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