record the interference pattern caused by the summation of the two
fields. The result on the film then is
H(u,v) = 1 + IF(u,v) 2 + F(u,v)exp j2Tav + F*(u,v)exp -j2Trav, (2.35)
which contains a constant, the power spectral density, and two terms
due to a spatial carrier fringe formed due to interference with the
plane wave. The two spatially modulated terms contain the original
image and Fourier transform information. With this Fourier transform
recorded on the film, it is placed in the optical filter arrangement
and illuminated with the Fourier transform G(u,v) of the test image
g(x,y). The output of the film transparency is the product of its
transmittance and the illuminating Fourier transform.
O(u,v) = G(u,v) H(u,v) (2.36)
= G(u,v) + G(u,v)IF(u,v)l2
+ G(u,v)F(u,v)exp j2rav + G(u,v)F*(u,v)exp -j2rrav
The product of the transforms from the reference and test images is
then Fourier transformed by another lens to obtain the correlation of
the two images.
o(x,y) = g(x,y) + g(x,y)*h(x,y)*h*(x,y) (2.37)
+ g(x,y)*f(x,y)*6(x,y-a)
+ g(x,y)*f*(x,y)*6 (x,y+a)
The first two terms are formed on axis or at the origin of the output
plane. The third term is the convolution of the reference and test