31-08-2012, 04:17 PM
Tutorial on Gabor Filter
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Gabor Energy Filters
The real and imaginary components of a complex Gabor filter are phase sensitive,
i.e., as a consequence their response to a sinusoid is another sinusoid (see Figure
1.2). By getting 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 response to a target sinusoid input (see Figure 1.2). In some
cases it is useful to compute the overall output of the two out of phase filters.
One common way of doing so is to add the squared output (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
Gabor functions as models of simple cell receptive fields
Jones and Palmer (1987) showed that the real part of complex Gabor functions
fit very well the receptive field weight functions found in simple cells in cat striate
cortex.
Here are some useful pieces information for designing biologically inspired Gabor
filters.
• To a first approximation the orientation of the Gaussian envelop !0 can be
modeled as being equivalent to the orientation of the carrier.8 0 = !0.
The actual absolute deviations between 0 and !0 have a Median of about
10 degrees (see Jones and Palmer, 1987, p. 1249).
• In macaque V1, most cells have a half magnitude spatial frequency bandwidth
between 1 and 1.5 octaves. The median is about 1.4 octaves (see
De Valois et al., 1982a, p. 551).
• In macaque V1, the range of half-magnitude orientation bandwidths among
cells is very large: From 8 degrees to the most narrowly tuned. At the other
end there were cells with no orientation selectivity at all. (see De Valois
et al., 1982b, p. 535 and 541) reports the following statistics for the orientation
bandwidth: mean = 65 degrees, median = 42 degrees, mode =
30 degrees. However they point out that others have reported significantly
larger numbers. For example () reports a 71 % from max median bandwidth
of 38.5 degrees. This would correspond to a median half magnitude
bandwidth of 66 degrees. The median bandwidth of simple cells in the cat
is a bit smaller than in the macaque, with a typical median half magnitude
orientation bandwidth of 30 degrees (see De Valois et al., 1982b, p. 535 and
541).
• In macaque V1 the peak frequencies range from as low as 0.5 cycles per
degree of visual angle, to as large as 15 cycles per degree of visual angle.
Mean values are 2.7 cycles per degree for cells mapping into the parafoveal
and 4.25 cycles per degree for cells mapping into the fovea.
• The spatial frequency bandwidth (in octaves) tends to be a bit larger for
cells with low peak frequency than for cells with large peak frequency. For
example, the median half magnitude bandwidth of cells tuned to frequencies
higher than 5 cycles/degree is 1.2 octaves, whereas the median for
cells tuned to frequencies smaller than 2 cycles/degree is 1.7 octaves. (see
De Valois et al., 1982a, p. 552).
• Orientation selective simple cells in V1 show minimum response at about
30 to 40 degrees away from the optimal orientation, not at 90 degrees away
from the optimal orientation (see De Valois et al., 1982b, p. 539).
• The spiking rate of simple cells neurons in macaque V1 is between close
to 0 Hz, at rest, to about 120 Hz, when maximally excited (see De Valois
et al., 1982a, p. 547).
• In the area mapping the fovea, there are more kernels oriented vertically
and horizontally than oriented diagonally (about 3 to 2). (see De Valois
et al., 1982b, p. 537).