H(u,v) = IA12 + F(u,v)@F(u,v) + A F(u,v+a) + A F(u,v-a) (3.14)
where F(u,v) is the Fourier transform of f(x,y) and denotes
convolution.
The first term is a delta function at (0,0). The second term is
centered on axis (0,0) but has twice the width as the spectrum F(u,v).
The third and fourth terms are the Fourier transforms of the f(x,y)
but centered off axis at plus and minus a. To prevent frequency
overlap, the second term and the heterodyned terms must not overlap.
This requires that the spatial carrier frequency, a, used to
heterodyne the information must be sufficiently large. Specifically,
this carrier frequency must be larger than three times the one-sided
bandwidth of the information spectrum.
In the case of the Vander Lugt filter and the subsequent
correlation, the output of the holographic matched filter has the form
o(x,y) = g(x,y) + g(x,yf)f(x,yY)f*(x,y)
+g(x,y)f(x,y) 6 (x,y-a)
+g(x,y~@f (x,y) 6(x,y+a). (3.15)
The output, shown in Figure 3.5, contains a replica of the test image
g(x,y) centered on-axis along with a term consisting of the test image
convolved with the autocorrelation of the reference image f(x,y).
This term consumes a width of twice the filter size plus the test
image size. In addition to the on-axis terms, there are two
heterodyned terms centered at plus and minus a. These heterodyned
terms have a width equal to the sum of the widths of the test image
g(x,y) and reference image f(x,y). Again to prevent overlap of the