U. Glaeser

FIGURE 28.22 The (stylized) power spectra of a set of 3-D Gabor filters.
FIGURE 28.23 Velocity estimation in the frequency domain via estimation of the slope of the spectrum.
Optical Flow Using 3-D Gabor Filters
Heeger [29] proposed the use of 3-D Gabor filters to determine this slope. Following the definition
discussed for 2-D, a 3-D Gabor filter has the impulse response
where
© 2002 by CRC Press LLC
ξt
g( x, y, t)
= gˆ x, y, t
gˆ ( x, y, t)
ξy
ξt
(28.53)
(28.54)
To detect motion in different directions, a family of these filters is defined, as shown in Fig. 28.22.
In order to capture velocities at different scales (high velocities can be thought of as occurring over large
scales, because a large distance is covered per unit time), these filters are applied to a Gaussian pyramidal
decomposition of the sequence. Given the energies of the outputs of these filters, which can be thought
of as sampling spatiotemporal/spatiotemporal-frequency space, the problem is analogous to that shown
in Fig. 28.23. The slope of the line (corresponding to the slope of the plane which characterizes motion)
ξx
( )e j2π[ξ x (x − x
0 0 ) + ξ (y − y y0 0 ) + ξ (t − t t0 0 )]
1
( 2π)
3/2 =
--------------------------------e
σxσ yσt 1 x − x
– -- ⎛ 0
------------- ⎞
2 ⎝ σ ⎠
x
2 y − y ⎛ 0
------------- ⎞
⎝ σ ⎠
y
2 t − t ⎛ 0
----------- ⎞
⎝ σ ⎠
t
2
+ +
ξx