Graphics Reference
In-Depth Information
B
A
Figure 3.8. Discrete sets required for solving the Poisson equation using digital images. A small
image region is shown. The lightly shaded squares comprise ; the darker-shaded squares
comprise .
2 I
2 I
2 I
where we have used the common notation of
=
+
y 2 for the Laplacian
x 2
operator.
An equation of the form ( 3.11 ) (with a generic right-hand side) is called a Poisson
equation , anda constraint of the form( 3.12 ) is calleda Dirichlet boundary condition .
If the right-hand side of Equation ( 3.11 ) is zero, it is called a Laplace equation 2 ; ifthe
right-hand side of Equation ( 3.12 ) is zero, it is called a Neumannboundary condition .
Before we discuss how the Poisson equation is solved in practice, we mention
an important generalization. In Equation ( 3.11 ), we assumed that the Laplacian
of the new image was equal to the Laplacian of another image (i.e., the source)
inside
. However, the technique is more powerful if we minimize the difference
between the Laplacian of the new image and some arbitrary guidance vector field
(
. The distinction is that the guidance vector field
neednot arise by taking the gradient of some original image (inwhich case it is called a
non-conservative field ). The Poisson equation in Equation ( 3.11 ) slightly changes to
S x (
x , y
)
, S y (
x , y
))
at every pixel
(
x , y
)
div S x
+
S y
(
x , y
)
=
S x
2 I
(
x , y
) =
in
(3.13)
S y
(
x , y
)
x
y
where div represents the divergence of an arbitrary vector field . 3
To solve thePoissonequation for a real-worldpixellated image, we create adiscrete
version of Equations ( 3.11 )-( 3.12 ). As illustrated in Figure 3.8 ,
is a user-defined
collection of pixels (i.e., the pixels where M
=
1 in the previous section) and
is the
set of pixels not in
that have one of their 4-neighbors in
. The source image S must
at least be defined on
plus a one-pixel-wide dilation of
.
2 The Laplace equation is also sometimes known as the heat equation or diffusion equation .
3 For a readable refresher on vector calculus and derivatives, see the topic by Schey [ 427 ].
 
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