Graphics Reference
In-Depth Information
(a)
(c)
(b)
Figure 3.7.
(a) The target image and (b) the source image, indicating the region
to be com-
posited. (c) Laplacian pyramid blending fails when the source and target regions' colors differ
by too much.
that we want to solve
2
dx dy
min
∇
I
(
x
,
y
)
−∇
S
(
x
,
y
)
I
(
x
,
y
)
∈
(3.7)
s
.
t
.
I
(
x
,
y
)
=
T
(
x
,
y
)
on
∂
If we denote the integrand as
∂
2
∂
2
x
−
∂
I
S
I
y
−
∂
S
2
F
(
x
,
y
)
=∇
I
(
x
,
y
)
−∇
S
(
x
,
y
)
=
+
(3.8)
∂
∂
x
∂
∂
y
then the calculus of variations implies that the
I
(
x
,
y
)
that solves Equation (
3.7
)isa
solution of the
Euler-Lagrange equation
:
∂
F
dx
∂
d
F
dy
∂
d
F
−
I
x
−
I
y
=
0in
(3.9)
∂
I
∂
∂
Plugging Equation (
3.8
) into Equation (
3.9
) yields
1
2
2
2
I
2
S
2
I
2
S
∂
x
2
−
∂
∂
y
2
−
∂
+
=
0in
(3.10)
x
2
y
2
∂
∂
∂
∂
or more simply,
2
I
2
S
∇
(
x
,
y
)
=∇
(
x
,
y
)
in
(3.11)
s
.
t
.
I
(
x
,
y
)
=
T
(
x
,
y
)
on
∂
(3.12)
1
Note that the term
∂
F
∂
I
in Equation (
3.9
) equates to 0 in this case, since the partial is treated with
respect to the symbol
I
, and
I
doesn't appear by itself in Equation (
3.8
).
