Mathematical representation of Glass patterns

Let the function, $g(x,y)$, define a pair of Glass pattern dots in two dimensions of space,

$$g(x,y) = \delta(x+\Delta x,y+\Delta y) + \delta(x-\Delta x,y-\Delta y),$$ (1)

where $\delta(x,y)$ is the Dirac delta function, sometimes called the unit impulse function or the impulse symbol. It is defined to be zero-valued everywhere except at $(x=0,y=0)$, where it is infinite so that its integral is one. The Fourier transform, $F(s_x,s_y)$, of a function, $f(x,y)$, in two dimensions is given by:

$$F(s_x,s_y) = \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y) e^{-i2\pi(xs_x+ ys_y)} dx dy.$$ (2)

Combining Eqn. 4-1 and Eqn. 4-2, the Fourier transform (FT) of the Glass pattern dot pair is:

$$G(s_x,s_y) = \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(x+\Delta x,y+\Delta y) e^{-i2\pi(xs_x + ys_y)} dx dy$$

$$+ \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(x-\Delta x,y-\Delta y) e^{-i2\pi(xs_x + ys_y)} dx dy,$$ (3)

which, because the $\delta$-functions are non-zero at one point only, is simply,

$$G(s_x,s_y) = e^{i2\pi(\Delta x s_x + \Delta y s_y)} + e^{-i2\pi(\Delta x s_x + \Delta y s_y)}.$$ (4)

Using Euler's formula, and letting $\phi = 2\pi(\Delta x s_x + \Delta y s_y)$, we get,

$$G(s_x,s_y) = [cos(\phi) + i sin(\phi)] + [cos(\phi) - i sin(\phi)]$$

$$= 2 cos(\phi),$$ (5)

which is simply a cosine whose frequency and orientation in two dimensions is determined by $\Delta x$ and $\Delta y$ through the definition of $\phi$. The power spectrum, the square of the modulus of the Fourier transform, is

$$P(s_x,s_y) = \vert F(s_x,s_y)\vert^2$$

$$= 4*cos^2(\phi)$$

$$= 2 + 2*cos(2*\phi),$$ (6)

a frequency-doubled cosine with a constant offset.

A translational Glass pattern can be generated by replacing each dot in a random field with a pair of dots. This can be modeled mathematically as the convolution of a dot pair, $g(x,y)$ from Eqn. 4-1 above, with a random field of $\delta$-functions. This convolution is equivalent to a multiplication in the frequency domain.

Thus, the FT of the Glass pattern is the product of Eqn. 4-5, a cosine, with the FT of the white noise process, and the expected value of the spectrum of the Glass pattern is the spectrum of the dot pair (Eqn. 4-6), which is non-random, times the expected value of the white noise spectrum, which is flat. In addition, a term must be added at zero, a constant times $\delta(x,y)$, to account for the area of dot pair function and the density of the random process (see Champeney (1973)).

In the actual patterns shown on computer displays, the Glass patterns are comprised of small square dots. The FT of a square pulse is a sinc function. In the frequency domain, the true representation of our Glass pattern must then result from a multiplication with this sinc function. This leads to an attenuation of high frequencies, with the cutoff moving lower as the square pulse grows larger. In our case, the dots were very small relative to the receptive field, and the frequencies which were attenuated were much higher than the peak sensitivity in cortical cells. Because of this and the effects of blurring in the retina and in the optics, the difference between square dots and true $\delta$-functions is not significant.

In addition, we presented Glass patterns within a circular aperture in most of our experiments. This windowing of the pattern is arrived at via a multiplication in space. The FT of a circular or cylinder function in two dimensions is a Bessel function. In the frequency domain, the convolution of this function with the cosinusoid will result in the correct frequency representation of a circularly windowed Glass pattern. However, the Glass pattern aperture extended to fill the CRF of the cell. The Gaussian envelope of the receptive field is more important to the model results than the circular boxcar function which windows the stimulus. We therefore present our results without including the effects of this aperture.