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TVM – EDGES/TEXTURES – REVIEW AND EXPLANATION OF JONES AND
PALMERS EVALUATION OF 2D GABOR FILTER MODEL OF SIMPLE
RECEPTIVE FIELDS IN THE STRIATE CORTEX OF CATS
Jørgen N. M. Hausted
Introduction
In 1987 Judson P. Jones and Larry A. Palmer tested the hypothesis that 2D Gabor filters can be used as an
analogous to the class of linear spatial filters simple receptive fields belongs to, according to Marcelja’s
hypothesis (1). They tested it on some simple receptive fields in the striate cortex on some cats. Jones and
Palmer made three predictions to decide if the hypothesis were right.
“First, simple cell 2D spatial profiles will be indistinguishable. Second, simple cell 2D spectral response
profiles and 2D Gabor filter amplitude spectre will be indistinguishable. Since the 2D Gabor filter model
presupposes linear spatial summation in 2D, we also predict that simple receptive fields will satisfy this
constraint.” 1
To understand this experiment and the conclusions of it, it’s necessary to have some understanding of how
the brain processes images and then how the 2D Gabor filter works.
Visual pathway
The retina contains a lot of photoreceptive cells, these neurons sends their signals, via the optic nerve, to
the Lateral Geniculate Nucleus (LGN), which is a part of Thalamus. There are two LGNs in Thalamus, one for
each monocular zone. In the LGN the signal is processed by six cellular layers. Two of the layers are
Magnocellar (M-cells) and the remaining four are Parvocellular (P-cells). The M-cells are rods and the
largest of the M- and P-cells. These rods detect depth, movement and small differences in the shade of a
colour is also detected here. The smaller P-cells are the counterpart and therefore cones. These cones
detects wavelength of light. Therefore is the P-cells necessary to see colours. Figure 1 illustrates the
pathway through LGN and on to the Primary Visual Cortex (PVC).
1 (1) p. 1237 ”Predictions”
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The difference between
the two types of layers is
described with results
from an experiment on
monkeys (2). This is
done by making some
damage on chosen areas
of one of the LGNs in a
monkey brain. One of
the LGNs is kept intact,
so it can be used ascontrol. With some
monkeys the P-cells
were damaged and with
others the M-cells. So by
checking the processed
signals from the
damaged LGN and then
compared them with the
control signals, the
features of the different
layers could be
determined (2).
Figure 1 – The visual pathway
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Receptive fields
On the path from the retina to PVC are there some different receptive fields. The first are the retinal
ganglions which is very similar to the ones found in LGN. These fields are round and seen at figure 2a.
There are two types of these fields, those which are on-center and those which is off-center. The on-areas
are sensitive to light, so they react when hit by light. But if the light at the same time hits the off-area, then
it will decrease the response. So if a bigger
off-area is hit than the on-area, then there
will be no response. This means, if only the
on-area is hit, the field give full response. Ifthe most of the hit area is off, then there will
be no response.
The receptive fields in PVC are different. As
seen on Figure 2b are these fields different
on both shape and where the on- and off-
areas are. On figure 2c are shown three
receptive fields connected to a simple cell.
This model is proposed by Hubel and Wiesel
(2). This model tells that a simple cortical
neuron in PVC receives signals from three or
more on-center cells. These combined signals
can be used to edge detection, since an edge
is a sudden change in the light. If one of the
three cells is stimulated and the next one
isn’t, then it could be an edge. On figure 3 is
shown an experiment with a bar of light and
a spot of light (2). On figure 3,3 is the tested
field shown. On figure 3,1 and 3,2 is the two
experiment shown. First an illustration of
where the light hits the field and then the
response. Above the response is shown how long the field where hit with light. In all these experiments is
the light on for 1s. As seen on the experiment with the bar of light, the bars starts horizontal with no
Figure 2 - Receptive fields
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response from the field. Calmly as the bar rotates and more and more of the on-area is hit, and less of the
off-area is, the more is the field responding. When only the on-area is hit, then is the response almost
constant. As the bar continues its rotating is the response decreased. The experiment with the spot of light
is used to locate the on- and off-areas. If there is response, an on-area is found and vice versa. If the whole
field is hit with light, then is the field not responding, because a majority of off-area is hit.
Figure 3 - Test with bar and spot of light
Then are the basics behind the visual part of Jones and Palmers explained and then it’s time for the more
mathematical part, the Gabor filter.
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Gabor filter introduction
The Gabor filter is used for edge detection, like Hubel and Wiesels models proposes. The Gabor filter uses
the frequencies in pictures to detect edges, therefore first something about the frequency domain.
The frequency domain
A picture can be described as a lot of sine and cosine
functions, with different frequencies. To clarify some
terms, then is figure 4 showing which parts of a sine
curve is the Length, L, the amplitude, A, and the phase,φ.
If a single sinusoidal function is used to make a picture, it
will only show a picture with stripes. The number of
stripes depends on the frequency. Figure 5 illustrates
different kind sinusoidal functions. 5a is with a higher
frequency than the others, 5b is with lower amplitude so
the picture becomes more blurry. 5c has a 90º phase
shift, so the picture starts with the middle of a white line instead of the border of a black line (3).
This is a little too simple to describe more complex images, so Fourier series is used. Fourier series is the
weighted sum of a set of sine and cosine functions. This can be summed up to:
(1)
(2) 2
The index n is here the number of cycles of the sine that fits
within one period of f(x). This means that an index value of
one is a normal sinusoidal curve. But with a higher index
value there will be more cycles within one period and it’s
illustrated on figure 6. The top example is with index of one,
the middle is three and the last is 15. At index value 15, it
2 (3) p. 192 Equation (8.2)
Fi ure 4 - A sinusoidal function
Fi ure 5 - Different kinds of sinusoidal
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Figure 7 - Some basic images of the Fourier
representation of an image
becomes clearer that this method can
make sudden changes in pictures, so
the shift from a positive amplitude to
a negative. This makes the edges
sharper and not fuzzy like figure 5b.
This is only one dimension and this
will only make lines. With Fourier
series they can have different width
and sharpness but they will all be
vertical. So by combining two dimensions of Fourier series, one for each axis (x, y) it’s possible to describe
complicated images. Combining two dimensions gives:
(3) 3
Here are the indexes, u and v, the number of cycles fitting into
respectively one horizontal and one vertical period. To show
how this can be used to make more complex images are 16
images shown at figure 7. The images show how different u
and v values can rotate the stripes. When a lot of small images
made this way are put together, more detailed pictures can be
made. For example does figure 7 look like a part of a bunch of
circles, if looked at from afar.
With some basic knowledge about Fourier series it’s time to
move on to the Gabor filter.
Gabor filter
As mentioned is the 2D Gabor filter used for edge detection. The filter is build up by a complex and a real
part, which both is a build on variant of the Fourier series (4), which can be seen here:
3 (3) p. 193 Equation (8.3)
Fi ure 6 - Exam les of Fourier series
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(4) (5)
(6) 4
Where is the aspect ratio, is a part of the bandwidth, is the wavelength, is the phase offset and is
the orientation(s). To describe factors fast; the aspect ratio specifies the ellipticity of the support of the
Gabor filter. For the value one is the support circular and for smaller values is the support stretched in the
parallel orientation as the stripes. A default support is 0.5. is changed through the bandwidth and is
connected with the wavelength. With a default bandwidth value one is . The wavelength from
the cosine factor of the Gabor filter kernel. It’s specified in pixel and must be a real number equal or
greater than two. The phase offset is specified en degrees and is the argument of the cosine factor. At last
is the orientation which is specified in degrees and more than one can be used at the same time. The
orientation determines how the picture is seen, so with more orientations it’s possible to see more edges.
If only one edge is used, then will parallel edges appear. Figure 8d shows original image. 8a and 8 b are
made with an orientation at respectively 0 and 90 degrees. Since there is only used one orientation, then
there are only lines in one direction. On 8c is used 0 and 90 degrees and it’s basically 8a and b combined. So
with many orientations will image be more and more clear.
Figure 8 - Examples with different orientation
Then is the 2D Gabor briefly explained and Palmer and Jones’ results can be reviewed.
The results
As told in the introduction, did Palmer and Jones made three predictions to test if the hypothesis were
right. These predictions were tested in the striate cortex in 14 cats. 25 profiles for both the 2D spatial and
the 2D spectral response were made in this experiment. These profiles were the test data for processing
the simple cells and a similar set of profiles were made by using the 2D Gabor filter. So to conclude if the
hypothesis were right the paired profiles where compared and the error where calculated. The three
predictions is here examined one by one.
4 (5)
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The first prediction where:
“ First, simple cell 2D spatial profiles will
be indistinguishable.” 5
The compared profiles and the error are
shown on figure 9, for the first three
results. As seen is there some noise in
the test data from the cats while the
Gabor results are clean. This noise hardly
affects anything which can be seen at the
error. The errors are small and barely
more than the noise. Therefore is the
first prediction approved.
The second prediction where:
“ Second, simple cell 2D spectral response
profiles and 2D Gabor filter amplitude
spectre will be indistinguishable.” 6
The first three results for this part of the
experiment are shown at figure 10. This
time is the test data a little more rough
than the Gabor result. The error is
therefore insignificantly little and this
prediction where also approved.
At last is the third prediction, which is:
“ Since the 2D Gabor filter model
presupposes linear spatial summation in
2D, we also predict that simple receptive
fields will satisfy this constraint.” 7
5 (1) p. 1237 ”Predictions”
6
(1) p. 1237 ”Predictions” 7 (1) p. 1237 ”Predictions”
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This is tested and shown on figure 11. The Gabor result is the diagonal line which the dots, the data, is
predicted to be on the line. For the spatial frequency is dots spread, but mostly above the line. This could
indicate some kind of systematic bias of unknown origin. To determine if the results are satisfying is the
correlation calculated and for these
result are r=0.91, which isn’t bad. For
the orientation is the fit as close as it
almost could be, which is also proved
by the correlation, which is r=0.99.
This means that this prediction also
could be approved.
So Palmer and Jones succeeded in
proving the hypothesis that 2D Gabor
filters can be used as an analogous to
the class of linear spatial filters
simple receptive fields belongs to.
Figure 9 - Test results for the first prediction
Figure 10 - Test results for the second prediction
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References
(1) J.P. Jones and L.A. Palmer. An evaluation of the two-dimensional gabor filter model of simple
receptive fields in cat striate cortex. J. Neurophysiol., 58(6):1233-1258, 1987
(2) R. H. Wurtz and E. R. Kandel. Central Visual Pathways. Principles of Neural Science. 4th Ed.: Ch 27,
(3) N. Efford. Digital Image Processing – A practical introduction using Java. Ch 8: 188-226, 2000
(4) J. R. Movellan. Tuorial on Gabor Filters, http://mplab.ucsd.edu/tutorials/gabor.pdf
(5) Gabor filter for image processing and computer vision,
http://matlabserver.cs.rug.nl/edgedetectionweb/web/edgedetection_params.html
Figures
1 – (2) Figure 27-4
2 – (2) Figure 27-12
3 – (2) Figure 27-11
4 – (3) Figure 8.1
5 – (3) Figure 8.3
6 – (3) Figure 8.5
7 – (3) Figure 8.7
8 – (5) Examples made with simulation program
9 – (1) Figure 2
10 – (1) Figure 5
11 – (1) Figure 7
Figur 11 - Test results for the third prediction