Vander Lugt filter is
0(u,v) = F'(u,v)G(u,v)ej2rav + F*,(u,v)G(u,v)e-j2rav (3.21)
or
o(x,y) = f'(x,y)g(x,y)6(x,y+a) + f'(-x,-yg(x,y) (x,y-a) (3.22)
= f'(x,y)g(x,y)@6(x,y+a) + Rf,g(x,y)@(x,y-a)
which gives the spectrum shown in Figure 3.10, assuming Bf=Bg=B.
This phase hologram reduces the number of points to 4B, a power of 2.
This is the smallest possible size in a spatially modulated hologram.
As will be shown later, the phase modulation process may significantly
affect the information, and the correlation obtained may be a poor
approximation to the ideal correlation.
The Vander Lugt filter is typically used to detect the presence of
a small object in a large scene. This implies that Bf may be much
smaller than B In any case, the least theoretical hologram size
using equation 3.20 is still twice the size of the reference image and
test image combined in the y direction. For example, a large scene
consisting of 1024 by 1024 points is to be searched for an object that
would occupy 32 by 32 points in that scene. The smallest continuous-
tone hologram to perform that correlation would contain 2112 points in
the y direction (at least 1088 in the x direction). For most
practical applications, the absorption hologram illustrated in Figure
3.9 would be used. For the same example consisting of a 1024 by 1024
test scene and a 32 by 32 reference image, a square hologram would be
at least 2144 by 2144.
Another practical consideration provides some relief in the size
of the correlation plane. The correlation of two images creates a