Page 108 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 89

of other phase-space rotators for signal processing tasks, which can
be elegantly expressed in the framework of generalized convolution.
The convolution operation between signals f and h, Eq. (3.51), can
be alternatively expressed via the Fourier transform as 9
−1
C f,h (r) = ( f ∗ h)(r) = F {F[ f (·)](u) F[h(·)](u)} (r) (3.82)
By analogy we can introduce the generalized (canonical) convolution
(GC) operation as 6,8,56
T 2
T 3
GC f,h (T 1 , T 2 , T 3 , r) = R {R [ f (·)](u) R [h(·)](u)}(r) (3.83)
T 1
T
where the FT operators F are substituted by the CT ones R .Inthe
T
U
widely used GCs, R corresponds to the RCT R and can be denoted
by GC f,h (U 1 , U 2 , U 3 , r).
6
It is easy to see that Eq. (3.83) reduces to common convolution,
Eq. (3.82), if the ray transformation matrices correspond to the direct/
−1
inverse FT ones U 1 = U 2 = U = U f ( /2, /2). Besides that the GC
3
includes, as particular cases, the correlation operation

! "
Cor f,h (r) = GC f,h ∗ U f , , U f − , − , U f − , − r
2 2 2 2 2 2
(3.84)
9
used as a measure of similarity between two signals f and h; the
fractional convolution
GC f,h [U f ( x , y ), U f ( x , y ), U f ( x , y ), r] (3.85)
applied for shift-variant filtering and pattern recognition; 4,15 the
Wigner distribution

!
2GC f, f ∗ U f x + , y + , U f − x + , − y + ,
2 2 2 2

"
U f − , − , 2 (3.86)
2 2

2
2
2
2
expressed in polar coordinates = ( x + p , y + p ); and the
x
y
U
2
−1
RCT power spectrum GC f, f (U, U , I, r) =|R [ f ](r)| , correspond-
∗
ing to the squared modulus of the RCT of the signal or Wigner distri-
bution projection, which in the case U = U f ( x , y ) is denoted as the
Radon-Wigner transform (see Chap. 4)
U f ( x , y ) 2
GC f, f [U f ( x , y ), U f (− x , − y ), I, r] =|R [ f ](r)| (3.87)
∗
The generalized convolution GC f,h (U 1 , U 2 , U 3 , r) of two-
dimensional signals f and h is a function of 2 variables (r) and 12