The Logic Based Spatiotemporal Filter Approach of Zorkoczy

The Boolean logical modules used in Zorkoczy's (1966) model for retinal feature extraction are clocked with a period T. A photoreceptor array lies in a plane with receptor centers on a rectangular grid spaced Ax = Ay = 8 units apart. The input to the receptor array is some pattern of black and white that changes in time. The Zorkoczy system produces outputs at the clock frequency that depend on the distribution of the pattern in space and its velocity (speed and direction). The receptor at x = k8, y = j8 has an output a(k, j) = akj of 0 or 1 at t = nT, depending on the distribution of the object in space, and its velocity. An object can also be stationary and change its intensity pattern, evoking an output.

First described are dyadic operations on two adjacent photoreceptors, a and b, lying on the y axis of the photoreceptor plane. a and b can each be 0 or 1 at each clock cycle.

1. The elementary contrast operator: c(a,b) = a ■ b = 1 @ t = nT IF a(nT) = 1 AND b(nT) = 0. This also can be written: c(a,b) = a ■ b = 1 @ t = nT IF a(nT) = 0 AND b(nT) = 1.

2. The ON operator: T1(a) = a ■ a* = 1 @ t = nT IF a[(n - 1)T] = 0 AND a(nT) = 1, ELSE 0. a* is the output of receptor a one sample period previously. The dot denotes a logical AND operation. The_ON operator can also be used with two adjacent receptors: T1(a,b) = a ■ b* = 1 @ t = nT IF b[(n - 1)T] = 0 AND a(nT) = 1, ELSE 0.

3. The OFF operator: T2(a) = a* ■ a = 1 @ t = nT IF a[(n - 1)T) = 1 AND a(nT) = 0, ELSE 0. For adjacent receptors: T2(a,b) = a* ■ b = 1 @ t = nT IF a[(n - 1)T) = 1 AND b(nT) = 0, ELSE 0.

4. The ON/OFF operator: B(a,b) = T1(a,b) + T2(a,b), or B(a) = T1(a) + T2(a). The plus sign denotes a logical OR operation.

5. Multiple input operations include the OR operations:

R = ^^ rj = 1 @ nT IF any one or more ri (nT) = 1 7.1-1

and i=N

P = ^^ pi = 1 @ nT IF any one or more pi (nT) = 1 7.1-2

6. A contrast operator C operates on two groups of receptors, P and R, separated by a boundary, f(x, y). C(P, R) = P ■ R = 1 @ nT if 1 or more elements of receptor set [P] = 1 AND none of set [R] = 1. See Figure 7.1-1.

7. To sense contrasts in both space and time, Zorcoczy defines the functions:

N1(P, R) = S T1(pi) ■ X T2 (ri )= 1 @ nT IF at least one element of [P] was turned ON at t = nT, AND 0 elements of set [R] were turned ON at nT.

N2(P, R) = X T2 (pi )-X T2(ri) = 1 @ nT IF 0 elements of [P] were turned

OFF at nT, AND at least one element of [R] was turned off at nT. Note that N1(P, R) and N2(P, R) will not fire for general ON or OFF over [P] and [R]. N1 and N2 operators apply to stationary patterns changing in time only.

8. Now consider the On-Center/Off-Surround operator, which will give a 1 @ nT IFF excitation over [P] (or a subset of it) changes from 0 ^ 1 at

FIGURE 7.1-1 Illustration of two possible receptor geometries for a Zorkoczy contrast operator. See text for description.

t = nT, AND simultaneously illumination over [R] remains the same or dims (1 ^ 0), OR, IF excitation over [R] dims (1 ^ 0) AND remains constant or brightens (0 ^ 1) over [P], ELSE 0. See Figure 7.1-2. The On-Center/Off-Surround operator is written N(P, R) = N1(P, R) + N2(P, R).

FIGURE 7.1-2 Illustration of the receptor geometry for an ON-center/OFF-surround Zorkoczy operation. See text for explanation of the Boolean expression.

FIGURE 7.1-2 Illustration of the receptor geometry for an ON-center/OFF-surround Zorkoczy operation. See text for explanation of the Boolean expression.

Similarly, one can create an Off-center/On-surround operator. Define:

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