Graphics Reference
In-Depth Information
given by
130
150
180
10 0 0
0 00
000
µ
=
=
(2.90)
F
2
F
2
Assuming
σ
d
=
2, what is the Bayesian matting solution for the best
F
,
B
,
and
α
using the strategy discussed on p.
19
?
2.11
Sketch configurations for pixels
i
and
j
such that there are 0, 1, 2, 3, 4,
or 6 windows
k
in the sum for the matting Laplacian element
L
(
i
,
j
)
in
Equation (
2.36
).
2.12
a) Verify that Equation (
2.34
) follows from Equation (
2.33
).
b)
If
G
j
is defined by Equations (
2.27
)-(
2.28
), show that
G
j
G
j
can be writ-
ten as a function of
µ
j
and
j
, the mean and covariance matrix of the
colors in window
j
.
2.13 Prove that, in the absence of any scribbles or user constraints, any constant
matte minimizes the closed-formmatting objective function.
2.14 Consider one of the matrices that makes up the matting Laplacian in
Equation (
2.34
), i.e.,
G
j
G
j
)
−
1
G
j
M
j
=
I
(
W
+
3
)
×
(
W
+
3
)
−
G
j
(
(2.91)
Show that
M
j
has a nullspace of at least dimension 4. (Hint: what is
M
j
G
j
?)
×
2.15
Suppose we compute the matting affinity
A
in Equation (
2.38
) using 3
3
windows. Determine the values in the
i
th
row of
A
if pixel
i
is at the center
ofa5
5 region of constant intensity. Howmany nonzero values are there?
2.16 Showthat the closed-formmattingminimizationprobleminEquation (
2.40
)
leads to the linear system in Equation (
2.41
).
2.17 We know that a constant matte (i.e.,
×
α
=
1) is a null vector of the matting
1
4
on its right-hand side is also a null vector. Compute a linear combination
of these two matting components that is as binary as possible (but is not
constant).
3
4
on its left-hand side and
Laplacian. Suppose the matte that has
α
=
α
=
2
inEquation (
2.42
) for a particular pixel
i
can
be written in the form
F
D
x
F
where
F
is a vector of all the foreground
values and
D
x
is a diagonal matrix with only two nonzero entries.
b) Hence, determine the linear system corresponding to Equation (
2.42
)
for computing
F
and
B
images from given
2.18
a)
Showhow the term
∇
x
F
i
and
I
images.
2.19 Show that solving Equation (
2.51
) in learning-based matting leads to
Equation (
2.52
).
2.20 Plot the smoothness energy in Equation (
2.56
) for belief-propagation-based
matting as a surface over the
α
(α
1
,
α
2
)
plane, where
σ
s
=
0.1.
