1 The Temporal (1-D) Gabor Filter
Gabor filters can serve as excellent band-pass filters for unidimensional sig nals (e.g.,
sp e e ch). A complex Gabor filter is defined as the product of a Gaussian kernel times
a complex sinusoid, i.e.
g(t) = ke
jθ
w(at)s(t) (1)
where
w(t) = e
−πt
2
(2)
s(t) = e
j(2πf
o
t)
(3)
e
jθ
s(t)e
j(2πf
o
t+θ)
=
sin(2πf
o
t + θ), j cos(2πf
o
t + θ)
(4)
Here k, θ, f
o
are filter parameters. We can think of the complex Gabor filter as two
out of phase filter s continently allocated in the real and complex part of a complex
function, the real part holds the filter
g
r
(t) = w(t) sin(2πf
o
t + θ) (5)
and the imaginary part holds the filter
g
i
(t) = w(t) cos(2πf
o
t + θ) (6)
1.1 Frequency Response
Taking the Fourier transform
ˆg(f) = ke
jθ
Z
∞
−∞
e
−j2πft
w(at)s(t) dt = ke
jθ
Z
∞
−∞
e
−j2π(f−f
o
)t
w(at)dt (7)
=
k
a
e
jθ
ˆw(
f − f
o
a
) (8)
where
ˆw(f) = w(f) = e
−πf
2
(9)
1.2 Gabor Energy Filters
The re al and imaginary components of a complex Gabor filter are phase sensitive,
i.e., as a consequence their respons e to a sinusoid is another sinusoid (see Figure
1.2). By ge tting the magnitude of the output (square root of the sum of squared
real and imaginary outputs) we can get a response that phase insensitive and thus
unmodulated positive respo ns e to a target sinusoid input (see Figure 1.2). In some
cases it is useful to compute the over all output of the two out of phase filters.
One commo n way of doing so is to add the squared o utput (the energy) of each
filter, equivalently we can get the magnitude. This corresponds to the magnitude
(more precisely the squared magnitude) of the complex Gabor filter output. In the
frequency domain, the magnitude of the response to a particular frequency is simply
the magnitude of the complex Fourier transform, i.e.
kg(f )k =
k
a
ˆw(
f − f
o
a
) (10)
Note this is a Gaussian function centered at f
0
and with width proportional to a.
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