## The Compound Eye as a Two Dimensional Spatial Sampling Array

This section analyzes the properties of the CE as whole as a spatial sampling array. Each ommatidium gathers light intensity from a corresponding point on a visual object, weighted by its DSF. Intensity information from each ommatidium can theoretically be used to reconstruct an "image" of the object in a sampling space lying in the animal's CNS.

To permit easy mathematical analysis of spatial sampling, assume that the surface of a model CE is a perfect hemisphere of radius r in which each ommatidium has an optical axis that diverges from its nearest neighbors by a uniform interommatidial angle, X. Further, assume that the ommatidia are arranged on the surface of the eye in an equally spaced, rectangular grid rather than a hexagonally close-packed array, which is the usual case. If a model CE has 180 ommatidia on an "equator," then the interommatidial angle is 1.0°. The object intensity is assumed to be displayed on the inside surface of a hemisphere of radius R > r, concentric with the CE model.

To examine spatial sampling mathematically, we will first consider the case of ideal, one-dimensional, spatial sampling. The ideal sampling process is mathematically equivalent to impulse modulation, in which the object intensity, g(x), is multiplied by an infinite impulse array formed by the receptor array. The one-dimensional impulse array can be written as

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