Page 112 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 93

reconstructed numerically from the phase-only data (parts a and b)
and the amplitude-only data (parts c and d) of the symmetric fractional
FT of the test image are displayed for two transformation angles
:

= /4 (parts a and c) and
= /2 (parts b and d). We observe that the
information about the structure of the image codified in the amplitude
of the fractional FT transform is poor for
= n . Meanwhile the frac-
tional FT phase contains essential information about the signal almost
for all ranges of
. Thus we can conclude that the phase information
of the fractional FT is more relevant than the amplitude one, where
we exclude rather exotic images whose fractional FTs have constant
phase. The same results are also valid for other phase-space rotators 32
excluding ones of the imaging type.
If the pattern on the scene is rotated with respect to the refer-
ence one, then we can apply the GC where U 1 = U f ( /2, /2)U r ( )
and U 2 = U 3 = U f (− /2, − /2), or U 1 = U −1 = U f ( /2, /2) and
3
U 2 = U f (− /2, − /2)U r ( ). The identification of the largest corre-
lation peak as a function of indicates the right orientation of the
pattern. This analysis for the two mentioned cases, respectively, can
be done by adding the flexible rotator system 55 just before the com-
mon correlator with an invariable filter mask or using the Van der
Lugt correlator with a variable filter mask, which can be obtained by
application of the spatial light modulator.
For rotation-invariant pattern recognition, the reference image h,
presented in the polar coordinates, is decomposed into a linear sum
of the circular harmonics. 58
∞ ∞
, ,
h(r,
) = h l (r) exp (il
) = c l (r,
)
l=−∞ l=−∞
2
1
h l (r) = h(r,
) exp (−il
) d
(3.88)
2
0
±
Since the Laguerre-Gaussian functions [see Eq. (3.73)] L m,n (r) =
l
L (r,
), where l = m − n and p = min{m, n}, form the complete
p
orthonormal set, then h(r,
) can be also represented as their linear
superposition
∞ ∞
, , l
h(r,
) = b l, p L (r,
) (3.89)
p
l=−∞ p=0
and therefore the circular harmonic c l (r,
) is a linear superposition
of the LG modes with the same index l = m − n
∞
,
l
c l (r,
) = b l, p L (r,
) (3.90)
p
p=0