Graphics Reference
In-Depth Information
Second, it's commutative, which we'll show for the continuous case, answer-
ing the inline problem above:
g
)(
t
)=
∞
−∞
(
f
f
(
x
)
g
(
t
−
x
)
dx
.
(18.17)
Substituting
s
=
t
−
x
,
ds
=
−
dx
, and
x
=
t
−
s
, we get
g
)(
t
)=
∞
−∞
(
f
f
(
x
)
g
(
t
−
x
)
dx
(18.18)
=
−∞
s
=
f
(
t
−
s
)
g
(
s
)(
−
ds
)
(18.19)
∞
=
∞
s
=
−∞
g
(
s
)
f
(
t
−
s
)
ds
(18.20)
=(
g
f
)(
t
)
.
(18.21)
The proofs for the discrete and mixed cases are very similar.
Third, convolution is associative. The proof, which we omit, involves multiple
substitutions.
Finally, continuous-continuous convolution has some special properties
involving derivatives, such as
f
g
(under some fairly weak assump-
tions). It also generally increases smoothness: If
f
is continuous and
g
is piece-
wise continuous, then
f
g
=
f
g
is differentiable; similarly, if
f
is once differentiable,
then
f
g
is twice differentiable. In general, if
f
is
p
-times differentiable and
g
is
k
-times differentiable, then
f
g
is
(
p
+
k
+
1
)
-times differentiable (again under
some fairly weak assumptions).
Alas, for a fixed function
f
,themap
g
→
g
is usually not invertible—you
can't usually “unconvolve.” We'll see why when we examine the Fourier trans-
form shortly.
f
Convolution appears in other places as well. Consider the multiplication of 1231
by 1111:
1231
x1111
-----
1231
1231
1231
1231
-------
1367641
In computing this product, we're taking four shifted copies of the number 1231,
each multiplied by a different 1 from the second factor, summing them; this is
essentially a convolution operation.
Figure 18.20: The square
occluder casts a shadow with
both umbra and penumbra when
illuminated
As another example, consider how a square occluder, held above a flat table,
casts a shadow when illuminated by a round light source (see Figure 18.20). The
brightness at a point
P
is determined by how much of the light source is visible
by
a
round
light
source.
