Image Processing Reference
In-Depth Information
ω
y
|F|
2
ω
ω
x
d
2
, used in the MS estimate. The error is not measured as the shortest
distance between a frequency coordinate
Fig. 10.14. The error,
and the
k
-axis. This should be contrasted with the
TLS error, which does measure the shortest distance, as shown in Fig. 10.9
ω
tensor, via its minor eigenvector, encodes the direction in which a small translation of
the image causes it the least departure from the original. To see this, we perform the
expansion, i.e., we express the image
f
at
r
+
k
, where the direction
k
=(
k
x
,k
y
)
T
has the unit length, by using the partial derivatives of
f
at
r
f
(
r
+
k
)=
f
(
r
)+
k
x
D
x
+
k
y
D
y
f
(
r
)+
2
2
k
x
D
x
+
k
y
D
y
2
f
(
r
)+
···
(10.58)
Accordingly,
k
x
D
x
+
k
y
D
y
f
(
r
)=
f
(
r
+
k
)
f
(
r
) (10.59)
is the linear approximation of the difference between the function
f
(
r
) and its trans-
lated version,
f
(
r
+
k
). In consequence,
(
k
x
D
x
+
k
y
D
y
f
(
r
))
2
−
(10.60)
is the magnitude of the rate of the change in the direction of
k
, which can be viewed
as the error rate or resistance rate when translating the image in the direction
k
.
Integrating this function and using the (Parseval-Plancherel) theorem 7.2, yields
k
T
f
(
x, y
)
2
dxdy
=
k
T
T
f
(
x, y
)
dxdy
k
=
k
T
Sk
(10.61)
∇
∇
f
(
x, y
)
∇
which is the dynamic part of our original error function,
e
(
k
), because
