Page 113 - Phase Space Optics Fundamentals and Applications

94 Chapter Three

Thus for rotation-invariant pattern recognition, only one circular har-
monic c l (r,
) (usually with l =±1) substitutes the reference image.
The application of the combination of the various harmonics limits
the rotation invariance for a certain angle range.

3.7.3 Chirp Signal Analysis
Chirp, given, e.g., by Eq. (3.65), is often a part of medical and indus-
trial signals. It may contain valuable information or may correspond
to a noise. Then chirp detection, localization, estimation, and, if neces-
sary, elimination are important tasks in signal processing. The chirp,
Eq. (3.65), can be easily localized applying the RCT parameterized by
U −1 because the output signal becomes a function. In particular, the
applicationoftheFT,thefractionalFT,andthegyratorallowsonetolo-
calize plane, elliptic, and hyperbolic waves, respectively. Thus, the GC
−1 U 2
GC f, f (U, U , I, r) corresponding to the RCT spectra |R [ f (r i )](r)|
∗
with modifying parameters of U, associated with the intensity dis-
tributions of the output signal, is suitable for the detection of chirps
presented in the signal f (r i ). Here r and the parameters of U are vari-
−1
ables of the GC GC f, f (U, U , I, r).
∗
For example, if U = U f ( x , y ), then elliptic-type chirps can be
detected as a local maxima of the Radon-Wigner transform map 59
U f ( x , y ) 2
|R [ f (r i )](r)| for x , y ∈ [0, ]. The appropriate filtering in
the fractional FT domains has been used for elimination of elliptic
4
chirplike noise and, therefore, image quality improvement. Analo-
gously, the hyperbolic chirps can be localized by analyzing the gyrator
U g (ϑ) 2
power spectra |R [ f (r i )](r)| (Ref. 32).
3.7.4 Signal Encryption
The phase-space rotators are also used for signal encryption. The sim-
ple algorithm for optical image encryption consists of random phase
filtering in the position and FT domains. 60 It has been recently gen-
eralized to the case of random phase filtering in different fractional
Fourier 16 and gyrator 32 domains. In these cases, not only the ran-
dom phase masks but also the orders of the phase-space domains
(fractional or gyrator angles) where they are located play the role of
encryption keys. It was demonstrated that it is impossible to recon-
struct the image by using the correct masks but the wrong phase-space
domains.
In general, other phase-space rotators can also be used for signal
encryption. Indeed the simple encryption procedure of signal f using
phase-space rotators consists of a cascade of N operations: the RCT
transform parameterized by matrix U n with further resultant multi-
plication at a random phase mask exp(i n ) for n = 1, 2, ... ,N, which