Example 721

As a first example of an SMF calculation, consider a one-dimensional, continuous SMF for an image that is a light spot of radius r, centered at x = 0. Thus, s(x) = Io[U(x + r) - U(x - r)] is a pulse of height Io, which has the well-known Fourier transform, S(u) = (Io2r) sin(ru)/(ru), which is real. g(x) is given by real convolution.

FIGURE 7.2-1 Plots of the one-dimensional, input signal intensity, s(x), reversed and translated in the form for real convolution. s(x) is made asymmetrical for illustrative purposes.

The SMF is also a pulse of radius r centered at x = 0, and height KIo. In the real convolution process, shown in Figure 7.2-2, g(x) emerges as a triangle of base 4r and a peak of height Kio 2r at x = 0. Interestingly, if the input object has width 2w, where w > r, the height of gs(x) will still be KI2o 2r, maximum at x = 0. The triangular shape of this g(x) is 2(r + w) wide, however. The ms noise output of the SMF is

No = (n2)^J ™lHm((u) du = (l/2) —J \l02r[sin(ruV(ru)]| du = (n/2)(Klo )22r The peak MS SNR is at x = xo = 0:

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