Image Processing Reference
In-Depth Information
g
(
x, y
)=
g
(
x
−
t
v
x
,y
−
t
v
y
)=
f
(
x
,y
,t
)
(12.59)
This yields a spatio-temporal image function
f
defined on
E
3
that contains a volume
of image frames originating from the continuously varying time,
t
. Accordingly, the
CT is written in matrix form as:
⎛
⎞
10
v
x
01
v
y
00 1
⎝
⎠
·
r
=
Ar
,
with
r
=(
x, y, t
)
,
A
=
(12.60)
Using
s
=(
x
,y
)
T
, and
s
=(
x, y
)
T
for convenience, the inverse of this CT is
easily found by observing that
s
=
s
+
t
v
, Eq. (12.58),
⎛
⎞
t
v
=
x
=
x
− t
v
x
y
=
y
−
10
−v
x
01
⎝
⎠
(12.61)
s
=
s
−
t
v
=
s
−
A
−
1
=
t
v
y
⇒
v
y
00 1
−
Now, the FT of
f
(
x
,y
,t
) is
F
(
k
x
,k
y
,k
t
)=
i
k
T
r
)
d
r
f
(
r
) exp(
−
=
g
(
x
−
t
v
x
,y
−
t
v
y
) exp(
i
k
T
r
)
d
r
−
=
g
(
x, y
) exp(
−i
k
T
Ar
)
|J
(
r
,
r
)
|d
r
=
i
k
T
r
)
J
(
r
,
r
)
g
(
x, y
) exp(
−
|
|
d
r
=
ik
t
t
)
d
s
dt
=
G
(
k
x
,k
y
)
δ
(
k
t
)=
G
(
k
x
,k
y
)
δ
(
k
x
v
x
+
k
y
v
y
+
k
t
) (12.62)
i
(
k
x
x
+
k
y
y
)) exp(
g
(
x, y
) exp(
−
−
where
k
=(
k
x
,k
y
,k
t
)
T
J
(
r
,
r
)
and
|
|
, the determinant of the Jacobian, are
⎡
⎛
⎞
⎤
∂x
∂x
∂x
∂y
∂x
∂t
⎣
⎝
⎠
⎦
k
T
∂y
∂x
∂y
∂y
∂y
∂t
=
k
T
A
,
J
(
r
,
r
)
and
|
|
=det
=det(
A
)=1
∂t
∂x
∂t
∂y
∂t
∂t
(12.63)
We can see from Eq. (12.62) that the 3DFTofa2D pattern in translation is an
intersection of an oblique plane having the normal
w
=(
v
T
,
1)
T
,
k
x
v
x
+
k
y
v
y
+
k
t
=
w
T
k
=0
(12.64)
and a cylinder,
6
G
=
G
6
Note that the cylinder is given by the 2D FT of the pattern stacked in the depth, i.e.,
for all values of
k
t
.
