The fact that a three-dimensional image (intensity, x, y) can be processed using coherent light by successive layers of Fourier optical transform filters to extract image features was described in the engineering literature by Cutrona et al. (1960) and by vander Lugt (1964).

The idea that visual feature extraction can proceed in an analogous manner using layers of discrete neurons was first set forth by Fukushima (1969; 1970). Fukushima's models were inspired by a long series of papers on the neurophysiology of vision in various vertebrates, including frogs (Lettvin et al. 1959), rabbits (Barlow, 1963; Barlow and Hill, 1964; Levick, 1967), cats (Rodiek and Stone, 1965; Rodiek, 1965; Hubel and Wiesel, 1959; 1962; 1965), and also in the horseshoe crab, Limulus (Ratliff, 1964). Fukushima's (1969) model for visual feature extraction used six layers of neural signal space, including a two-dimensional receptor layer. It was a static model; i.e., no object motion was assumed, and the object did not change in time.

Although the signal nodes in the layers were discrete, Fukushima's approach was to assume their density sufficiently high to justify the simplification of continuous, analog representation of signals and their transformations. Because the signal at each node is assumed to be a spike frequency, a non-negativity operator, <(u), is assumed, following the diagram in Figure 7.1-9. Figure 7.1-10 illustrates the signal node layers. Operations on visual information are performed by the connection weights connecting the nodes between layers. These operations are shown more clearly in the block diagram of Figure 7.1-11. The system is truly nonlinear because the signal at every node must be non-negative (specifically, the <{*} operator).

In Fukushima's (1970) model, the visual object is assumed to have bright details only, i.e., white lines on a dark field. The signals in the first, u0(x, y), receptor layer are processed by the weighting function, c1(x, y) to yield u1(x, y), where low spatial frequencies are attenuated and object contrast is enhanced. A node at some xo, yo in the u1 plane is connected to nodes in the receptor plane such that a point source of light, Io 8(x, y), moved over u0(x, y) maps the weighting function, c1(x - xo, y -yo). This weighting function is of the ON-center/OFF-surround type, shown for the u1(0, 0) node in Figure 7.1-12. The signal at the nodes in the u1(x, y) layer is given by the two-dimensional, real convolution:

u1 (x, y) = J JJ u0( n) c1 (x - y - n) dn | = 9K (x, y) (x, y)} 7.1-20

FIGURE 7.1-9 Treatment of signals at a node in a layer of a Fukushima static visual feature extraction model. k non-negative signals {uj from nodes in previous layers are multiplied by k weights {cj and added together to form a signal s. s can be positive or negative, depending on the values of the k, {u cj values. Because the signals at the nodes represent "neuron" spike frequency, the actual signal at the output node is half-wave-rectified to form

FIGURE 7.1-9 Treatment of signals at a node in a layer of a Fukushima static visual feature extraction model. k non-negative signals {uj from nodes in previous layers are multiplied by k weights {cj and added together to form a signal s. s can be positive or negative, depending on the values of the k, {u cj values. Because the signals at the nodes represent "neuron" spike frequency, the actual signal at the output node is half-wave-rectified to form uo = 9(s) = k=1uici} .

If u1(x, y) S 0 over the surface, S1, then ui(x, y) = uo(x, y) ** ui(x, y) 7.1-21

Real convolution of two Fourier-transformable functions can be shown to be equal to the inverse Fourier transform of the product of their Fourier transforms providing u1(x, y) S 0.

Where u and v are the spatial frequencies in rad/mm.

There are several ways to model the "Mexican hat" weighting function mathematically. One can simply add two Gaussian functions with different peak amplitudes and standard deviations. For example, in one dimension,

2o 2

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