Image Processing Reference
In-Depth Information
k
s
>K
s
⇒
G
(
k
x
,k
y
)=0
(12.75)
Because of this
|
k
t
|
will have an upper bound for the meaningful spatial frequencies
k
s
≤
K
s
⇒|
k
t
|≤
vK
s
=
K
t
(12.76)
Then, if the translation speed
v
is limited, the spatio-temporal image is band-limited,
too. Furthermore, if the temporal axis is sampled with the sampling period
2
π
2
K
t
or
tighter, the speeds of points moving with speeds not greater than
v
will be recover-
able, because the motion planes generated by such translations will have a smaller
inclination angle with the
k
x
,k
y
plane. In consequence of this, it is sufficient that
v
max
,
K
t
, and
K
s
satisfy
K
t
K
s
v
max
≤
(12.77)
where
v
max
is the largest speed of any point in
f
(
x, y, t
), to recover the spatio-
temporal image, and thereby represent the motion without distortion, via the samples
f
(
x
j
,y
j
,t
j
). We summarize these results in the following lemma:
Lemma 12.8.
Let
g
(
x, y
)
be a band-limited function with the maximum spatial fre-
quency
K
s
and
f
be a translated version of it,
f
(
x, y, t
)=
g
(
x
−
v
x
t, y
−
v
y
t
)
, with
the velocity
v
=(
v
x
,v
y
)
T
. Then,
K
s
|
v
|
=
K
t
(12.78)
where
K
t
is the maximal temporal frequency that can occur in
F
. If the structure
tensor is to be sampled, then
K
s
and
K
t
must satisfy
π
2
π
2
K
s
≤
K
t
≤
(12.79)
assuming normalized spatial and temporal sampling periods.
If we assume that both
K
s
=
K
t
=
π
, which normalizes the sampling period to
unity to the effect that the sampling points are one unit apart in
x, y,
and
t
directions,
then the maximal speed of the translation is bounded by
K
t
K
s
v
max
≤
=1
(12.80)
To recover larger speeds from sampled images, either
K
t
must be increased (the tem-
poral axis is sampled more densely) or
K
s
should be decreased (
x, y
axes are low-
pass filtered or the image is enlarged). When the structure tensor is used to recover
the velocity, one must make sure that both
K
t
and
K
s
are less than
π/
2, assuming
normalized distances between the image pixels and image frames.
