S

where

11 ci(E, n) d£ dn = 0, and cp{u} = u for u > 0 Si =0 for u<0

FIGURE 7.1-11 Block diagram of a continuous, five-layer Fukushima static feature extractor. For mathematical convenience in using continuous Fourier transforms, let the spacing between nodes A ^ 0 in the limit, and also the number of nodes ^ x, generating a continuous system in x, y coordinates. Thus, a signal in the kth plane is related to that in the (k - 1)th plane by uk(x, y) = uk-1(x, y) ®® ck(x, y). ®® denotes the operation of two-dimensional, real convolution. All uk are non-negative by the ^j»} operation (half-wave rectification).

FIGURE 7.1-12 An ON-center/OFF-surround, one-dimensional weighting function, c1(x), can be shown to be the sum of a positive and a negative Gaussian function (see Equation 7.123).

Note that other models for ON-center/OFF-surround, c1 weighting functions can be used. For example, in one dimension:

Where A > B, a > p, and A/a = B/p for zero response to zero frequency (background) light on the U0 layer. For the dc stop condition, C1(u) can be shown to be bandpass:

At mid-frequencies,

FIGURE 7.1-13 Plot of the continuous Fourier transform of Oj(x), Q(u). Note that low spatial frequencies are attenuated. See text for discussion.

Next, consider Fukushima's model for simple line detection. A weighting function, c2(x, y, a) is defined connecting signal nodes in the bandpass-filtered image layer, uj, with nodes in the simple line detection layer, u2. Here a is the angle a bright line makes with the x-axis in the receptor plane. (To detect lines with 15° angular spacing would require j2 independent c2 filters.) As before, the signals in the u2 layer are given by u2(x, y, a) = 9 jjjc2 (x, y, a) u (x - y - n))dnj 7.1-29

Figure 7.1-14, taken from Fukushima (1970), illustrates a typical c2(x, y, a). Note that this figure is a contour plot, i.e., the contours of c2 are seen from "above." This weighting function is an elongated ON-center/OFF-surround type, symmetrical around the a-direction axis. This c2 can be made up from the difference of two, two-dimensional, elliptical, Gaussian functions. Remember that there are many parallel, overlapping, receptive fields (RFs) of u2 nodes having a given a, so that if an a-oriented line is presented anywhere over u0(x, y), at least one or more signal nodes in the u2 layer will be active. If the line lies outside a particular u2 RF boundary, or lies in the RF but is perpendicular to the a-axis, there will be no output from the node in the u2 plane that has that particular c2 RF. Maximum output occurs when the line axis is at a° and the line is centered over the particular c2 RF. If the line is centered over the RF but at an angle such as a ± 30°, there will be a slight output from the corresponding u2 unit.

FIGURE 7.1-14 Contour plot of Fukushima's c2(x, y, a). This weighting function (WF) selects white lines at an angle of a° with the horizontal (x-axis) on the ON-center/OFF-surround RF.

The next layer in Fukushima's model feature extractor is the complex-type line detector. This operation occurs as a result of signal connectivities, c3(x, y, a) receiving inputs from nodes in layer u2, and producing outputs at nodes in layer u3. One can again write the nonnegative convolution:

u3 (x, y, a) = 9 jjjc3(x, y, a) u2 (x - Ç, y - n) dnj 7.1-30

A node, u3(xo, yo, ao), responds to a bright line of orientation ao, but in contrast to nodes in the u2 layer is not sensitive to the location of the line in the receptor plane. Fukushima states, "These elements [nodes] correspond to complex cells in the [cat's] visual cortex." Each u3 node effectively adds the outputs of a large number of u2 nodes with similar a sensitivity. Thus, if a point source input is used, a u3 node is found that responds to light over a large, narrow area of u0, perpendicular to a. Thus, a u3 node receives a large number of u2 outputs responding to a line of orientation a over a large area of u0. The c3(x, y, a) weighting function for a u3 node is shown in Figure 7.1-15A and B.

C3(Xo, yo, cc)
FIGURE 7.1-15 (A) Contour plot of Fukushima's c3(x, y, a), the complex line detector of layer 3. u3 elements respond to a bright line of orientation a anywhere over the receptor layer. See text for discussion. (B) The same c3 WF as in A, portrayed as a three-dimensional surface.