The Continuous One Dimensional Spatial Matched Filter

An image of intensity s(x) is projected on the x-axis (assume it is a one-dimensional, continuous "retina"). Additive Gaussian white noise, n(x), also impinges on the x-axis. In general, f(x) = s(x) + n(x). f(x) is processed by a linear, SMF, hm(x), giving an output, g(x). Thus, because real convolution in the space domain is equivalent to multiplication in the frequency domain, g(xo) = £f (v) hm (xo - v)dv = ¿j (ju)exp(jux0) du 7.2-1

The noise is assumed to have a two-sided power density spectrum, ^m(f) = n/2 mean squared watts/cycle/mm. Thus the total noise output power of the SMF, No, can be written:

Note that u = 2nf (rad/mm = 2n cycles/mm). The SMF output due to the signal is given by the inverse Fourier transform:

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