U. Glaeser

where ‘ ∗’
denotes (discrete) convolution. Similarly, for the continuous case,
The 3-D Fourier Transform
The 3-D continuous Fourier transform can be expressed as
where ξx,
ξy,
and ξt
are the spatiotemporal frequency variables and f(
x,
y,
poral signal. As in the 2-D case, the 3-D Fourier transform is separable:
Also as in the 1-D and 2-D cases, if
then
© 2002 by CRC Press LLC
(28.3)
(28.4)
t)
is a continuous spatiotem-
(28.5)
(28.6)
(28.7)
If h(
x,
y,
t)
is the LSI system impulse response, then H(
ξx,
ξy, ξt) is the frequency response of the system.
The spatiotemporal discrete Fourier transform is defined as
where 0 ≤ h, k, l ≤ N − 1 and WN = .
The inverse transform is
where 0 ≤ m, n, p ≤ N − 1.
g( x, y, t)
= f( x′, y′, t′ )h( x– x′, y– y′, t– t′ ) dx′ dy′ dt′
Moving Images in the Frequency Domain
∫
∞
(28.8)
(28.9)
Following the discussion in [2], a moving monochrome image can be represented by an intensity
distribution f(x, y, t). The image is static if f(x, y, t) = f(x, y, 0) for all t. The velocity of the image can be
expressed via the image velocity vector
If the (initially static) image translates at a constant velocity r,
then
∫
∞
∫
∞
– ∞ – ∞ – ∞
∫
( )e j2π(xξ ∞ ∞ ∞
– x + yξy + tξt )
∫ ∫ – ∞ – ∞ – ∞
F( ξx, ξy, ξt) = f x, y, t
∫
∞
∫
∞
F( ξx, ξy, ξt) f( x, y, t)e
j2πxξ x
=
dx
– ∞
– ∞
∫
∞
– ∞
g( x, y, t)
= h( x, y, t)
∗ f( x, y, t)
–
e j2πyξ – y
G( ξx, ξy, ξt) = H( ξx, ξy, ξt)F( ξx, ξy, ξt) N−1 N−1 N−1
∑∑∑
v( h, k, l)
= u( m, n, p)
u( m, n, p)
e j2π/N –
m=0 n=0 p=0
N−1 N−1 N−1
1
= ----- v( h, k, l)WN
∑∑∑
N 3
h=0 k=0 l=0
r = ( rx, ry) dxdydt dy
hm kn lp
WN WN WN
fr( x, y, t)
=
f( x– rxt, y– ryt, t)
– hm – kn – lp
WN WN
e j2πtξ – t
dt
(28.10)
(28.11)