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For this attenuation to occur, and for this DSF, the ratio of DSF half-angle to interommatidial angle, 0m/2/X, must be > 1.465.

The mathematical development above was done for an ideal, one-dimensional, infinite, spatial sampling array for object intensity. In practice, interommatidial angles vary with the position of the ommatidium on the eye surface, and can differ in the x and y directions as well. It seems that the arthropod lives in a far less perfect visual world than do vertebrates with camera eyes. Still, they have survived through the ages.

A behavioral test for aliasing in CEs is to see if the optomotor response reverses sign when the period of a striped object moved past the animal reaches a small value where aliasing occurs. Ideally, this would be where the fundamental spatial frequency of the stripe square wave is greater than the Nyquist frequency set by interommatidial angle, X; i.e., 2n/Xo > n/X, or the stripe period, Xo , is less than 2X. (A time-domain analog of aliasing is when spoked wheels are perceived turning backward in the movies when they are moving a vehicle forward. In this case, the temporal sampling rate is the frames per second of the camera.) Bishop and Keehn (1967) have reported the phenomenon of optomotor reversal in the housefly Musca domestica. The crossover stripe period was 4 to 4.5°. Interommatidial angles in the Musca eye are variable, depending on position in the eye, and range from 1.0 to 5.4° with a mean X = 3.9°. Bishop and Keehn (1967) noted:

The orientation of the array of ommatidia with respect to the rotating pattern, results in a spectrum of interommatidial angles, some of which could contribute positively, and some negatively to the neural (optomotor) response.

Northrop (1974) also noted a loss of directional selectivity in the firing rates of the third cervical nerve (CN3) (see Section 5.4.2 below) of the locust, Schistocerca gregaria, when presented with moving stripes having a critical period. (N3 innervates muscles that move the animal's head.) In this experiment, the animal still sensed stripe motion, but the neural response was unable to distinguish between the preferred and null directions of motion. The loss of directional discrimination occurred for stripe periods less than 5 to 6°. See Figure 5.2-3 for an example. Presumably this loss was also due to aliasing.

A general property of CEs is that when they become DA, the DSFs of the ommatidia become broader (0m/2 increases). This broadening of s(x) means the spatial frequency response S(u) decreases, making aliasing less likely. Thus, one would not expect to see optomotor reversal for DA CEs.

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FIGURE 5.2-3 Log-log plot of the normalized response of the cervical motor nerve N3 of the locust when a restrained animal is shown moving rectilinear stripes. Stripes were viewed in a square window 2.5 periods on a side with its sides parallel with the stripes. Horizontal bar is the background firing rate for the "C" unit; circles: C unit responses to stripes moved in a PD direction; squares: C unit responses to anterio-ventral (AV) stripe velocity; hexagons: D-unit response to PD stripe movement. The D unit was silent for no stripe motion, and did not respond to AV stripe motion. Stripes were moved two periods in 1.4 s. A 20-s i.s.i. was used to avoid habituation. Note curious "knee" in the C unit responses. N3 C unit responses ceased for stripe period less than about 5°. (From Northrop, R.B., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, Oxford, 1975. With permission of Oxford University Press.)

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FIGURE 5.2-3 Log-log plot of the normalized response of the cervical motor nerve N3 of the locust when a restrained animal is shown moving rectilinear stripes. Stripes were viewed in a square window 2.5 periods on a side with its sides parallel with the stripes. Horizontal bar is the background firing rate for the "C" unit; circles: C unit responses to stripes moved in a PD direction; squares: C unit responses to anterio-ventral (AV) stripe velocity; hexagons: D-unit response to PD stripe movement. The D unit was silent for no stripe motion, and did not respond to AV stripe motion. Stripes were moved two periods in 1.4 s. A 20-s i.s.i. was used to avoid habituation. Note curious "knee" in the C unit responses. N3 C unit responses ceased for stripe period less than about 5°. (From Northrop, R.B., in The Compound Eye and Vision in Insects, G.A. Horridge, Ed., Clarendon Press, Oxford, 1975. With permission of Oxford University Press.)

5.2.2 Calculation of Intensity Contrast

As an object moves relative to the eye, the intensity of light on the rhabdoms, Ie, will change in time, producing temporal changes in the retinula cell resting potential, Avm(t). One theoretical measure of visual resolution is to calculate the intensity contrast, CIe, in a retinula cell.

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Simple test objects themselves can be described by a contrast function, Cobji

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