where i, k, l, m relate to positions on a regular sampling grid and h(x, y) is the point-spread function—it is the image of a single object point placed at (0,0). The relationship (A1.15) follows immediately from the properties of linearity and isoplanasy. The right-hand side of equation (A1.15) represents the convolution product of o(x, y) with h(x, y). We will make use of the symbolic notation o(x, y)o h(x, y). The delta-function is a particular point spread fuction consisting of a single point, h(xi — xi, ym — yk) = S(xi — xi, ym — yk), which has the value one at xi = xi, ym = yk and zero elsewhere. Use of this function in (A1.15) gives the result p(xi, yk) = o(xi, yk); i.e., convolution with the delta-function reproduces the original function.

The convolution theorem is of fundamental importance for the computation of convolution expressions. The theorem states that the Fourier transform of a convolution product of two functions is equal to the product of their Fourier transforms.

Hence, the computation of the convolution product of two functions o(x, y) and h(x, y) can be computed in the following way:

(iii) Form the scalar product of these two transforms

(iv) compute the inverse Fourier transform of the result:

This is one of the many instances where the seemingly complicated step of Fourier transformation proves much more efficient than the direct evaluation of a linear superimposition summation. The time saving is due to the very economic organization of the fast Fourier transform (FFT) algorithm in numerical implementations (see Cooley and Tukey, 1965). Another example will be found in the evaluation of the auto- and cross-correlation functions below.

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