Image Processing Reference
In-Depth Information
6
4
2
0
−2
−4
−6
−6
−4
−2
0
2
4
6
Fig. 8.9. The
red square
shows the basic area [
−π, π
]
⊗
[
−π, π
] that is preserved (and re-
peated) by a rectangular sampling grid. The
two squares behind
show the same frequency
content but rotated with
16
and
4
radians. The
magenta regions
show the areas that should be
suppressed if the original grid density is retained
r
=
Ar
+
r
0
(8.25)
where
A
is a constant invertible matrix, and
r
0
is a constant translation vector. Even
affine warping is a linear operator. Because a translation with the constant vector
r
0
corresponds to a multiplication with exp(
k
) in the frequency domain, the fre-
quency support of
F
is unchanged. Accordingly, only the effect of the matrix multi-
plication, i.e.,
r
0
=0above, on the spectral support is of relevance when discussing
sampling of an affine deformation of an image. Just like in the rotation transforma-
tion,
−
i
r
T
0
r
=
Ar
f
(
Ar
)
⇒
(8.26)
the result of affine warping is a band-limited function, albeit the boundary of the
characteristic function of
F
now undergoes a more general linear transformation
instead of a simple rotation,
r
=
Ar
k
=
A
−
1
k
⇔
(8.27)
An affine coordinate transformation is therefore absorbed by the interpolation func-
tion, and the result is a value of the image at the new grid coordinates but computed
