2-D Gabor function - interactive visualization

Below you can interactively generate intensity maps of 2-D Gabor functions and their power spectra. Such functions are used as models of the receptive fields of simple cells. They are also widely used in Gabor filters for image processing and computer vision.

Please, specify values of the parameters and press the 'Display' button. The corresponding Gabor function and its power spectrum will be displayed as intensity map images. In the Gabor function image (shown on the left), light and dark gray colors correspond to positive and negative function values, respectively. In the image of the power spectrum of the Gabor function (shown on the right), a dark color marks the spatial frequencies for which the power spectrum is zero or very small, whereas a light color marks spatial frequencies for which the power spectrum takes substantial positive values. The generated images are of size 256X256.

Remark: For correct transfer of parameter values, your browser may require the use of a decimal comma instead of a decimal point.

Here is a description of the parameters:

 Wavelength (lambda) specifies the wavelength of the cosine factor of the Gabor function. The wavelength is given in pixels. Valid values are real numbers between 2 and 256. Orientation (theta) specifies the orientation of the normal to the parallel stripes of the Gabor function. The orientation is specified in degrees. Valid values are real numbers between 0 and 180. Phase offset (phi) specifies the phase offset of the cosine factor of the Gabor function. It is specified in degrees. Valid values are real numbers between -180 and 180. The values 0 and 180 correspond to symmetric 'centre-on'  and 'centre-off' functions, respectively, while -90 and 90 correspond to antisymmetric functions. Aspect ratio (gamma) specifies the ellipticity of the Gaussian factor. Values typical of the receptive fields of simple cells lie between 0.2 and 1 but the applet accepts other values as well. Bandwidth (b) specifies the spatial-frequency bandwidth of the filter when the Gabor function is used as a filter kernel. The bandwidth is specified in octaves. Values typical of the receptive fields of simple cells lie between 0.4 and 2.5 but the applet accepts other values as well.

2-D Gabor function as models of the receptive fields of simple cells

The following family of two-dimensional Gabor functions was proposed by Daugman [1] to model the spatial summation properties (of the receptive fields) of simple cells in the visual cortex:

where the arguments x and y specify the position of a light impulse in the visual field and sigma, gamma, lambda, theta and phi are parameters as follows (Greek letters are spelled out):

The standard deviation sigma of the Gaussian factor determines the (linear) size of the receptive field. Its ellipticity and herewith the ellipticity of the receptive field ellipse is determined by the parameter gamma, called the spatial aspect ratio. It has been found to vary in a limited range of 0.23 < gamma < 0.92. Sigma cannot be controlled directly in the applet. Its value is determined by the choice of the parameters lambda and b.

The parameter lambda is the wavelength and 1 / lambda the spatial frequency of the cosine factor in Eq. (1). The ratio sigma / lambda determines the spatial frequency bandwidth of simple cells and thus the number of parallel excitatory and inhibitory stripe zones which can be observed in their receptive fields. The half-response spatial frequency bandwidth b (in octaves) and the ratio sigma / lambda are related as follows:

Neurophysiological research has shown that the half-response spatial-frequency bandwidths of simple cells vary in the range of 0.5 to 2.5 octaves in the cat (weighted mean 1.32 octaves) and 0.4 to 2.6 octaves in the macaque monkey (median 1.4 octaves). While there is a considerable spread, the bulk of cells have bandwidths in the range 1.0-1.8 octaves. Some researchers propose that this spread is due to the gradual sharpening of the orientation and spatial frequency bandwidth at consecutive stages of the visual system and that the input to higher processing stages is provided by the more narrowly tuned simple cells with half-response spatial frequency bandwidth of approximately one octave. Since lambda and sigma are not independent when the bandwidth is fixed, only one of them, lambda,  is considered as a free parameter which is used in the applet.

The angle parameter theta specifies the orientation of the normal to the parallel excitatory and inhibitory stripe zones - this normal is the axis x' in Eq. (1) - which can be observed in the receptive fields of simple cells.

Finally, the parameter phi, which is a phase offset in the argument of the cosine factor in Eq. (1), determines the symmetry of the concerned Gabor function: for phi=0 degrees and phi=180 degrees the function is symmetric, or even; for phi=-90 degrees and phi=90 degrees, the function is antisymmetric, or odd, and all other cases are asymmetric mixtures of these two.

Further details can be found in references [2] through [7].

Here is a link to a web enabled Gabor filter for image processing and computer vision.

References

1. J.G. Daugman: Uncertainty relations for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters, Journal of the Optical Society of America A, 1985, vol. 2, pp. 1160-1169.
2. N. Petkov: Biologically motivated computationally intensive approaches to image pattern recognition, Future Generation Computer Systems, 11 (4-5), 1995, 451-465.
3. N. Petkov and P. Kruizinga: Computational models of visual neurons specialised in the detection of periodic and aperiodic oriented visual stimuli: bar and grating cells, Biological Cybernetics, 76 (2), 1997, 83-96.
[bibTex], [pdf 470 KB © Springer]
4. P. Kruizinga and N. Petkov: Non-linear operator for oriented texture, IEEE Trans. on Image Processing, 8 (10), 1999, 1395-1407.
[bibTex], [pdf 1775 KB © IEEE]
5. S.E. Grigorescu, N. Petkov and P. Kruizinga: Comparison of texture features based on Gabor filters, IEEE Trans. on Image Processing, 11 (10), 2002, 1160-1167.
[bibTex], [pdf 286 KB © IEEE]
6. N. Petkov and M. A. Westenberg: Suppression of contour perception by band-limited noise and its relation to non-classical receptive field inhibition, Biological Cybernetics, 88, 2003, 236-246.
[bibTex], [pdf 636 KB © Springer - original publication available at http://link.springer.de DOI 10.1007/s00422-002-0378-2]
7. C. Grigorescu, N. Petkov and M. A. Westenberg: Contour detection based on nonclassical receptive field inhibition, IEEE Trans. on Image Processing, 12 (7), 2003, 729-739.
[bibTex], [pdf 843 KB © IEEE],

Contact: simona@iwinet.rug.nl, petkov@cs.rug.nl