d where Ib is the background intensity of illumination, and kj and Id are positive constants.

It is mathematically convenient to work in the spatial frequency domain to find the effective absorbed intensity, Ie, for various objects. That is, one takes the two-dimensional Fourier transforms of p(x, y) and s(x, y). u and v are spatial frequencies with the dimensions of radian/mm. Thus,

F{ko[p(x, y)**s(x, y)]} = k P(u, v) S(u, v) Real convolution theorem 5.2-4C

To find Ie, take the inverse Fourier transform of Equation 5.2.4C with x = y = 0. (x = y = 0 corresponds to the center of the coordinate system for the one retinula cell under study. It is tacitly assumed that because of axial symmetry, all retinula cells in a given ommatidium have the same DSF.) The effective intensity is found by le = F-1[koP(u,v)S(u, v)] = -ko2 f f P(u, v)S(u,v)dudv 5.2-5

Still further mathematical simplification occurs if it is assumed that s(x, y) is symmetrical and independent in x and y, i.e., s(x, y) ^ s(x) s(y), and s(x) = s(y). Now analysis can be carried out in one-dimensional with little loss of generality. Two models frequently used for one-dimensional DSFs are s(x) =--, n > 2 Hill function 5.2-6

Gaussian function

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