Page 120 - Introduction to Information Optics

2.6. Algorithms for Processing 105

which are of the same scaling. Thus, we see that the Mellin transform of an
object function is scale invariant; that is,

(2,73)

However, unlike the Fourier transform, the magnitude of the Mellin transform
is not shift invariant; that is,

|M[/(x, y}] * |M[/(.x - x 0, y - y 0)] . (2.74)

In correlation detection, we assume that a Fourier domain filter of
M[/(x, >')] — M(p, q) has been constructed; for example, by using holographic
technique as given by

2
H(p,q) = K l+K 2\M(p,q)\ + 2K 1K 2\M(p,q)\ cos[a 0p - ftp,*?)],

where M(p,q) = \M(p,q)\ exp[i0(p, q}]. If the filter is inserted in the Fourier
y
x
plane of which the input SLM function is f(e , e ), the output light distribution
can be shown to be
x
y
y
x
y
x
0(a, jff) = KJ(e , e ) + K 2f(e*, e )*f(e , e )*f*(e~ , ?">)
x
y
y
+ K^K 2f(e\ e ) * f(e ~*°, e )
x y x a y
+ K lK 2f(e , e ] * f*(e~ ~ °, e~ ),
in which the last term represents the correlation detection, and it is diffracted
at a = a 0. In other words, the implementation of a Mellin transform in a
conventional linear processor requires a nonlinear coordinate transformation
of the input object, as illustrated in the block diagram of Fig. 2.30.
2.6.2. CIRCULAR HARMONIC PROCESSING

Besides the scale-variant problem, the optical processor is also sensitive to
object rotation. To mitigate this difficulty, we will discuss circular harmonic
processing, for which the rotation variant can be alleviated. It is well known
that if a two-dimensional function f(r, 9) is continuous and integrable over the
region (0, 2n), it can be expanded into a Fourier series, such as

ime
/(r,0)= F m(r}e , (2.75)