1
j
1
.
1
α
a
j
b
j
.
.
N
J
(
α
,
a
,
b
)
=
−
(2.27)
I
j
W
j
W
0
3
×
1
1
α
j
=
1
√
ε
I
3
×
3
0
