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

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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”



2

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)



8

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

http://mplab.ucsd.edu/tutorials/gabor.pdf







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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