Graphics Reference
In-Depth Information
Each pixel p
= (
x , y
)
generates a linear equation in the unknown values of
I
(
x , y
)
. There are two cases, depending on the 4-neighborhood of p (denoted
N(
p
)
):
1.
N(
. In this case — such as pixel A in Figure 3.8 — the neighborhood of
the pixel is fully contained in
p
)
. There are no boundary conditions and we use
the usual approximations of the Laplacian:
I
(
x
+
1, y
) +
I
(
x
1, y
) +
I
(
x , y
+
1
) +
I
(
x , y
1
)
4 I
(
x , y
)
=
S
(
x
+
1, y
) +
S
(
x
1, y
) +
S
(
x , y
+
1
) +
S
(
x , y
1
)
4 S
(
x , y
)
(3.14)
2.
N(
. In this case—such as pixel B in Figure 3.8 — the pixel is on the edge
of the source region, and the estimate of the Laplacian includes pixels from
the target that are specified by the boundary condition:
p
)
+
I
(
q
)
T
(
q
)
4 I
(
x , y
)
q
N (
p
)
q
N (
p
)
=
S
(
x
+
1, y
) +
S
(
x
1, y
) +
S
(
x , y
+
1
) +
S
(
x , y
1
)
4 S
(
x , y
)
(3.15)
is well inside the target image (i.e., surrounded by a
healthy border of target pixels). However, if
Typically, the region
runs all the way to the image bor-
der, Equation ( 3.14 ) and Equation ( 3.15 ) need to be modified to avoid querying pixel
values outside the image. For example, if the upper left-hand corner
(
1, 1
)
,we
would modify Equation ( 3.14 )to
I
(
2, 1
) +
I
(
1, 2
)
2 I
(
1, 1
) =
S
(
2, 1
) +
S
(
1, 2
)
2 S
(
1, 1
)
(3.16)
Collecting together all the equations for each p
results in a large, sparse linear
system. There are asmany unknowns as pixels in
, but atmost five nonzero elements
per row, with a regular structure on where these elements occur. 4
Solving the Poisson equation for the example images in Figure 3.7 results in
the improved composite in Figure 3.9 . As with the Laplacian pyramid, the Poisson
equation was applied to each color channel independently. We can see that the over-
all colors of the target image merge naturally into the source region, while keeping
the sharp detail of the source region intact.
We can obtain a slightly different interpretation of Equations ( 3.11 )-( 3.12 )by
defining E
(
x , y
) =
I
(
x , y
)
S
(
x , y
)
and rearranging:
2 E
(
x , y
) =
0in
(3.17)
s . t . E
(
x , y
) =
T
(
x , y
)
S
(
x , y
)
on
(3.18)
That is, E
is a “correction” that we add to the source pixels to get the final image
pixels. We can think of E
(
x , y
)
as a smooth membrane that interpolates the samples
of the difference between the target and source pixels around the boundary of
(
x , y
)
.
Now Equation ( 3.17 ) is a Laplace equation, which implies that the solution E
(
x , y
)
is
a harmonic function . Once we compute E
(
x , y
)
, we recover I
(
x , y
) =
S
(
x , y
) +
E
(
x , y
)
.
4 In fact, the same kinds of systems occurred when we considered the matting problem in Sections
2.4 and 2.6 .
 
Search WWH ::




Custom Search