Graphics Reference
In-Depth Information
3.5
Suppose we use a Laplacian pyramid to blend two images along a vertical
boundary using the 5
5 kernel in Equation (
3.3
). If we use five levels of the
pyramid, howwide is the transition band at the lowest level (with respect to
the original image resolution)?
3.6 Show how Equations (
3.11
)-(
3.12
) result from Equation (
3.7
) based on the
Euler-Lagrange equation.
3.7 Prove that when
×
is a conservative vector field, Equation (
3.13
)isthe
same as Equation (
3.11
).
(
S
x
,
S
y
)
3.8
Suppose the region
for a Poisson compositing problem is given by
Figure
3.41
, with the pixels labeled from 1 to 30 as shown.
1
23456
789 011
12
Figure 3.41.
An example region
for a
18
13
14
15
16
17
Poisson compositing problem.
19
20
21
22
23
24
25
26
27
28
29
30
a) Which pixels make up the set
?
∂
b) Which pixels make up the set
?
c) Explicitly determine the linear system relating the knowns (the source
S
and target
T
) and unknowns (composite pixels
I
) for this region. Your
answer should be in the formof a linear system
Ax
=
b
, where
x
contains
the unknown values of
I
.
3.9 Prove that
the harmonic interpolant
E
(
x
,
y
)
obtained by solving
Equations (
3.17
)-(
3.18
) satisfies
|
E
(
x
,
y
)
|≤
max
∂
|
S
(
x
,
y
)
−
T
(
x
,
y
)
|
(3.47)
(
x
,
y
)
on
(Hint: use a concept from complex analysis.)
3.10 Compute the mean-value coefficients
λ
(
p
)
for the point
p
and
∂
specified
i
in Figure
3.42
.
3.11 Can Farbman's approach using mean-value coordinates be applied to
the mixed-gradient compositing scenario of Equation (
3.19
)? Explain your
reasoning.
by the points
{
p
1
,
...
,
p
4
}
3.12
Suppose we solve a compositing problemusing graph cuts, where the user-
specified source and target pixels are given in Figure
3.43
. Prove that none
of the pixels in the striped region can come from
T
in the composite.
3.13
-expansion label changes to get from the initial
labeling to the final labeling in Figure
3.44
.
3.14 Compute the discrete approximation
U
n
(
Sketch a plausible series of
α
)
x
,
y
in Equation (
3.32
)tothe
continuous derivative
∂
I
∂
t
in Equation (
3.31
).
