Page 177 - Phase Space Optics Fundamentals and Applications

158 Chapter Four

conventional correlation is a shift-invariant operation, the correlation
outputsimplymovesiftheobjecttranslatesattheinputplane.Inmany
cases this property is necessary, but there are situations in which the
position of the object provides additional information such as in im-
age coding or cryptographic applications, and so shift invariance is a
disadvantage.
The fractional correlation 58,59 is a generalization of the classic cor-
relation that employs the optical FrFT of a given fractional order p
instead of the conventional FT. Conventionally, the fractional corre-
lation is obtained as the inverse Fourier transform of the product of
the FrFT of both the reference and the input objects, but for a sin-
gle fractional order p at a time. The fractional order involved in the
FrFT controls the amount of shift variance of the correlation. As is
well known, the shift-variance property modifies the intensity of the
correlation output when the input is shifted. In several pattern recog-
nition applications this feature is useful, for example, when an object
should be recognized in a relevant area and rejected otherwise, or
when the recognition should be based on certain pixels in systems
with variable spatial resolution. However, the optimum amount of
variance for a specific application is frequently difficult to predict,
and therefore more complete information would certainly be attained
from a display showing several fractional correlations at the same
time. Ideally, such a display should include the classic shift-invariant
correlation as the limiting case. In this section we will show that such a
multichannel fractional correlator could be easily implemented from
the RWD system presented in Sec. 4.2.2. The resulting optical system
generates a simultaneous display of fractional correlations of a one-
dimensional input for a continuous set of fractional orders in the range
p ∈ [0, 1].
We start by recalling 58 the definition of the fractional correlation
between two one-dimensional functions f (x) and f (x)

−1 ∗
C p (x) = F {F p ( )F ( ),x} (4.111)
p
It is important to note that with the above definition the classic cor-
relation is obtained if we set p = 1. The product inside the brackets
of Eq. (4.111) can be optically achieved simultaneously for all frac-
tional orders, ranging between p = 0 and p = 1, following a two-step
process. In the first stage, the RWD of the input is obtained with the
experimental configuration shown in Sec. 4.2.2. A matched filter can
be obtained at the output plane if, instead of recording the intensity,
we register a hologram of the field distribution at this plane with a
reference wavefront at an angle . (see Fig. 4.33).
In the second stage, the obtained multichannel matched filter is
located at the filter plane, and the input function to be correlated is
located at the input plane (see Fig. 4.34).