or more precisely 
t GG 9S (xtp..v) (3) 
where 
D, 7 x parallax shift 
This formulation is the crux of the coherent opti- 
cal parallel processor. The detection problem is one 
of detecting p, as a function of y. This can be done 
by making a filter which contains the Fourier trans- 
form in the x direction as a function of y. The output 
of the processor will then display the x parallax, p,, 
as a function of y photocoordinate. 
The implication of this formulation of the problem 
is simply that the optical correlator is designed as a 
multichannel one-dimensional correlator rather than a 
two-dimensional correlator. When using the correlator 
with a scanning slit, the y photocoordinate is parallel 
to the long dimension of the slit and the x photo- 
coordinate is given by the slit position. 
RECORDING THE COMPLEX FOURIER TRANSFORM 
To implement the Fourier transform representa- 
tion of the correlation process as stated by equation 
2, it is necessary to record the amplitude and phase of 
oo 
T (p,q) = f f (x,y) e? * Y9 qxay (4) 
—00 
It should be noted that the transform kernel is writ- 
ten with a positive exponent rather than the conven- 
tional negative exponent. This is done because lenses 
do not take inverse transforms® but only successive 
transforms. This results only in a rotation of coordi- 
nates in the appropriate planes. 
Conventional photographic recording, which re- 
cords only intensity, is insensitive to phase. However, 
a Mach-Zehnder interferometer can be used to deter- 
mine phase in a light distribution by combining the 
distribution with a reference wave of known ampli- 
tude and phase as shown in Figure 4. The signal and 
Beam Splitter 
Mirror 
  
  
Input Photo 
P4 P2 
T den 
Mirror TR—-- 
N Beam Splitter 
Virtual Image 
Of Reference Source 
| Matched Filter 
1 
Correlation 
  
  
  
  
  
  
  
Convolution 
Figure 4 Modified Mach-Zehnder Interferometer 
for Image/Matched-Filter Correlation 
62 
reference beams are designated 1 and 2, respectively. 
The reference beam is tilted slightly with respect to 
the optic axis by means of the second beam splitter 
so that the light appears to come from a point in the 
input plane displaced a distance b from the optic axis. 
In operation, the first stereo photograph is placed in 
the input plane and its Fourier transform recorded 
photographically as a Fourier transform hologram in 
plane P5. With the recording put back in place after 
development, the second stereo photograph is placed 
in the input plane. The conjugate imagery can be 
anywhere on this second photograph. The correlator 
output will display the spatial position of the conju- 
gate imagery. 
Let the reference beam at plane P, in Figure 4 be 
given by 
K elP (5) 
The one dimensional transform of one of the stereo 
photographs may be written as 
t (x- xy) T, (ye (6) 
The light distribution in plane P, may be ex- 
pressed as the sum of equations (5) and (6). Since a 
photographic emulsion is used as the recording 
medium and it is sensitive only to intensity, not 
amplitude and phase, it may be considered as a square 
law detector. The function actually recorded may 
then be expressed as 
G(,y)s [K e? 4 T, (py) Jol = 
IK)? + IT, (py)? + 
K T, (p,y) eX*xpP + 
kT, (py) J&P (7) 
The bar denoting complex conjugate is redundant 
notation since the exponential terms are being re- 
tained. However, it is a help in visualizing and inter- 
preting the equations. 
The last term of equation (7) is the necessary 
complex conjugate Fourier transform needed to per- 
form correlation in accord with equation (2).16 Con- 
sider the second stereo photograph as 
{2 (X-X3,Y)= 1, [x-(X, * p). y] 
Its one dimensional Fourier transform may be ex- 
pressed as 
G[x-G4 p yl T, (py) e 189p. (y 
The complex transform appears in plane P, , Figure 4, 
the back focal plane of the transform lens. 
KRULIKOSKI, KOWALSKI, AND WHITEHEAD 
) 
| 
Th 
equat 
plane 
to the 
The s 
at 
Th 
relati 
Since 
termi 
PAR