Image Processing Reference
In-Depth Information
To summarize, to achieve translation, rotation, and scale-invariance jointly, one
can ignore
B
(0) and normalize all other FDs by another
B
(
n
) that has a large
magnitude, usually
B
(1). In that one ignores
B
(0) and
B
(1), containing four free-
doms. This is economical because it corresponds precisely to translation- (two free-
doms), rotation- (1 freedom), and scale-invariance (one freedom). Note that the in-
dex
n
1 runs over the full range of the coefficients so that the negative
frequencies have the indices
N
∈
0
···
N
−
−
n
according to the cyclical translation rules of
DFT. Whereas
|
B
(
n
)
|
=
|
B
(
−
n
)
|
because
b
(
k
) are not real, in practical applications
|
. Accordingly,
when truncating
B
(
n
), the coefficients are kept symmetrical w.r.t.
n
=0, as is done
in Fig. 17.7.
B
(
n
)
|
still decays as the index
n
increases through 0
,
±
1
,
±
2
,
±
3
,
···
