Image Processing Reference
In-Depth Information
filter is a pure Gaussian and has been omitted in the center of the illustration but can
be included in the filter bank. In certain applications, such as direction estimation,
the DC-filter is not needed. As before, the filters need tessellate only one half of the
frequency plane because the image f is assumed to be real.
9.7 Design of Gabor Filters on Nonregular Grids
It has been shown that both in audio and in visual signal processing pathways of
primates the sensitivity is not uniformly distributed across the perceivable frequen-
cies. We start to hear low-frequency audio at lower amplitudes than we do for high-
frequency audio. In images, we start to perceive low spatial-frequencies at lower
contrasts (amplitudes) than high spatial-frequencies. Therefore, noise in high fre-
quencies does not influence our audio or image quality assessment as much as it
does at the low frequencies of the respective spectra. The nonuniformity in sensitiv-
ity is attributable to the limited signal processing and communication resources, such
as the number of specialized cells, the connections between the cells and the statis-
tics of real-world signals that matter for the organism. Images that humans and other
primates encounter have a decreasing spectral power with increased frequencies. It
is therefore plausible that an organism devotes its limited processing resources in
proportion to their actual use.
As to human-made systems, these observations help to design more efficient sys-
tems, such as in compression of audio and video, where the largest part of the limited
vocabulary represents the frequencies to which humans are most sensitive. Pattern
recognition in audio and video increasingly uses filters with bandwidths that increase
with tune-on frequencies . 6 Applications such as biometric person authentication also
increasingly use such filter banks. Here, we will discuss Gabor filter design on mul-
tidimensional and nonuniform grids, [21], to yield higher sensitivies at certain bands
than others.
We illustrate the approach first in 1D for convenience, and then we generalize the
approach by extending it to 2D. To reduce the amount of manual labor, we suggest
using an analytic coordinate transformation:
ω = u 1 ( ξ ) ,
ξ = u ( ω ) ,
or
(9.42)
where u is yet to be specified. The transformation u is an injective reparametrization
of ω such that u is invertible and establishes a differentiable (continuous) map be-
tween ω
[ ω min max ] and ξ [0 , 1]. In consequence, we require that u satisfies the
boundary conditions:
ω (0) = u 1 (0) = ω min
ω (1) = u 1 (1) = ω max
and
(9.43)
Additionally, we demand that there will be higher sensitivity at low frequencies. We
must translate sensitivity to a mathematical concept by defining it as a property of
6 The tune-on frequency of a bandpass filter is the frequency where the Fourier transform of
the filter has the maximum amplitude.
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