Graphics Reference
In-Depth Information
(a)
(b)
Figure 2.14.
(a) An original image and (b) the eight eigenvectors corresponding to the smallest
eigenvalues of its matting Laplacian.
smallest eigenvalues of an input image. We can see that these eigenvector images
tend to be locally constant in large regions of the image and seem to follow the con-
tours of the foreground object. Any single eigenvector is generally unsuitable as a
matte, because mattes should be mostly binary (i.e., solid white in the foreground
and solid black in the background). On the other hand, since any linear combination
of null vectors is also a null vector, we can try to find combinations that are as binary
as possible in the hopes of creating “pieces” useful for matting.
Levin et al. [
272
] subsequently proposed an algorithm based on this natural idea
called
spectral matting
. We begin by computing the matting Laplacian
L
and its
eigenvectors
E
corresponding to the
K
smallest eigenvalues (since the
matrix is positive semidefinite, none of the eigenvalues are negative). Each
e
i
thus
roughly satisfies
e
i
Le
i
=[
e
1
, ...,
e
K
]
0 and thus roughly minimizes Equation (
2.30
), despite
being a poor matte. We then try to find
K
linear combinations of these eigenvectors
called
matting components
that are as binary as possible by solving the constrained
optimization problem
=
K
N
k
i
|
ρ
+|
k
i
|
ρ
min
1
|
α
1
−
α
(2.43)
K
,
k
∈
R
y
k
=
1,
...
,
K
k
=
1
i
=
k
Ey
k
s
.
t
.
α
=
(2.44)
K
k
i
1
α
=
1,
i
=
1,
...
,
N
(2.45)
k
=
That is, Equation (
2.44
) says that each matting component must be a linear com-
bination of the eigenvectors, while Equation (
2.45
) says that thematting components
must sumto1 at eachpixel. Figure
2.15
illustrates the score function inEquation (
2.43
)
with
is either 0 or 1.
The result of applying this process to the eigenvectors in Figure
2.14
is illustrated
in Figure
2.16
a. At this point, the user can simply view a set of matting components
and select the ones that combine to create the desired foreground (this step takes the
place of the conventional trimap or scribbles). For example, selecting the highlighted
components in Figure
2.16
a results in the good initial matte in Figure
2.16
b. User
scribbles can be used to further refine the matte by forcing certain components to
contribute to the foreground or the background.
ρ
=
0.9; we can see that it's lowest when
α
