Resolution can be thought of as the ability of a visual system to resolve small, low-contrast objects as separate entities without error. For example, to resolve two adjacent black spots as two spots, rather than one big fuzzy spot. This type of test is analogous to the time-domain resolution of two closely spaced pulses as two separate pulses after they have propagated through a low-pass filter or a transmission line.
It is also possible to test the resolution of a visual system in the frequency domain, i.e., its steady-state, sinusoidal spatial frequency response. It is possible to generate an object having a one-dimensional, spatial, sinusoidal intensity variation given by p(x) = I0 + Im sin(2nf x), I0 S Im 5.2-1
Such an object allows interpretation of visual resolution in terms of the spatial frequency response in one direction, i.e., x. At very high spatial frequencies, the pattern disappears, and only Io is perceived. This loss of high spatial frequency information indicates that all visual systems are low-pass in nature. Resolution tests are generally carried out on the responses single cells, e.g., ganglion cells, retinula cells, or on the eye as a whole.
Arthropod CEs are not noted for their high resolution and spatial frequency response. Their resolution is usually tested behaviorally, or by neurophysiological recording, as will be seen. To derive a quantitative model describing the resolution of CEs, it is necessary to define first a coordinate system (object space). Most CEs view over 2n steradians of solid angle (over a half of a hollow sphere, viewed from the inside). The "view" of individual retinula cells is rather narrow, however, and is characterized by the DSF. The surface of a CE is convex, so that the optical axis of the DSF of each ommatidium diverges slightly from its neighbors. Consider Figure 5.2-1. A CE views a point source of light on a concave spherical surface of radius R. The angle subtended by the light source at the eye is much smaller than the halfmax angle, 0m/2, of an ommatidium DSF. Thus, the point source of light behaves like a two-dimensional spatial impulse function of intensity Io. That is, p(^, 8) = Io8(^ - 8 - 8o), where the spot is located at spherical coordinates, (R, 8o). In general, any object viewed by the eye can be described by its intensity as a function of ^ and 8. For purposes of demonstration, it is mathematically more convenient to abandon spherical coordinates to describe the object intensity and instead use two-dimensional rectangular coordinates (x, y). The arc lengths x and y are shown in Figure 5.2-1. If 8 and ^ are small, then the arc lengths approach the linear dimensions, x and y, given ^ and 8. That is, x = R8 and y = R^, 8 and ^ in radians.
To establish the relationship between the two-dimensional, spatial distribution of intensity of an object, p(x, y), and a retinula cell depolarization voltage, vm, assume that the superposition of spatial impulse responses occurs, which leads to the following linear relation:
where Ie is the effective intensity causing the photoreaction in the rhabdom of a retinula cell. (Ie is unobservable.) ko is an "un-normalizing" scale factor required to get the "true" value of Ie, noting that the DSF, s(x, y), is normalized; i.e., s(0, 0) = 1. The ** denotes the operation of two-dimensional, real convolution. Once computed, vm can be found from the log relation:
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