Page 304 - Introduction to Information Optics

5.10. Geometric Transform 289

generalized geometric transformations should be defined as

F(u) =-- f(x) dx. (5.44)
JA

Example 5.3
The ring-to-point geometric transformation is useful to transform concen-
trically distributed signals into a linear detector array, by mapping rings in the
x- y plane to the points along the v axis in the u-v plane. We execute the polar
transformations first. For an input f(u, v) we have

F p(r, 8) = f(r cos 0, r sin 0).

According to the definition of the generalized geometric transformation, the
ring-to-point transform can be written as

F, p(r)=<bFJr,0)rd9,

where r = v x + y and the integral is computed along a circle of radius r.

5.10.3. OPTICAL IMPLEMENTATION

Many techniques exist to implement geometric transformations with optics.
A typical optic system for geometric transformation is shown in Fig. 5.6. The
input image is represented by a transparency and is illuminated by a collimated
coherent beam. A phase mask is placed behind the input plane, which
implements the geometric transformation.
The complex amplitude transmittance of the phase mask

t = exp[j</>(x, .y)]

is computed in such a way that the required coordinate system mapping is
proportional to its derivatives as:

( \~ . '
u — (f>i\x, y) — - -
2n ox
(5.45)
Af cl(f)(x, y)
v = (p 2(x, y) =
2n ay